Faster Penalized Stochastic Gradient Methods for Strongly Convex Optimization with Linear Constraints
Meng Li, Paul Grigas, Alper Atamturk
We consider a penalty function method for minimizing a strongly convex objective subject to linear inequality constraints, which is especially effective when there is a large number of constraints as one can apply stochastic gradient methods to the penalty reformulation. We introduce a class of smooth penalty functions that includes the previously studied one-sided Huber penalty as a special case. We give a nested version of the stochastic gradient descent method and propose a novel way of updating the smoothness parameter of the penalty function and the step-size. We demonstrate that the new update scheme requires at most O(1/e) stochastic gradient iterations to produce a solution that is within an expected squared distance of e to the optimal solution of the constrained problem. To the best of our knowledge, this is the first complexity result of this type that holds generally for a wide class of smooth penalty functions and improves upon the previously best known O(1/e^2) complexity result that was developed in the special case of the one-sided Huber penalty.
Finally, we present computational results that demonstrate the effectiveness and robustness of the algorithm on synthetically generated quadratic programming problems and multivariate convex regression problems.
[
BCOL Research Report 21.02, IEOR, University of California-Berkeley.
]
Enhanced Modeling of Contingency Response in Security-constrained Optimal Power Flow
Tuncay Altun, Ramtin Madani, Alper Atamtürk, Ross Baldick, and Ali Davoudi
This paper provides an enhanced modeling of the
contingency response that collectively reflects high-fidelity phys-
ical and operational characteristics of power grids. Integrating
active and reactive power contingency responses into the security-
constrained optimal power flow (SCOPF) problem is challeng-
ing, due to the nonsmoothness and nonconvexity of feasible
sets in consequence of piece-wise curves representing generator
characteristics. We introduce a continuously-differentiable model
using functions that closely resemble PV/PQ switching and the
generator contingency response. These models enforce physical
and operational limits by optimally allocating active power
imbalances among available generators and deciding the bus
type to switch from the PV type to the PQ type. The efficacy of
this method is numerically validated on the IEEE 30-bus, 300-
bus, and 118-bus systems with 12, 10, and 100 contingencies,
respectively.
[
BCOL Research Report 21.01, IEOR, University of California-Berkeley.
]
pv-pq-switch.pdf
Supermodularity and Valid Inequalities for Quadratic Optimization with Indicator Variables
Alper Atamturk and Andres Gomez
We study the minimization of a rank-one quadratic with indicators and show that the underlying set function obtained by projecting out the continuous variables is supermodular. Although supermodular minimization is, in general, difficult, the specific set function for the rank-one quadratic can be minimized in linear time. We show that the convex hull of the epigraph of the quadratic can be obtaining from inequalities for the underlying supermodular set function by lifting them into nonlinear inequalities in the original space of variables. Explicit forms of the convex-hull description are given, both in the original space of variables and in an extended formulation via conic quadratic-representable inequalities, along with a polynomial separation algorithm. Computational experiments indicate that the lifted supermodular inequalities in conic quadratic form are quite effective in reducing the integrality gap for quadratic optimization with indicators.
[
BCOL Research Report 20.03, IEOR, University of California-Berkeley.
]
supermodular-rankone.pdf
2x2-Convexifications for Convex Quadratic Optimization with Indicator Variables
Shaoning Han, Andres Gomez and Alper Atamturk
In this paper, we study the convex quadratic optimization problem with indicator variables. For the bivariate case, we describe the convex hull of the epigraph in the original space of variables, and also give a conic quadratic extended formulation. Then, using the convex hull description for the bivariate case as a building block, we derive an extended SDP relaxation for the general case. This new formulation is stronger than other SDP relaxations
proposed in the literature for the problem, including Shor's SDP relaxation, the optimal perspective relaxation as well as the optimal rank-one relaxation.
Computational experiments indicate that the proposed formulations are quite effective in reducing the integrality gap of the optimization problems.
[
BCOL Research Report 20.02, IEOR, University of California-Berkeley.
]
two-by-two.pdf
Safe Screening Rules for L0-Regression from Perspective Relaxations
Alper Atamturk and Andres Gomez
We give safe screening rules to eliminate variables from regression with L0
regularization or cardinality constraint. These rules are based on guarantees that a feature may or
may not be selected in an optimal solution.
The screening rules can be computed from a convex relaxation solution in linear time,
without solving the L0 optimization problem.
Thus, they can be used in a preprocessing step to safely remove variables from consideration apriori.
Numerical experiments on real and synthetic data indicate that,
on average, 76% of the variables can be fixed to their optimal values, hence, reducing the computational burden
for optimization substantially. Therefore, the proposed fast and effective screening rules extend the scope of
algorithms for L0-regression to larger data sets.
[
Proceedings of the 37th International Conference on Machine Learning, PMLR 119:421-430, 2020.
]
http://proceedings.mlr.press/v119/atamturk20a/atamturk20a.pdf
icml-2020.bib
Submodular Function Minimization and Polarity
Alper Atamturk and Vishnu Narayanan
Using polarity, we give an outer polyhedral approximation for the epigraph of set functions.
For a submodular function, we prove that the corresponding polar relaxation is exact;
hence, it is equivalent to the Lov\'asz extension.
The polar approach provides an alternative proof for the convex hull description
of the epigraph of a submodular function.
Computational experiments show that the inequalities from outer approximations can be effective as
cutting planes for solving submodular as well as non-submodular set function minimization problems.
[
BCOL Research Report 19.02, IEOR, University of California-Berkeley. Forthcoming in Mathematical Programming.
]
submodular-polarity.pdf
https://doi.org/10.1007/s10107-020-01607-w
Rank-one Convexification for Sparse Regression
Alper Atamturk and Andres Gomez
Sparse regression models are increasingly prevalent due to their ease of interpretability and superior out-of-sample performance. However, the exact model of sparse regression with an L0 constraint restricting the support of the estimators is a challenging non-convex optimization problem.
In this paper, we derive new strong convex relaxations for sparse regression. These relaxations are based on the ideal (convex-hull) formulations
for rank-one quadratic terms with indicator variables. The new relaxations can be be formulated as semidefinite optimization problems in an
extended space and are stronger and more general than the state-of-the-art formulations, including the perspective reformulation and formulations with the reverse Huber penalty
and the minimax concave penalty functions. Furthermore, the proposed rank-one strengthening can be interpreted as a non-separable, non-convex sparsity-inducing regularizer,
which dynamically adjusts its penalty according to the shape of the error function.
In our computational experiments with benchmark datasets, the proposed conic formulations are solved within seconds and result in near-optimal solutions (with 0.4% optimality gap) for non-convex L0 problems. Moreover, the resulting estimators also outperform alternative convex approaches from a statistical viewpoint, achieving high prediction accuracy and good interpretability.
[
BCOL Research Report 19.01, IEOR, University of California-Berkeley.
]
rank-one.pdf
Sparse and Smooth Signal Estimation: Convexification of L0 Formulations
Alper Atamturk, Andres Gomez and Shaoning Han
Signal estimation problems with smoothness and sparsity priors can be naturally modeled as quadratic optimization with L0-``norm" constraints. Since such problems are non-convex and hard-to-solve, the standard approach is, instead, to tackle their convex surrogates based on L1-norm relaxations. In this paper, we propose new iterative (convex)
conic quadratic relaxations that exploit not only the L0-``norm" terms, but also the fitness and smoothness functions. The iterative convexification approach substantially closes the gap between the L0-``norm" and its L1 surrogate. These stronger relaxations lead to significantly better estimators than L1-norm approaches and also allow one to utilize affine sparsity priors. In addition, the parameters of the model and the resulting estimators are easily interpretable. Experiments with a tailored Lagrangian decomposition method indicate that the proposed iterative convex relaxations yield solutions within 1% of the exact L0 approach, and can tackle instances with up to 100,000 variables under one minute.
[
Journal of Machine Learning Research 22(52):1-43, 2021.
]
https://jmlr.org/papers/volume22/18-745/18-745.pdf
sparse-signal.bib
Penalized Conic Relaxations for Quadratically-Constrained Quadratic Programming
R. Madani, M. Kheirandishfard, J. Lavaei and A. Atamturk
This paper revisits conic programming relaxations for the class of quadratically-constrained quadratic programs (QCQPs). We present penalty terms, whose incorporation into the objective of convex relaxations enables the retrieval of feasible and near-optimal solutions for non-convex QCQPs. We introduce a generalized linear independence constraint qualification (GLICQ) criterion and prove that any GLICQ regular initial point that is sufficiently close to the original QCQP feasible set
can be used to construct an appropriate penalty term. As a consequence, a sequential conic optimization method is developed that preserves feasibility and aims to improve the solution at every round. Numerical experiments on large-scale system identification problems as well as benchmark examples from the library of quadratic programming (QPLIB) instances demonstrate the ability of the proposed framework in finding feasible and near-globally optimal points.
[
BCOL Research Report 18.04, IEOR, University of California-Berkeley. Forthcoming in Journal of Global Optimization.
]
penalty.pdf
Submodularity in conic quadratic mixed 0-1 optimization
Alper Atamturk and Andres Gomez
We describe strong convex valid inequalities for conic quadratic mixed 0-1 optimization.
The inequalities exploit the submodulity of the binary restrictions and are based on the polymatroid inequalities over binaries for the
diagonal case. We prove that the convex inequalities completely describe the convex hull of a single conic constraint over binary
variables and unbounded continuous variables. We then generalize and strengthen the inequalities by incorporating additional constraints
of the optimization problem. Computational experiments on a number of conic quadratic mixed 0-1 optimization applications indicate that
the new inequalities strengthen the convex relaxations substantially for the diagonal as well as the non-diagonal cases and lead to
significant performance improvements.
[
Operations Research 68, 609-630, 2020.
]
poly-mip.pdf
Successive Quadratic Upper-Bounding for Discrete Mean-Risk Minimization and Network Interdiction
Alper Atamturk, Carlos Deck and Hyemin Jeon
The advances in conic optimization have led to its increased utilization for modeling data uncertainty. In
particular, conic mean-risk optimization gained prominence in probabilistic and robust optimization. Whereas the
corresponding conic models are solved efficiently over convex sets, their discrete counterparts are intractable. In
this paper, we give a highly effective successive quadratic upper-bounding procedure for discrete mean-risk
minimization problems. The procedure is based on a reformulation of the mean-risk problem through the perspective of
its convex quadratic term. Computational experiments conducted on the network interdiction problem with stochastic
capacities show that the proposed approach yields solutions within 1-2% of optimality in a small fraction of the time
required by exact search algorithms. We demonstrate the value of the proposed approach for constructing efficient
frontiers of flow-at-risk vs. interdiction cost for varying confidence levels.
[
INFORMS Journal on Computing 32, 346-355, 2020.
]
interdiction.pdf
Simplex QP-based Methods for Minimizing a Conic Quadratic Function over Polyhedra
Alper Atamturk and Andres Gomez
We consider minimizing a conic quadratic objective over a polyhedron.
Such problems arise in parametric
value-at-risk minimization, portfolio optimization, and robust
optimization with ellipsoidal objective uncertainty;
and they can be solved by polynomial interior point algorithms for conic
quadratic optimization. However,
interior point algorithms are not well-suited for branch-and-bound
algorithms for
the discrete counterparts of these problems due to the lack of effective
warm starts necessary for the efficient
solution of convex relaxations repeatedly at the nodes of the search
tree.
In order to overcome this shortcoming, we reformulate the problem using
the perspective of its objective. The perspective
reformulation lends itself to simple coordinate descent and bisection
algorithms utilizing the simplex
method for quadratic programming, which makes the solution methods
amenable to warm starts and suitable
for branch-and-bound algorithms. We test the simplex-based quadratic
programming algorithms to solve convex
as well as discrete instances and compare them with the state-of-the-art
approaches.
The computational experiments indicate
that the proposed algorithms scale much better than interior point
algorithms and return higher precision solutions.
In our experiments, for large convex instances, they provide up to 22x
speed-up.
For smaller discrete instances, the speed-up is about 13x over a
barrier-based branch-and-bound
algorithm and 6x over the
LP-based branch-and-bound algorithm with extended formulations.
[
Mathematical Programming Computation 11, 311-340, 2019.
]
_published/mpc11-2019.pdf
cq-simplex.bib
https://doi.org/10.1007/s12532-018-0152-7
Lifted Polymatroid Inequalities for Mean-Risk Optimization with Indicator Variables
Alper Atamturk and Hyemin Jeon
We investigate a mixed 0-1 conic quadratic optimization problem with indicator variables arising in
mean-risk optimization. The indicator variables are often used to model non-convexities such as fixed
charges or cardinality
constraints. Observing that the problem reduces to a submodular function minimization for its binary
restriction,
we derive three classes of strong convex valid inequalities by lifting the polymatroid inequalities on
the binary variables. Computational experiments demonstrate the effectiveness of the inequalities in
strengthening the convex
relaxations and, thereby, improving the solution times for mean-risk problems with fixed charges and
cardinality
constraints significantly.
[
Journal of Global Optimization 73, 677-699, 2019.
]
conicvub.pdf
conicvub.bib
https://doi.org/10.1007/s10898-018-00736-z
Accomodating New Flights into an Existing Airline Flight Schedule
Ozge Safak, Alper Atamturk and M. Selim Akturk
We present two novel approaches for airline rescheduling to respond to increasing passenger demand. In both approaches, we alter an existing flight schedule to accommodate new flights while maximizing the airline's profit. A key feature of the first approach is to adjust the aircraft cruise speed to compensate for the block times of the new flights, trading off flying time and fuel burn. In the second approach, we introduce aircraft swapping as an additional mechanism to provide a greater flexibility in reducing the incremental fuel cost and adjusting the capacity. The nonlinear fuel-burn function and the binary aircraft swap and assignment decisions complicate the optimization problem significantly. We propose strong mixed-integer conic quadratic formulations to overcome the computational difficulties. The reformulations enable solving large-scale instances from a major U.S. airline optimally within reasonable compute times.
[
Transportation Research Part C 104, 265-286, 2019.
]
_published/trc104-2019.pdf
newflights.bib
https://doi.org/10.1016/j.trc.2019.05.010
A Bound Strengthening Method for Optimal Transmission Switching in Power
Systems
Salar Fattahi, Javad Lavaei and Alper Atamturk
This paper studies the optimal transmission switching
(OTS) problem for power systems, where certain lines are
fixed (uncontrollable) and the remaining ones are controllable
via on/off switches. The goal is to identify a topology of the
power grid that minimizes the cost of the system operation
while satisfying the physical and operational constraints. Most
of the existing methods for the problem are based on first
converting the OTS into a mixed-integer linear program (MILP)
or mixed-integer quadratic program (MIQP), and then iteratively
solving a series of its convex relaxations. The performance of
these methods depends heavily on the strength of the MILP or
MIQP formulations. In this paper, it is shown that finding the
strongest variable upper and lower bounds to be used in an
MILP or MIQP formulation of the OTS based on the big-M or
McCormick inequalities is NP-hard. Furthermore, it is proven
that unless P = NP, there is no constant-factor approximation
algorithm for constructing these variable bounds. Despite the
inherent difficulty of obtaining the strongest bounds in general, a
simple bound strengthening method is presented to strengthen the
convex relaxation of the problem when there exists a connected
spanning subnetwork of the system with fixed lines. The proposed
method can be treated as a preprocessing step that is independent
of the solver to be later used for numerical calculations and can
be carried out offline before initiating the solver. A remarkable
speedup in the runtime of the mixed-integer solvers is obtained
using the proposed bound strengthening method for mediumand
large-scale real world systems.
[
IEEE Transactions on Power Systems 34, 280-291, 2019.
]
_published/ots.pdf
ots.bib
https://doi.org/10.1109/TPWRS.2018.2867999
Strong Formulations for Quadratic Optimization with M-matrices and Indicator Variables
Alper Atamturk and Andres Gomez
We study quadratic optimization with indicator variables and an M-matrix, i.e., a PSD matrix with non-positive
off-diagonal entries, which arises directly in image segmentation and portfolio optimization with transaction costs, as well as a substructure of general quadratic optimization problems. We prove, under mild assumptions, that the minimization problem is solvable in polynomial time by showing its equivalence to a submodular minimization problem. To strengthen the formulation,
we decompose the quadratic function into a sum of simple quadratic functions with at most two indicator variables each, and provide the convex-hull descriptions of these sets.
We also describe strong conic quadratic valid inequalities.
Preliminary computational experiments indicate that the proposed inequalities can substantially improve the strength of the continuous relaxations with respect to the standard perspective reformulation.
[
Mathematical Programming 170, 141-176, 2018.
]
_published/mathprog170-2018.pdf
mmatrix.bib
https://doi.org/10.1007/s10107-018-1301-5
A Mixed-Integer Linear Programming Method for Optimal Orificing in Breed-and-Burn Cores
Chris Keckler, Alper Atamturk, Massimiliano Fratoni and Ehud Greenspan
In the design of nuclear reactors, it is important to ensure that each fuel assembly can be adequately
cooled so that fuel melting does not occur. The assemblies are cooled by
flowing coolant around and through
the assemblies during normal operations. The amount of coolant that
flows through each assembly can be
controlled through the use of assembly orificing, where a specified pressure drop is induced by placing an
"orifice" at the inlet of each assembly. To reduce manufacturing complexity, it is desirable to have as few
unique orifices as possible (i.e. many assemblies have the same orifice, and therefore the same coolant
flowrate). However, the orificing scheme must be designed so that each assembly can be adequately cooled
during the course of operation, even though the power of each assembly varies throughout its lifetime.
Additionally, the orificing must account for a number of engineering and safety constraints. Standard
reactors are able to achieve these goals using typically only a handful of orifice groups (5-10). Furthermore,
the orificing scheme can typically be determined through simple trial-and-error. However, a specific class
of reactors called breed-and-burn reactors present a much larger challenge for the orifice designer. This
paper explores the use of mathematical programming to determine a suitable orificing scheme
which adheres to the outlined engineering constraints while simultaneously minimizing the number of orifice groups
required.
[
Transactions of the American Nuclear Society, Vol. 118, 887-890, 2018.
]
orificing.pdf
On Capacity Models for Network Design
Alper Atamturk and Oktay Gunluk
In network design problems capacity constraints are modeled in three different ways
depending on the application and the underlying technology for installing capacity:
directed, bidirected, and undirected. In the literature, polyhedral investigations for
strengthening mixed-integer formulations are done separately for each model.
In this paper, we examine the relationship between these models to provide a unifying
approach and show that one can indeed translate valid inequalities from one to the others.
In particular, we show that the projections of the undirected and bidirected models
onto the capacity variables are the same. We demonstrate that valid inequalities
previously given for the undirected and bidirected models can be derived as a
consequence of the relationship between these models and the directed model.
[
Operations Research Letters 46, 414-417, 2018.
]
_published/orl46-2018.pdf
netcap.bib
https://doi.org/10.1016/j.orl.2018.05.002
Multi-Commodity Multi-Facility Network Design
Alper Atamturk and Oktay Gunluk
We consider multi-commodity network design models, where capacity can be
added to the edges of the network using multiples of facilities that may have different capacities. This class of mixed-integer optimization models appears frequently in telecommunication network capacity expansion problems, train scheduling with multiple locomotive options, supply chain and service network design problems. Valid inequalities used as cutting planes in branch-and-bound algorithms have been
instrumental in solving their large-scale instances. We review
the progress that has been done in polyhedral investigations in this area by emphasizing three fundemantal techniques. These are the metric inequalities for projecting out continuous flow variables, mixed-integer rounding from appropriate base relaxations, and shrinking the network to a small k-node graph. The basic inequalities derived from arc-set, cut-set and partition relaxations of the network
are also extensively utilized with certain modifications in robust and survivable network design problems.
[
BCOL Research Report 17.04, IEOR, University of
California-Berkeley. Forthcoming in Network Design with Appl. to Transportation and Logistics, T. G. Crainic, M. Gendreau, B. Gendron (eds.)
]
mcmfnd.pdf
Scalable Unit Commitment with AC Power Flow via Semidefinite Programming Relaxation
Ramtin Madani, Alper Atamturk and Ali Davoudi
Determining the most economic strategies for supply and transmission of electricity is a daunting computational challenge. The amount of effort to optimize the schedule of generating units and route of power, can grow exponentially with the number of decision variables. Practical approaches to this problem involve legacy approximations and ad-hoc heuristics that may undermine the efficiency and reliability of power system operations, that are ever growing in scale and complexity. Therefore, developing powerful optimization methods for detailed power system scheduling is critical to the realization of smart grids and has received significant attention recently. In this paper, we propose a computational method, which is capable of solving large-scale power system scheduling problems with thousands of generating units, while accurately incorporating the nonlinear equations that govern the flow of electricity on the grid. We design a polynomial-time solvable third-order semidefinite programming (TSDP) relaxation, with the aim of finding a near globally optimal solution for the unit commitment problem with AC power flow constraints. The proposed method is demonstrated on large-scale benchmark instances from real-world European grid data, for which provably optimal or near-optimal solutions are obtained.
[
BCOL Research Report 17.03, IEOR, University of
California-Berkeley.
]
uc-acopf.pdf
Network Design with Probabilistic Capacities
Alper Atamturk and Avinash Bhardwaj
We consider a network design problem with random arc capacities and give a formulation
with a probabilistic capacity constraint on each cut of the network. To handle the
exponentially-many probabilistic constraints a separation procedure that solves a
nonlinear minimum cut problem is introduced. For the case with independent arc
capacities, we exploit the supermodularity of the set function defining the constraints
and generate cutting planes based on the supermodular covering knapsack polytope.
For the general correlated case, we give a reformulation of the constraints that
allows to uncover and utilize the submodularity of a related function. The
computational results indicate that exploiting the underlying submodularity and
supermodularity arising with the probabilistic constraints provides significant
advantages over the classical approaches.
[
Networks 71, 16-30, 2018.
]
_published/networks-2018.pdf
prob-netdes.bib
http://dx.doi.org/10.1002/net.21769
A Conic Integer Programming Approach to Constrained Assortment Optimization under the Mixed Multinomial Logit Model
Alper Sen, Alper Atamturk and Philip Kaminsky
We consider the constrained assortment optimization problem under the mixed multinomial
logit model. Even moderately sized instances of this problem are challenging to solve
directly using standard mixed-integer linear programming formulations. This has
motivated recent research exploring customized optimization strategies and approximation
techniques. In contrast, we develop a novel conic quadratic mixed-integer formulation.
This new formulation, together with McCormick inequalities exploiting the capacity
constraints, enables the solution of large instances using commercial
optimization software.
[
Operations Research 66, 994-1003, 2018.
]
_published/or66-2018.pdf
assortment.bib
https://doi.org/10.1287/opre.2017.1703
Conic Relaxations of the Unit Commitment Problem
Salar Fattahi, Morteza Ashraphijuo, Javad Lavaei, Alper Atamturk
The unit commitment (UC) problem aims to find an optimal schedule of generating
units subject to demand and operating constraints for an electricity grid.
The majority of existing algorithms for the UC problem rely on solving a series
of convex relaxations by means of branch-and-bound and cutting-planning
methods. The objective of this paper is to obtain a convex model of polynomial
size for practical instances of the UC problem. To this end, we develop a convex
conic relaxation of the UC problem, referred to as a strengthened semidefinite
program (SDP) relaxation. This approach is based on first deriving certain
valid quadratic constraints and then relaxing them to linear matrix inequalities.
These valid inequalities are obtained by the multiplication of the linear constraints
of the UC problem, such as the flow constraints of two different lines.
The performance of the proposed convex relaxation is evaluated on several hard
instances of the UC problem. For most of the instances, globally optimal integer
solutions are obtained by solving a single convex problem. For the cases
where the strengthened SDP does not give rise to a global integer solution, we
incorporate other valid inequalities, including a set of boolean quadratic polytope
constraints. We prove that the proposed convex relaxation correctly finds
the optimal status of each generator that has a positive reliable lower bound.
The proposed technique is extensively tested on various IEEE power systems in
simulations.
[
Energy 134, 1079-1095, 2017.
]
UC_Energy.pdf
conic-uc.bib
https://doi.org/10.1016/j.energy.2017.06.072
Maximizing a Class of Utility Functions over the Vertices of a Polytope
Alper Atamturk and Andres Gomez
Given a polytope X, a monotone
concave univariate function g, and two vectors c and d, we consider the discrete optimization problem of
finding a vertex of X that maximizes the utility function c'x + g(d'x). This problem has numerous
applications in combinatorial optimization with a probabilistic objective, including estimation of project
duration with stochastic times, in reliability models and in multinomial logit models. We show that the
problem is NP-hard for any strictly concave function g even for simple polytopes, such as the uniform
matroid, assignment and path polytopes; and propose a 1/2-approximation algorithm for it.
We discuss improvements for special cases where g is the square root, log utility, negative exponential
utility and multinomial logit probability function.
In particular, for the square root function, the approximation ratio is 4/5.
Although the worst case bounds are tight,
computational experiments with an implementation using Lagrangian relaxations indicate that the suggested
approach finds solutions within 1-2% optimality gap
for most of the instances very quickly and can be considerably faster than existing alternatives.
[
Operations Research 65, 433-445, 2017.
]
_published/or65-2017.pdf
vertexopt.bib
http://dx.doi.org/10.1287/opre.2016.1570
A Spatial Branch-and-Cut Algorithm for Nonconvex QCQP with Bounded Complex Variables
Chen Chen, Alper Atamturk and Shmuel Oren
We develop a spatial branch-and-cut approach for nonconvex Quadratically Constrained Quadratic
Programs with bounded complex variables (CQCQP). Linear valid inequalities are added at each node
of the search tree to strengthen semidefinite programming relaxations of CQCQP. These valid
inequalities are derived from the convex hull description of a nonconvex set of 2x2
positive semidefinite Hermitian matrices subject to rank-one constraint. We propose branching
rules based on an alternative to a rank-one constraint that allows for local measurement of
constraint violation. Closed-form bound tightening procedures are used to reduce the domain of the
problem. We apply the algorithm to solve
the Alternating Current Optimal Power Flow problem with complex variables and the
Box-constrained Quadratic Programming problem with real variables.
[
Mathematical Programming 165, 549-577, 2017.
]
_published/mathprog165-2017.pdf
sbc.bib
http://dx.doi.org/10.1007/s10107-016-1095-2
Path Cover and Path Pack Inequalities for Capacitated Fixed-Charge Network Flow Problems
Alper Atamturk, Simge Kucukyavuz and Birce Tezel
Capacitated fixed-charge network flows are used to model a variety of problems in telecommunication, facility location,
production planning and supply chain management.
In this paper, we investigate capacitated path substructures and derive strong and easy-to-compute path cover and path
pack inequalities. These inequalities are based on an explicit characterization of the submodular inequalities through a fast
computation of parametric minimum cuts on a path, and they generalize the well-known flow cover and flow pack inequalities for
the single-node relaxations of fixed-charge flow models. We provide necessary and sufficient facet conditions. Computational
results demonstrate the effectiveness of the inequalities when used as cuts in a branch-and-cut algorithm.
[
SIAM Journal on Optimization 27, 1943-1976, 2017.
]
_published/siopt27-2017.pdf
pathcover.bib
https://doi.org/10.1137/15M1033009
Three-partition Inequalities for Constant Capacity Fixed-Charge Network Flow Problems
Alper Atamturk, Andres Gomez and Simge Kucukyavuz
Flow cover inequalities are among the most effective valid inequalities for solving capacitated
fixed-charge network flow problems. These valid inequalities are implications
on the flow quantity on the cut arcs of a two-partitioning of the network, depending on
whether some of the cut arcs are open or closed. As the implications are only on the
cut arcs, flow cover inequalities can be modeled by collapsing the corresponding subset of
nodes defining the cut into a single node. In this work we give new valid inequalities for
the capacitated fixed-charge network flow problem by exploiting additional information
of the network. In particular, the new inequalities are based on a three-partitioning of
the nodes and they can be modeled by collapsing the each partition into a single node.
The three-partitions inequalities include the flow cover inequalities as a special case. We
discuss the constant capacity case and give a polynomial separation algorithm for the inequalities.
Finally, we report computational results with the new inequalities for networks
with different characteristics.
[
Networks 67, 299-315, 2016.
]
_published/tpf-inequalities.pdf
tpf.bib
http://dx.doi.org/10.1002/net.21677
A Polyhedral Study of Production Ramping
Pelin Damci, Simge Kucukyavuz, Deepak Rajan and Alper Atamturk
We consider a relaxation of the unit commitment problem with ramping constraints
-- used to control the change in production level for a generator from one time period to the next --
and production limits.
This relaxation also arises in lot sizing with production smoothing constraints.
We first show that there is a polynomial algorithm for optimization over this relaxation.
For the two-period case, we give the first complete description of the convex
hull of feasible solutions. The two-period inequalities can be readily used to strengthen ramping formulations without the need for separation.
For the general case, we define exponential classes of multi-period variable upper bound and multi-period ramping inequalities,
and give conditions under which these inequalities define facets of ramping polyhedra. Finally, we present exact polynomial separation algorithms for
these inequalities and report computational experiments on using them in a branch-and-cut algorithm.
[
Mathematical Programming 158, 175-205, 2016.
]
_published/ramping.pdf
ramping.bib
http://dx.doi.org/10.1007/s10107-015-0919-9
Bound Tightening for Alternating Current Optimal Power Flow Instances with Duality Gap
Chen Chen, Alper Atamturk and Shmuel S. Oren
We consider the Alternating Current Optimal
Power Flow (ACOPF) problem, formulated as a nonconvex
Quadratically-Constarined Quadratic Program (QCQP) with
complex variables. ACOPF may be solved to global optimality
with a semidefinite programming (SDP) relaxation for cases for
which its QCQP formulation attains zero duality gap. However,
when there is positive duality gap, no optimal solution to the
SDP relaxation is feasible for ACOPF. To solve cases with duality
gap we implement a spatial branch-and-bound (SBB) algorithm
that uses a sparse strengthened SDP relaxation. SBB methods
rely on partitioning the feasible space; consequently, tightening
upper and lower variable bounds can improve solution times. We
propose closed-form bound tightening methods to tighten limits
on nodal powers, line flows, phase angle differences, and voltage
magnitude limits. We also construct variants of IEEE test cases
with high duality gaps to demonstrate the effectiveness of the
bound tightening procedures.
[
IEEE Transactions on Power Systems 31, 3729-3736, 2015.
]
_published/spbbieee.pdf
spbieee.bib
http://dx.doi.org/10.1109/TPWRS.2015.2497160
Supermodular Covering Knapsack Polytope
Alper Atamturk and Avinash Bhardwaj
The supermodular covering knapsack set is the discrete upper level set of a non-decreasing supermodular function.
Submodular and supermodular knapsack sets arise naturally when modeling utilities, risk and probabilistic
constraints on discrete variables.
In a recent paper Atamturk and Narayanan (2009) study the lower level set of a non-decreasing submodular function.
In this complementary paper we describe pack inequalities for the supermodular covering knapsack set and investigate their separation, extensions and lifting. We give sequence-independent upper bounds and lower bounds on the lifting coefficients. Furthermore, we present a computational study on using the polyhedral results derived for solving 0-1 optimization problems over conic quadratic constraints with a branch-and-cut algorithm.
[
Discrete Optimization 18, 74-86, 2015.
]
_published/supmodcovknap.pdf
SupModCovKnapsack.bib
http://dx.doi.org/10.1016/j.disopt.2015.07.003
Exact Reachability Analysis for Planning Skew-Line Needle Arrangements for Automated Brachytherapy
A. Garg, T. Siauw, G. Yang, S. Patil, J.A.M. Cunha, I.-C. Hsu, J. Pouliot, A. Atamturk and K. Goldberg
When planning skew-line needle arrangements for automated brachytherapy, one objective is to identify a set of candidate needles that enter from a specified entry region, avoid specified organs-at-risk and sufficiently cover the target (tumor) volume. Existing methods use uniform or random sampling to generate a set of candidate needles, which may not adequately cover the target volume. In this paper we present an exact reachability analysis that can be used to guide the selection of candidate needles and to identify which
subset of the target volume may not be reachable. Assuming linear needles, convex
polyhedral representations of entry zone, organs-at-risk and target volume,
we give an exact polynomial time algorithm for checking existence and calculation of the non-reachable set in the target volume. We perform experiments using patient data from 18 brachytherapy cases and found that 11 cases had non-empty occluded volume inside the target ranging from 0.01% to 4.3% of target volume. We also report a sensitivity study showing the change in the occluded volume with dilation of the avoidance volume and entry zone.
[
Proceedings of IEEE International Conference on Automation Science and Engineering (CASE), pp 524-532,
2014.
]
_published/reachability.pdf
reach.bib
http://dx.doi.org/10.1109/CoASE.2014.6899376
Aircraft Rescheduling with Cruise Speed Control
Selim Akturk, Alper Atamturk and Sinan Gurel
Airline operations are subject to frequent disruptions typically
due to unexpected aircraft maintenance requirements and undesirable
weather conditions. Recovery from a disruption often involves propagating delays
in downstream flights and increasing cruise stage speed when possible in an effort to contain
the delays. However, there is a critical tradeoff between fuel consumption
(and its adverse impact on air quality and greenhouse gas emissions) and cruise speed.
Here we consider delays caused by such disruptions
and propose a flight rescheduling model that includes
adjusting cruise stage speed on a set of affected and unaffected flights
as well as swapping aircraft optimally.
To the best of our knowledge, this is the
first study in which the cruise speed is explicitly included as a
decision variable into an airline recovery optimization model along
with the environmental constraints and costs. The proposed
model allows one to investigate the tradeoff between flight
delays and the cost of recovery. We show that the optimization approach leads
to significant cost savings compared to the popular recovery method delay
propagation.
Flight time controllability, nonlinear delay, fuel burn and CO2
emission cost functions, and binary aircraft swapping decisions complicate the aircraft recovery problem
significantly. In order to mitigate the computational difficulty we
utilize the recent advances in conic mixed-integer
programming and propose a strengthened formulation so that
the nonlinear mixed-integer recovery optimization model
can be solved efficiently. Our computational tests on realistic cases indicate that the
proposed model may be used by operations controllers to manage disruptions in
real time in an optimal manner instead of relying on ad-hoc heuristic
approaches.
[
Operations Research 62, 829-845, 2014.
]
_published/or62-2014.pdf
adm.bib
http://dx.doi.org/10.1287/opre.2014.1279
Separation and Extension of Cover Inequalities for Conic Quadratic Knapsack Constraints with Generalized Upper Bounds
Alper Atamturk, Laurent Flindt Muller and David Pisinger
Motivated by addressing probabilistic 0-1 programs we study the conic quadratic
knapsack polytope with generalized upper bound (GUB) constraints. In particular,
we investigate separating and extending GUB cover inequalities. We show that,
unlike in the linear case, determining whether a cover can be extended with a
single variable is NP-hard. We describe and compare a number of exact and
heuristic separation and extension algorithms which make use of the structure
of the constraints. Computational experiments are performed for comparing the
proposed separation and extension algorithms. These experiments show that a
judicious application of the extended GUB cover cuts can reduce the solution
time of conic quadratic 0-1 programs with GUB constraints substantially.
[
INFORMS Journal on Computing 25, 420-431, 2013.
]
_published/ijoc25-2013.pdf
conicknap-gub.bib
http://dx.doi.org/10.1287/ijoc.1120.0511
A Conic Integer Programming Approach to Stochastic Joint
Location-Inventory Problems
Alper Atamturk, Gemma Berenguer, and Zuo-Jun Max Shen
We study several joint facility location and inventory management problems with stochastic retailer demand. In particular, we
consider cases with uncapacitated facilities, capacitated facilities, correlated retailer demand, stochastic lead times, and
multi-commodities. We show how to formulate these problems as conic quadratic mixed-integer problems. Valid inequalities,
including extended polymatroid and extended cover cuts, are added to strengthen the formulations and improve the computational
results. Comparing with the existing modeling and solution methods, the new conic integer programming approach not only
provides a more general modeling framework but also leads to fast solution times in general.
[
Operations Research 60, 366-381, 2012.
]
_published/or60-2012.pdf
riskpool.bib
http://dx.doi.org/10.1287/opre.1110.1037
n-step Mingling Inequalities: Facets for Mixed-Integer Knapsack Sets
Alper Atamturk and Kiavash Kianfar
The n-step mixed integer rounding (MIR) inequalities of Kianfar and Fathi (Math Programing, 120:313-346, 2009)
are valid inequalities for the general mixed integer knapsack set that are derived based on MIR and define
facets for group problems. The mingling and 2-step mingling inequalities of Atamturk and Gunluk
(forthcoming in Math Programming) are also derived based on MIR and incorporate upper bounds on integer variables.
It is shown that
these inequalities are facet-defining under certain conditions and generalizes several well-known valid inequalities.
In this paper, we introduce new classes of valid inequalities for the mixed-integer knapsack set with bounded
integer variables, which we call n-step mingling inequalities (for each positive integer n).
These inequalities incorporate upper bounds on integer variables
into n-step MIR, and therefore, unify the concepts of n-step MIR and mingling in a single class of inequalities.
Furthermore, we show that for each n, the n-step mingling inequality defines a facet
for the mixed integer knapsack set under certain conditions. In the case of n=2, our results extend the
results of Atamturk and Gunluk on facet-defining properties of 2-step mingling. For n >= 3, our results
present new facets for the mixed integer knapsack set. We also derive as a special case conditions under which the n-step
MIR inequalities of Kianfar and Fathi define facets for the mixed integer knapsack set. In addition, we
show that n-step mingling can be used to generate new valid inequalities and facets based on covers and
packs defined over a mixed integer knapsack set.
[
Mathematical Programming 132, 79-98, 2012.
]
_published/mathprog132-2012.pdf
nstepMingling.bib
http://dx.doi.org/10.1007/s10107-010-0382-6
NPIP: A Skew Line Needle Conguration Optimization System for HDR
Brachytherapy
T. Siauw, A. Cunha, D. Berenson, A. Atamturk, I-C. Hsu, J. Pouliot, K. Goldberg
Purpose: In this study, we introduce skew line needle configurations for high dose rate (HDR)
brachytherapy and Needle Planning by Integer Program (NPIP), a computational method for
generating these configurations. NPIP generates needle configurations that are specific to the anatomy
of the patient, avoid critical structures near the penile bulb and other healthy structures, and avoid
needle collisions inside the body.
Methods: NPIP consisted of three major components: a method for generating a set of candidate
needles, a needle selection component that chose a candidate needle subset to be inserted, and a dose
planner for verifying that the final needle configuration could meet dose objectives. NPIP was used
to compute needle configurations from prostate cancer data sets from patients previously treated
at our clinic. NPIP took two user-parameters: a number of candidate needles, and needle coverage
radius, delta. The candidate needle set consisted of 5000 needles, and a range of values was used to
compute different needle configurations for each patient. Dose plans were computed for each needle
con.guration. The number of needles generated and dosimetry were analyzed and compared to the
physician implant.
Results: NPIP computed at least one needle con.guration for every patient that met dose
objectives, avoided healthy structures and needle collisions, and used as many or fewer needles than
standard practice. These needle configurations corresponded to a narrow range of delta values, which
could be used as default values if this system is used in practice. The average end-to-end runtime
for this implementation of NPIP was 286 seconds, but there was a wide variation from case to case.
Conclusions: We have shown that NPIP can automatically generate skew line needle con.gurations
with the aforementioned properties, and that given the correct input parameters, NPIP can generate
needle con.gurations which meet dose objectives and use as many or fewer needles than the current
HDR brachytherapy workflow. Combined with robot assisted brachytherapy, this system has the
potential to reduce side effects associated with treatment. A physical trial should be done to test
the implant feasibility of NPIP needle configurations.
[
Medical Physics 39, 4339-4346, 2012.
]
_published/MedPhys39-2012.pdf
npip.bib
http://dx.doi.org/10.1118/1.4728226
Maximizing a Class of Submodular Utility Functions
Shabbir Ahmed and Alper Atamturk
Given a finite ground set N and a value vector a in R^N, we
consider optimization problems involving maximization of a submodular
set utility function of
the form h(S)= f(sum_{i in S} a_i), S subseteq N, where f
is a strictly concave, increasing, differentiable function. Such problems
arise, for instance, in the context of risk-averse capital budgeting
under uncertainty, competitive facility location, and combinatorial auctions.
These problems can be formulated
as linear mixed 0-1 programs. However, the standard formulation of
these problems using submodular inequalities is
ineffective for their solution, except for very small instances.
In this paper, we perform a polyhedral analysis of a relevant mixed-integer
set and, by exploiting the structure of the utility function h,
strengthen the standard submodular formulation significantly.
We show the lifting problem of the submodular inequalities to be
a submodular maximization problem with a special structure solvable
by a greedy algorithm, which leads to an easily-computable strengthening
by subadditive lifting of the inequalities. Computational experiments
on expected utility maximization in capital budgeting show the
effectiveness of the new formulation.
[
Mathematical Programming 128, 149-169, 2011.
]
_published/mathprog128-2011.pdf
submodular-utility.bib
http://dx.doi.org/10.1007/s10107-009-0298-1
Lifting for Conic Mixed-Integer Programming
Alper Atamturk and Vishnu Narayanan
Lifting is a procedure for deriving valid inequalities for mixed-integer sets
from valid inequalities for suitable restrictions of those sets. Lifting has been shown to be
very effective in developing strong valid inequalities for linear integer programming and
it has been successfully used to solve such problems with branch-and-cut algorithms.
Here we generalize the theory of lifting to conic integer programming, i.e., integer programs
with conic constraints. We show how to derive conic valid inequalities for a conic
integer program from conic inequalities valid for its lower-dimensional restrictions.
In order to simplify computations, we also discuss sequence-independent lifting
for conic integer programs. When the cones are restricted to nonnegative orthants,
conic lifting reduces to the lifting for linear integer programming.
[Mathematical Programming 126, 351-363, 2011.
]
_published/mathprog126-2011.pdf
coniclift.bib
http://dx.doi.org/10.1007/s10107-009-0282-9
IPIP: A New Approach to Inverse Planning for HDR Brachytherapy
by Directly Optimizing Dosimetric Indices
T. Siauw, A. Cunha, A. Atamturk, I-C. Hsu, J. Pouliot, K. Goldberg
Purpose: Many planning methods for high dose rate (HDR) brachytherapy require an iterative
approach. A set of computational parameters are hypothesized that will give a dose plan that meets
dosimetric criteria. A dose plan is computed using these parameters, and if any dosimetric criteria
are not met, the process is iterated until a suitable dose plan is found. In this way, the dose distribution
is controlled by abstract parameters. The purpose of this study is to develop a new approach
for HDR brachytherapy by directly optimizing the dose distribution based on dosimetric criteria.
Methods: The authors developed inverse planning by integer program (IPIP), an optimization
model for computing HDR brachytherapy dose plans and a fast heuristic for it. They used their heuristic
to compute dose plans for 20 anonymized prostate cancer image data sets from patients previously
treated at their clinic database. Dosimetry was evaluated and compared to dosimetric criteria.
Results: Dose plans computed from IPIP satisfied all given dosimetric criteria for the target and
healthy tissue after a single iteration. The average target coverage was 95%. The average computation
time for IPIP was 30.1 s on an Intel(R) CoreTM2 Duo CPU 1.67 GHz processor with 3 Gib RAM.
Conclusions: IPIP is an HDR brachytherapy planning system that directly incorporates dosimetric
criteria. The authors have demonstrated that IPIP has clinically acceptable performance for the
prostate cases and dosimetric criteria used in this study, in both dosimetry and runtime. Further
study is required to determine if IPIP performs well for a more general group of patients and
dosimetric criteria, including other cancer sites such as GYN.
[
Medical Physics 38, 4045-4051, 2011
]
_published/MedPhys38-2011.pdf
ipip.bib
http://dx.doi.org/10.1118/1.3598437
Parallel Match-up Scheduling with Manufacturing Cost Considerations
M. Selim Akturk, Alper Atamturk, and Sinan Gurel
Many scheduling problems in practice involve rescheduling of disrupted
schedules. In this study, we show that in contrast to fixed processing times, if we
have the flexibility to control the processing times of the jobs, we can
generate alternative reactive schedules considering the manufacturing cost
implications in response to disruptions.
We consider a non-identical parallel machining environment where processing
times of the jobs are compressible at a certain manufacturing cost, which is a
convex function of the compression on the processing time.
In rescheduling it is highly desirable to catch up the original schedule as
soon as possible by reassigning the jobs to the machines
and compressing their processing times. On the
other hand, one must also keep the manufacturing cost due to compression of the jobs low.
Thus, one is faced with a tradeoff between match-up time and manufacturing cost criteria.
We introduce alternative match-up scheduling problems for finding schedules on the
efficient frontier of this time/cost
tradeoff. We employ the recent advances in conic mixed-integer programming to
model these problems effectively. We further provide a fast heuristic algorithm
driven by dual prices of convex subproblems for generating approximate efficient schedules.
[
Journal of Scheduling 13, 95-110, 2010.
]
_published/jsched13-2010.pdf
matchup.bib
http://dx.doi.org/10.1007/s10951-009-0111-2
Conic Mixed-Integer Rounding Cuts
Alper Atamturk and Vishnu Narayanan
A conic integer program is an integer programming problem with conic
constraints. Many important problems in finance, engineering,
statistical learning, and probabilistic optimization are modeled
using conic constraints.
Here we study mixed-integer sets defined by second-order conic
constraints. We introduce general-purpose cuts for conic
mixed-integer programming based on polyhedral conic substructures of
second-order conic sets. These cuts can be readily incorporated in
branch-and-bound algorithms that solve continuous conic programming
or linear programming relaxations of conic integer programs at the
nodes of the branch-and-bound tree.
Central to our approach is a reformulation of the second-order conic
constraints with polyhedral second-order conic constraints in a
higher dimensional space. In this representation the cuts we develop
are linear, even though they are nonlinear in the original space of
variables. This feature leads to computationally efficient
implementation of nonlinear cuts for conic mixed-integer
programming. The reformulation also allows the use of polyhedral
methods for conic integer programming.
We report computational results on solving unstructured second-order conic mixed-integer problems as well as
mean-variance capital budgeting problems and least-squares estimation problems with binary inputs.
Our computational experiments show that conic mixed-integer rounding
cuts are very effective in reducing the integrality gap of
continuous relaxations of conic mixed-integer programs and, hence,
improving their solvability.
[
Mathematical Programming 122, 1-20, 2010.
]
_published/mathprog122-2010.pdf
conicmip.bib
http://dx.doi.org/10.1007/s10107-008-0239-4
Mingling: Mixed-Integer Rounding with Bounds
Alper Atamturk and Oktay Gunluk
Mixed-integer rounding (MIR) is a simple, yet powerful procedure
for generating valid inequalities for mixed-integer programs.
When used as cutting planes, MIR inequalities are very effective
for problems with unbounded integer variables.
For problems with bounded integer variables, however,
cutting planes based on lifting techniques appear to be
more effective.
This is not surprising as lifting techniques make explicit use of
the bounds on variables, whereas the MIR procedure does not.
In this paper we describe a simple procedure, which we call
mingling, for incorporating variable bound information into
mixed-integer rounding. By explicitly using the variable bounds, the
mingling procedure leads to strong inequalities for mixed-integer
sets with bounded variables.
We show that facets of the
mixed-integer knapsack sets derived earlier by superadditive lifting
techniques are mingling inequalities.
In particular, the mingling inequalities developed in this paper subsume the
continuous cover and reverse continuous cover inequalities of
Marchand and Wolsey as well as the
continuous integer knapsack cover and pack inequalities of
Atamturk.
In addition, mingling inequalities give a generalization of
the two-step MIR inequalities of Dash and Gunluk
under some conditions.
[
Mathematical Programming 123, 315-338, 2010.
]
_published/mathprog123-2010.pdf
mingle.bib
http://dx.doi.org/10.1007/s10107-009-0265-x
An Economic Analysis of Wind Energy Harvest
Alper Atamturk, Jane W. Lai, Leon E. Richartz, and Kevin Gu
In this paper we develop a novel approach that simultaneously optimizes a wind farm
and its financing for an improved economic analysis of wind energy projects.
In our analysis we compare the energy yield of optimal wind energy
projects with different turbine technologies under varying electricity price and wind
speed scenarios. We perform extensive simulation studies and build statistical confidence
intervals for the annual energy yield as well as the project's net present value at risk.
The proposed approach eliminates the need for a trial-and-error financial feasibility study
and, consequently, improves the harvest of the wind energy.
[
Proceedings of IEEE Asia-Pacific Power and Energy Conference, pp 1-5, 2009.
]
_published/IEEEPublished.pdf
wind.bib
http://dx.doi.org/10.1109/APPEEC.2009.4918131
A Strong Conic Quadratic Reformulation for
Machine-Job Assignment with Controllable Processing Times
M. Selim Akturk, Alper Atamturk, and Sinan Gurel
We consider a machine-job assignment problem with separable convex cost.
A major source of difficulty with solving the problem using relaxation methods
is that optimal solutions to its continuous relaxations are highly fractional
as they are typically found in the interior of the relaxation due to the convex cost function.
Here we give a polynomial-size conic quadratic reformulation of the problem
so as to strengthen the bounds from its continuous relaxation.
The results in the paper are sufficiently general so that they can also be applied
to other mixed 0-1 optimization problems with separable convex cost functions.
Our computational results on the machine-job assignment problem with controllable
processing times demonstrate that the proposed conic reformulation is very effective
for solving the problem to optimality.
[
Operations Research Letters 37, 187-191, 2009.
]
_published/orl37-2009.pdf
conicsch.bib
http://dx.doi.org/10.1016/j.orl.2008.12.009
The Submodular Knapsack Polytope
Alper Atamturk and Vishnu Narayanan
The submodular knapsack set is the discrete lower level
set of a submodular function. The modular case reduces to the classical
linear 0-1 knapsack set. One motivation for studying the submodular knapsack
polytope is to address 0-1 programming problems with uncertain coefficients.
Under various assumptions, a probabilistic constraint on 0-1 variables
can be modeled as a submodular knapsack set.
In this paper we describe cover inequalities for the submodular knapsack set and investigate
their lifting problem. Each lifting problem is itself an optimization problem over
a submodular knapsack set. We give sequence-independent upper and lower bounds on the
valid lifting coefficients and show that whereas the upper bound can be computed in polynomial time,
the lower bound problem is NP-hard. Furthermore, we present polynomial algorithms based on parametric
linear programming and computational results
for the conic quadratic 0-1 knapsack case.
[
Discrete Optimization 6, 333-344, 2009.
]
subknap.pdf
subknap.bib
http://dx.doi.org/10.1016/j.disopt.2009.03.002
Polymatroids and Mean-Risk Minimization in Discrete Optimization
Alper Atamturk and Vishnu Narayanan
In financial markets high levels of risk are associated with
large returns as well as large losses, whereas with lower levels of
risk,
the potential for either return or loss is small. Therefore,
risk management is fundamentally concerned with finding an optimal
tradeoff between risk and return matching an investor's risk tolerance.
Managing risk is studied mostly in a financial context; nevertheless,
it is certainly relevant in any area with a significant source of
uncertainty.
The mean-risk tradeoff is well-studied for problems with a convex
feasible set.
However, this is not the case in the discrete setting, even though, in
practice, portfolios are often restricted to discrete choices.
In this paper we study mean-risk minimization for problems with discrete
decision variables. In particular, we consider discrete optimization
problems with a submodular mean-risk minimization objective.
We show the connection between extended polymatroids and the convex
lower envelope of this mean-risk objective. For 0-1 problems a complete linear
characterization of the convex lower envelope is given. For mixed 0-1
problems we derive an exponential class of conic quadratic inequalities.
We report preliminary computational experiments
on a risk-aware capital budgeting problem with uncertain returns on
investments with discrete choices. The results show significant improvements
in the solvability of the problem with the introduced convexification method.
[
Operations Research Letters 36, 618-622, 2008.
]
_published/orl36-2-2008.pdf
conicobj.bib
http://dx.doi.org/10.1016/j.orl.2008.04.006
An O(n^2) Algorithm for Lot Sizing with Inventory Bounds and Fixed Costs
Alper Atamturk and Simge Kucukyavuz
Lot-sizing problems with inventory bounds and fixed charges have not received much
attention
in the literature, even though there are many emerging applications in which highly
specialized
storage is the main activity of interest. The traditional
infinite capacity and variable cost assumptions on inventory that have
been prevalent in the literature are inappropriate in situations for which tight
storage capacity and high fixed cost of specialized storage are critical.
Here we present an O(n^2) dynamic programming algorithm for the lot-sizing problem
with inventory bounds and fixed costs, where n is the number of time periods.
The algorithm operates on a hierarchy of two layers of
value functions to solve the problem efficiently.
It improves the complexity bound of the classic O(n^3) algorithm
of Love (1973) for lot sizing with concave cost and bounded
inventory.
[
Operations Research Letters 36, 297-299, 2008.
]
lsbi-alg.pdf
lsbi-alg.bib
http://dx.doi.org/10.1016/j.orl.2007.08.004
The Flow Set with Partial Order
Alper Atamturk and Muhong Zhang
The flow set with partial order is a mixed-integer set described by
a budget on total flow and a partial order on the arcs that may
carry positive flow. This set is a common substructure of resource
allocation and scheduling problems with precedence constraints and
robust network flow problems under demand/capacity uncertainty.
We give a polyhedral analysis of the convex hull of the flow set
with partial order. Unlike for the flow set without partial order,
cover-type inequalities based on partial order structure are a
function of a lifting sequence. We study the lifting sequences and
describe structural results on the lifting coefficients for general
and simpler special cases. We show that all lifting coefficients can be
computed in polynomial time
by solving maximum weight closure problems in general. For
the special case of induced-minimal covers, we give a
sequence-dependent characterization of the lifting coefficients. We
prove, however, if the partial order is defined by an arborescence,
then lifting is sequence-independent and all lifting coefficients
can be computed in linear time. On the other hand, if the partial
order is defined by a path (total order), then the coefficients can
be expressed explicitly. We also give a complete polyhedral
description of the flow set with partial order for the
polynomially-solvable total order case. We show that finding an
optimal lifting order for a given induced-minimal cover and a given
fractional solution is a submodular optimization problem, which is
solved greedily. Finally, we present preliminary computational
results with a cutting-plane algorithm based on the lifting and
separation results.
[
Mathematics of Operations Research 33, 730-746, 2008.
]
_published/mor33-2008.pdf
pcflow.bib
http://dx.doi.org/10.1287/moor.1080.0316
Partition Inequalities for Capacitated
Survivable Network Design Based on Directed P-Cycles
Alper Atamturk and Deepak Rajan
We study the design of capacitated survivable networks using directed p-cycles. A p-cycle is a
cycle with at least three arcs, used for rerouting disrupted flow during edge failures.
Survivability of the network is accomplished by reserving sufficient slack on directed p-cycles
so that if an edge fails, its flow can be rerouted along the p-cycles. We describe a model
for designing capacitated survivable networks based on directed p-cycles. We motivate this
model by comparing it with other means of ensuring survivability, and present a mixed-integer
programming formulation for it. We derive valid inequalities for the model based on the minimum
capacity requirement between partitions of the nodes and give facet conditions for them. We
discuss the separation for these inequalities and present results of computational experiments
for testing their effectiveness as cutting planes when incorporated in a branch-and-cut
algorithm. Our experiments show that the proposed inequalities reduce the computational effort
significantly.
[
Discrete Optimization 5, 415-433, 2008.
]
_published/do5-2008.pdf
sdp.bib
http://dx.doi.org/10.1016/j.disopt.2007.08.002
Network Design Arc Set with Variable Upper Bounds
Alper Atamturk and Oktay Gunluk
In this paper we study the network design arc set with
variable upper bounds. This set appears as a common substructure
of many network design problems and is a relaxation of several
fundamental mixed-integer sets studied earlier independently. In
particular, the splittable flow arc set, the unsplittable flow arc
set, the single node fixed-charge flow set, and the binary
knapsack set are facial restrictions of the network design arc set
with variable upper bounds. Here we describe families of strong
valid inequalities that cut off all fractional extreme points of
the continuous relaxation of the network design arc set with
variable upper bounds. Interestingly, some of these inequalities
are also new even for the aforementioned restrictions studied
earlier.
[
Networks 50, 17-28, 2007.
]
_published/networks50-2007.pdf
mixset.bib
http://dx.doi.org/10.1002/net.20162
Two-Stage Robust Network Flow and Design under Demand Uncertainty
Alper Atamturk and Muhong Zhang
We describe a two-stage robust optimization approach for solving
network flow and design problems with uncertain demand. In two-stage
network optimization one defers a subset of the flow decisions until
after the realization of the uncertain demand. Availability of such a
recourse action allows one to come up with less conservative solutions
compared to single-stage optimization. However, this advantage often
comes at a price: two-stage optimization is, in general, significantly
harder than singe-stage optimization.
For network flow and design under demand uncertainty we give a
characterization of the first-stage robust decisions with an exponential
number of constraints and prove that the corresponding separation
problem is NP-hard even for a network flow problem on a bipartite
graph. We show, however, that if the second-stage network topology is
totally ordered or an arborescence, then the separation problem is
tractable.
Unlike single-stage robust optimization under demand uncertainty,
two-stage robust optimization allows one to control conservatism of the
solutions by means of an allowed ``budget for demand uncertainty.''
Using a budget of uncertainty we provide an upper bound on the
probability of infeasibility of a robust solution for a random demand
vector.
We generalize the approach to multi-commodity network flow and design,
and give applications to lot-sizing and location-transportation
problems. By projecting out second-stage flow variables we define an
upper bounding problem for the two-stage min-max-min optimization
problem. Finally, we present computational results comparing the
proposed two-stage robust optimization approach with single-stage robust
optimization as well as scenario-based two-stage stochastic
optimization.
[ Operations Research 55, 662-673, 2007. ]
_published/or55-2007.pdf
rn.bib
http://dx.doi.org/10.1287/opre.1070.0428
Valid Inequalities for Mixed-Integer Knapsack from Two-Integer Variable Restrictions
Alper Atamturk and Deepak Rajan
In this paper, we present new inequalities for the
mixed-integer knapsack set. First, we analyze the mixed-integer
knapsack set with two integer variables and one continuous
variable, and develop a polynomial algorithm that enumerates all
the facets in its convex hull. Then, we study the exact lifting
function for the facets of the convex hull of this set, and
describe super-additive lower bounds for it. These super-additive
lower bounds are obtained from partial LP relaxations of the exact
lifting function, and used in a sequence independent lifting
framework to develop strong valid inequalities for the
mixed-integer knapsack polyhedron. We present some sufficient
conditions under which these lifted inequalities define facets of
mixed-integer knapsack sets with at most one continuous variable.
Finally, we summarize our computational experience with these
inequalities.
[
BCOL Research Report 04.02, IEOR, University of California-Berkeley.
December 2004.
]
Strong Formulations of Robust Mixed 0-1 Programming
Alper Atamturk
We introduce strong formulations
for robust mixed 0-1 programming with uncertain objective
coefficients. We focus on a polytopic uncertainty
set described by as "a budget constraint" for allowed uncertainty
in the objective. We show
that for a robust 0-1 problem, there is an alpha-tight linear
programming formulation with size polynomial in the size of an
alpha-tight
linear programming formulation for the nominal 0-1 problem. We
give extensions to robust mixed 0-1 programming and present
computational experiments with the proposed formulations.
[
Mathematical Programming 108, 235-250, 2006.
]
./_published/mathprog108-2006.pdf
robobj.bib
http://dx.doi.org/10.1007/s10107-006-0709-5
Lot Sizing with Inventory Bounds and Fixed Costs: Polyhedral Study and Computation
Alper Atamturk and Simge Kucukyavuz
We investigate the polyhedral structure of the lot-sizing
problem with inventory bounds. We consider two models, one with
linear costs on inventory, the other with linear and fixed costs
on inventory. For both models, we identify facet-defining
inequalities that make use of the inventory capacities explicitly
and give exact separation algorithms. We also give a linear
programming formulation of the problem when the order and
inventory variable costs satisfy the Wagner-Whitin nonspeculative
property. We present computational experiments that show the
effectiveness of the results in tightening the linear programming
relaxations of the lot-sizing problem with inventory bounds.
[
Operations Research 53, 711-730, 2005.
]
_published/or53-2005.pdf
bif.bib
http://dx.doi.org/10.1287/opre.1050.0223
Integer Programming Software Systems
Alper Atamturk and Martin Savelsbergh
We review the recent developments in integer
programming software systems that have improved tremendously our
ability to solve large-scale instances. Our objective is to
highlight the capabilities and advanced features of
state-of-the-art optimization systems and discuss advances
towards integrated modeling and solving environments. We conclude
with perspectives on new features that we expect to see in
integer programming software systems in the near
future.
[
Annals of Operations Research 140, 67-124, 2005.
]
_published/aor140-2005.pdf
ipsoftware.bib
http://dx.doi.org/10.1007/s10479-005-3968-2
Cover and Pack Inequalities for (Mixed) Integer Programming
Alper Atamturk
We review strong inequalities for fundamental knapsack relaxations
of (mixed) integer programs. These relaxations are the 0-1
knapsack set, the mixed 0-1 knapsack set, the integer knapsack
set, and the mixed integer knapsack set. Our aim is to give a
common presentation of the inequalities based on covers and packs
and highlight the connections among them. The focus of the paper
is on recent research on the use of superadditive functions for
the analysis of knapsack polyhedra.
We also present some new results on integer knapsacks. In
particular, we give integer version of cover inequalities and
describe the necessary and sufficient facet condition for them.
This condition generalizes the well-known facet condition of
minimality of covers for 0-1 knapsacks.
[
Annals of Operations Research 139, 21-38, 2005.
]
_published/aor139-2005.pdf
cover-pack.bib
http://dx.doi.org/10.1007/s10479-005-3442-1
Polyhedral Methods in Discrete Optimization
Alper Atamturk
In the last decade our capability of solving integer programming problems
has increased dramatically due to the effectiveness of cutting plane
methods based on polyhedral investigations. Polyhedral cutting planes
have become central features in optimization software packages for integer
programming. Here we present some of the important polyhedral
methods used in discrete optimization. We discuss applications to knapsack
problems and robust combinatorial optimization.
[
]*Trends in Optimization*. S. Hosten, J. Lee, R. Thomas (eds.),
Proc. of Symposia in Applied Mathematics
61, 21-37, 2004, American Mathematical Society.
poly.pdf
poly.bib
A Study of the Lot-Sizing Polytope
Alper Atamturk and Juan Carlos Munoz
The lot-sizing polytope is a fundamental structure
contained in many practical production planning problems.
Here we study this polytope and identify facet-defining inequalities that
cut off all fractional extreme points of its linear programming relaxation,
as well as liftings from those facets.
We give a polynomial-time combinatorial separation algorithm for
the inequalities when capacities are constant.
We also report computational experiments on solving the
lot-sizing problem
with varying cost and capacity characteristics.
[
Mathematical Programming 99, 443-465, 2004.
]
_published/mathprog99-2004.pdf
ls.bib
http://dx.doi.org/10.1007/s10107-003-0465-8
A Directed Cycle based Column-and-Cut Generation Method for
Capacitated Survivable Network Design
Deepak Rajan and Alper Atamturk
A network is said to be survivable if it has sufficient capacity for rerouting
all of its flow under the failure of any one of its edges. Here
we present a polyhedral approach for designing survivable
networks. We describe a mixed-integer programming model,
in which sufficient slack is explicitly introduced on the directed
cycles of the network while flow routing decisions are made.
In case of a failure, flow is rerouted along the slacks reserved on directed cycles.
We give strong valid inequalities that use the survivability requirements.
We present a computational study with a column-and-cut generation algorithm
for designing capacitated survivable networks.
[
Networks 43, 201-211, 2004.
]
_published/networks43-2004.pdf
surv-cyc.bib
http://dx.doi.org/10.1002/net.20004
Sequence Independent Lifting for Mixed-Integer Programming
Alper Atamturk
We show that superadditive lifting functions lead to sequence independent
lifting of inequalities for general mixed integer programming.
As an application, we note that
mixed-integer rounding (MIR) may be viewed as sequence independent lifting.
Consequently, we
obtain facet conditions for MIR inequalities for mixed-integer knapsacks.
[
Operations Research 52, 487-490, 2004.
]
_published/or52-2004.pdf
sil.bib
http://dx.doi.org/10.1287/opre.1030.0099
Deferred Item and Vehicle Routing within Integrated Networks
Karen Smilowitz, Alper Atamturk, and Carlos Daganzo
Rapid growth in the package delivery industry has led carriers to offer a wider range of transportation
services defined by guaranteed delivery time. This paper studies the possible integration of long haul
operations by transportation mode and service level. Specifically, we consider the allocation of
deferred items to excess capacity on alternative modes in ways that allow all transportation modes to be
utilized better. Model formulation and solution techniques are discussed. The solution techniques
presented produce efficient solutions for large-scale problem instances. Allowing deferred items to
travel by air reduces long haul transportation costs. These savings increase with the amount of excess
air capacity.
[
Transportation Research: Logistics and Transportation Review
39, 305-323, 2003.
]
_published/tre39-2003.pdf
divrp.bib
http://dx.doi.org/10.1016/S1366-5545(02)00048-0
On the Facets of the Mixed-Integer Knapsack Polyhedron
Alper Atamturk
We study the mixed-integer knapsack polyhedron, that is,
the convex hull of the mixed-integer set defined by an
arbitrary linear inequality and the bounds on the variables.
We describe facet-defining inequalities of this polyhedron that can be obtained
through sequential lifting of inequalities containing
a single integer variable.
These inequalities strengthen and/or generalize
known inequalities for several special cases.
We report computational results on using the inequalities
as cutting planes for mixed-integer programming.
[
Mathematical Programming 98, 145-175, 2003.
]
_published/mathprog98-2003.pdf
mip.bib
http://dx.doi.org/10.1007/s10107-003-0400-z
On Capacitated Network Design Cut-Set Polyhedra
Alper Atamturk
This paper provides an analysis of capacitated network design cut-set polyhedra.
We give a complete linear description of the cut-set polyhedron
of the single commodity - single facility capacitated network design problem.
Then we extend the analysis to single commodity - multifacility and
multicommodity - multifacility capacitated network design problems.
The valid inequalities described here
have coefficients for both inflow and outflow arcs of a cut-set and
are applicable to network design problems with an arbitrary number
of facility types and arbitrary capacities.
We report a computational study to test the effectiveness of the new
inequalities.
[
Mathematical Programming 92, 425-437, 2002.
]
_published/mathprog92-3-2002.pdf
nd.bib
http://dx.doi.org/10.1007/s101070100284
On Splittable and Unsplittable Capacitated Network Design Arc-Set Polyhedra
Alper Atamturk and Deepak Rajan
We study the polyhedra of splittable and unsplittable
arc sets of multicommodity flow capacitated network design problems.
We investigate the optimization problems over these polyhedra and
the separation and lifting problems of valid inequalities for them.
In particular, we give a linear-time separation algorithm for
the residual capacity inequalities (Magnanti et al., 1993) and
show that the separation problem of
c-strong inequalities (Brockmuller et al., 1996) is NP-hard,
but can be solved over the subspace of fractional variables only.
We introduce two new classes of inequalities that generalize the
c-strong inequalities
and show that the lifting of one of them can be done in polynomial time.
We present a summary of computational experiments with a branch-and-cut algorithm for
multicommodity flow capacitated network design problems
illustrating the effectiveness of the results presented here empirically.
[
Mathematical Programming 92, 315-333, 2002.
]
_published/mathprog92-2-2002.pdf
arcset.bib
http://dx.doi.org/10.1007/s101070100269
Survivable Network Design: Routing of Flows and Slacks
Deepak Rajan and Alper Atamturk
A network is said to be survivable if all of the demands on the nodes
can be met under the failure of any one of its links.
In order to ensure that the flow
on the network can be rerouted in the case of a failure,
sufficient spare (excess) capacity must be
available on the working links of the network.
Since over-provisioning of capacity is a major concern
due to the high investment costs required
for installing capacity, designing capacity-efficient survivable
networks is a highly critical problem in the telecommunication industry.
In this paper we present a new mixed-integer programming model and a column generation
method for the survivable design of telecommunication networks.
In contrast with the failure scenario models, the new model has almost the
same number of constraints as the regular network design problem, which makes it
effective for large instances.
Even though the complexity of pricing the exponentially many variables of the model
is NP-hard, in our computational experiments,
we are able to produce capacity-efficient survivable networks
for dense graphs up to 70 nodes.
[
Telecommunications Network Design and Management, G. Anandalingam and S.
Raghavan (eds.), 65-82, Kluwer Academic Publishers, 2002.
]
rajan-atamturk-kluwer.pdf
surv-colgen.bib
ISBN 978-1-4020-7318-2
Capacity Acquisition, Subcontracting, and Lot Sizing
Alper Atamturk and Dorit Hochbaum
The fundamental question encountered in acquiring capacity
to meet nonstationary demand
over a multi-period horizon is how to balance the tradeoff
between having insufficient capacity
in some periods and excess capacity in others. In the former
situation part of the demand
is subcontracted, while in the latter capacity that has been paid for is rendered idle.
Capacity and subcontracting decisions arise in many economic activities ranging from
production capacity planning in semiconductor fabs to leasing communication networks,
from transportation contracts to staffing of call centers. In this paper,
we investigate the tradeoffs between acquiring capacity,
subcontracting, production, and holding inventory
to satisfy nonstationary demand over a finite horizon.
We present capacity acquisition models
with holding and without holding inventory and
identify forecast-robust properties of the models that
restrict the dependence of optimal capacity decisions on the demand forecasts.
We develop algorithms for numerous practical cost structures
involving variable and fixed charges
and prove that they all have polynomial time complexity. For models
with inventory, we solve a sequence of
constant capacity lot-sizing and subcontracting subproblems,
which is also of independent interest.
[
Management Science 47, 1081-1100, 2001.
]
_published/ms47-2001.pdf
cap.bib
http://dx.doi.org/10.1287/mnsc.47.8.1081.10232
Valid Inequalities for Problems with Additive Variable Upper Bounds
Alper Atamturk, George L. Nemhauser, and Martin W. P. Savelsbergh
We study the facial structure of a polyhedron associated with
the single node relaxation of network flow problems
with additive variable upper bounds. This type of structure
arises, for example, in production planning problems with
setup times and in network certain expansion problems.
We derive several classes of valid inequalities for this polyhedron and
give conditions under which they are facet--defining.
Our computational experience with large network expansion problems indicates
that these inequalities are very effective in improving the quality of the
linear programming relaxations.
[
]*Mathematical Programming* 91, 145-162, 2001.
_published/mathprog91-2001.pdf
avub.bib
http://dx.doi.org/10.1007/s101070100235
Flow Pack Facets of the Single Node Fixed-Charge Flow Polytope
Alper Atamturk
We present a class of facet-defining
valid inequalities for the single node fixed-charge flow polytope.
We provide a comparison of the new inequalities with others from the literature.
We also present computational results that show the
effectiveness of these inequalities in solving fixed-charge network
flow problems with a branch-and-cut algorithm.
[
Operations Research Letters 29, 107-114, 2001.
]
_published/orl29-2001.pdf
fp.bib
http://dx.doi.org/10.1016/S0167-6377(01)00100-6
The Mixed Vertex Packing Problem
Alper Atamturk, George L. Nemhauser, and Martin W. P. Savelsbergh
We study a generalization of the vertex packing problem having
both binary and bounded continuous variables, called the mixed vertex
packing problem (MVPP). The well-known vertex packing model arises as
a subproblem or relaxation of many 0-1 integer problems, whereas the mixed
vertex packing model arises as a natural counterpart of vertex
packing in the context of mixed 0-1 integer programming.
We describe strong valid inequalities for the convex hull of solutions to the MVPP
and separation algorithms for these inequalities. We give
a summary of computational results with a branch-and-cut algorithm
for solving the MVPP and using it to solve general mixed-integer problems.
[
]*Mathematical Programming* 89, 35-53, 2000.
_published/mathprog89-2000.pdf
mvp.bib
http://dx.doi.org/10.1007/s101070000154
Conflict Graphs in Solving Integer Programming Problems
Alper Atamturk, George L. Nemhauser, and Martin W. P. Savelsbergh
We report on our investigation of the use of
conflict graphs in solving integer programs. A conflict graph represents
logical relations between binary variables appearing in an integer program.
We construct an extended conflict graph by using probing techniques
based on feasibility as well as optimality considerations.
Each node in the search tree of an LP based branch-and-bound algorithm has its own
associated conflict graph. We develop algorithms and data structures that
allow the effective and efficient construction, management, and use of
dynamically changing conflict graphs.
Our computational experiments show that the techniques presented
work very well.
[
European Journal of Operational Research 121, 40-55, 2000.
]
_published/ejor121-2000.pdf
cg.bib
http://dx.doi.org/10.1016/S0377-2217(99)00015-6
A Relational Modeling System for Linear and Integer Programming
Alper Atamturk, Ellis Johnson, Jeff Linderoth, and Martin W. P. Savelsbergh
We discuss an integer linear programming modeling system based on relational
algebra. In this system, all modeling related activities, such as
model formulation, model instantiation, and model and instance management,
are done using simple operations such as selection, projection, and
predicated join.
[
Operations Research 48, 846-857, 2000.
]
_published/or48-2000.pdf
armos.bib
http://dx.doi.org/10.1287/opre.48.6.846.12388
A Combined Lagrangian, Linear Programming and Implication Heuristic
for Large-Scale Set Partitioning Problems
Alper Atamturk, George L. Nemhauser, and Martin W.P. Savelsbergh
Given a finite ground set, a set of subsets and costs on the subsets, the
set partitioning problem is to find a minimum cost partition of the ground
set. Many combinatorial optimization problems can be formulated as set
partitioning problems. We present an approximation algorithm that produces
high quality solutions in an acceptable amount of computation time. The
algorithm is iterative and combines problem size reduction techniques,
such as logical implications derived from feasibility and optimality
conditions and reduced cost fixing, with a primal heuristic based on cost
perturbations embedded in a Lagrangian dual framework. Computational
experiments illustrate the effectiveness of the approximation algorithm.
[
Journal of Heuristics 1, 247-259, 1996.
]
_published/.pdf
sp.bib
http://dx.doi.org/10.1007/BF00127080