2014 Workshop on Mixed Integer Programming (MIP 2014) Abstracts!

Transcription

1 2014 Workshop on Mixed Integer Programming (MIP 2014) Abstracts MONDAY, JULY 21: 10:00a-10:30a: Simge Küçükyavuz, Ohio State University, USA Title: Cut Generation in Optimization Problems with Multivariate Risk Constraints Abstract: We consider a class of multicriteria stochastic optimization problems that feature benchmarking constraints based on conditional value-at-risk and second-order stochastic dominance. We develop alternative mixed-integer programming formulations and solution methods for cut generation problems arising in optimization under such multivariate risk constraints. We give the complete linear description of two non-convex substructures common in these cut generation problems. We present computational results that show the effectiveness of our proposed models and methods. Joint work with Nilay Noyan. 11:00a-11:30a: François Margot, Carnegie Mellon University, USA Title: Solving Quadratic Chance Constrained Problems with Random Technology Matrix Abstract: Two broad classes of Stochastic Programming (SP) models are SPs with recourse (where decisions taken prior to realization of the stochastic events can be corrected) and chance constrained programming (CCP) (where no recourse is possible). Most of the solution approaches for CCP models deal with the case where random variables only appear in the right-hand side of the constraints and are unable to tackle problems having random variables impacting constraint coefficients (a.k.a. "random technology matrix"). We develop methods to solve CCP problems with a multi-row random technology matrix, using an exact Boolean reformulation method based on the binarization of probability distributions. For linear stochastic constraints, the method derives an equivalent mixed-integer linear programming reformulation. We extend the method to problems with bilinear stochastic constraints, deriving deterministic reformulations with trilinear terms. Such problems appear naturally in many different fields, including response model to propagation of epidemics, facility location problem with random demand, and pooling systems with uncertain quality of reservoirs and flows.

2 We report on experiments with Couenne (a COIN-OR software project) to solve the resulting trilinear deterministic problems. We discuss practical issues with several mathematically equivalent formulations for some propagation of epidemic and facility location problems. Joint work with M. Lejeune. 11:30a-12:00p: Giacomo Nannicini, SUTD, Singapore Title: An algorithm for nonlinear chance-constrained problems with applications to hydro scheduling Abstract: In the spirit of MIP, I will present very recent work on midterm hydro scheduling, which is the problem of optimizing the performance of a hydro network over a period of a few months. This decision problem is affected by uncertainty on energy prices, demands and rainfall, and we model it as a nonlinear chance-constrained mathematical program. Akin to this problem, the talk is affected by uncertainty at the present stage. There is a probability bounded away from zero that I will discuss a Branch-and-Cut algorithm based on the approach recently proposed by Jim Luedtke, which uses Benders-type cuts. However, in our case the feasible region induced by each scenario is a general convex set instead of a polyhedron. I will almost surely present a separation algorithm for the corresponding scenario subproblems that exploits projection and KKT conditions, and has some clear advantages over generalized Benders decomposition. With unknown probability I will provide a preliminary computational evaluation of the proposed method for the scheduling of a hydro network in Greece. Finally, with probability one I will apologize for giving a talk that does not correspond to this abstract. Joint work with Andrea Lodi, Enrico Malaguti and Dimitri Thomopulos. 2:00p-2:45p: Daniel Bienstock, Columbia University, USA Title: Solving QCQPs Abstract: Quadratically constrained quadratic programs, or QCQPs are a natural generalization of linear programs and, in recent years, have received growing interest, motivated by both fundamental reasons and also by many applications in science and engineering. Typically QCQPs are continuous, but the similarity with linear programming ends there, because QCQPs are usually nonconvex, and even in simple cases, are NP-hard. In fact, QCQPs are quite general, because any polynomial optimization problem can be reduced to an equivalent (and polynomially larger) QCQPs. In this talk we will review classical results and describe recent results, including (hopefully) some applications of linear mixed integer programming ideas. Joint work with Gonzalo Munoz and Irene Lo.

3 2:45p-3:15p: Poster Teasers, Part I 3:45p-4:15p: Stefan Vigerske, GAMS, Germany Title: Analyzing the computational impact of individual MINLP solver components Abstract: General-purpose solvers that address large and heterogeneous problem classes like mixed-integer nonlinear programming (MINLP) necessarily combine a variety of algorithmic techniques in their solution process. In this talk, we present recent advances in the constraint integer programming-based MINLP solver SCIP with a special focus on analyzing the computational impact of individual solver components such as branching strategies, separation routines, bound tightening techniques, and primal heuristics. Joint work with Ambros Gleixner. 4:15p-5:00p: Andreas Wächter, Northwestern University, USA Title: Hot starting NLP solvers Abstract: One reason for the efficiency of the branch-and-bound method for MILP is that LP nodes can often be solved very quickly, especially when only a small number of variables bounds are changed during strong-branching or diving. The crucial observation here is that the factorization of the basis matrix from a previously solved LP can be reused and updated in the dual Simplex method for the new LP. NLP algorithms (such as sequential quadratic programming) also require the factorization of matrices that involve parts of the constraint Jacobian during the step computation with an active-set QP solver. However, due to the nonlinearity of the constraints, these matrices change with every iterate, so that the factorization from a previously solved NLP is not the one required for the step computations during the solution of a new NLP. In this talk, we present one approach that attempts to "hot-start" the solution of a new NLP by reusing the matrix factorization available from a previously solved NLP. In this way, the work of factorizing the new constraint matrix from scratch is avoided, at the cost of multiple refinement iterations that compensate for the error. Numerical results will be presented. 5:00p-5:30p: Poster Teasers, Part II 5:30p-8:00p: Poster Session and Reception

4 TUESDAY, JULY 22: 9:30a-10:00a: Juliane Dunkel, IBM Research, Zurich, Switzerland Title: Mixed-integer programming for real-time railway control Abstract: Railway networks are operated more and more at their capacity limits, disturbances and delays propagate quickly and affect the service level experienced by customers. As a consequence, railway traffic management and, in particular, real-time railway control has become an increasingly challenging task. Revised schedules have to be computed within a few seconds, making the need of computer-aided systems to support dispatchers more and more evident. However, due to the combinatorial complexity of the underlying optimization problems and the immense sizes of instances, this problem is difficult to solve in acceptable time. Macroscopic models generally neglect important technical details as train dynamics and safety regulations, and microscopic models that incorporate many technical aspects become practically infeasible. We suggest a mixed-integer programming formulation for the problem of computing revised schedules that combines macroscopic and microscopic aspects. The formulation allows us to model bottleneck areas of a network up to microscopic detail, while uncritical areas can be considered with only macroscopic granularity. Hence, the model produces practically workable solutions. We enhance our formulation with several variable fixing procedures and show how very large instances can be solved efficiently via a rolling-horizon approach. 10:30a-11:15a: Karen Aardal, TU Delft, Netherlands Title: GMI/Split cuts based on lattice information Abstract: Cutting planes incorporated in a branch-and-bound framework is the most dominant solution approach for (mixed)-integer optimization problems. One important family of cutting planes is the family of split cuts. A computational study by Balas and Saxena indicates that the first closure associated with the family of split inequalities is a very good approximation of the convex hull of feasible solution. It is, however, NP-hard to optimize a linear function over the split closure, so achieving these results is computationally expensive. A special case of the split cuts, which can trivially be generated, is the family of GMI-inequalities that can be obtained from optimal basic feasible solutions. The computational effectiveness of these inequalities is however much more modest (Bixby, Gu, Rothberg, and Wunderling). The discrepancy between the potential effectiveness of GMI/split inequalities indicated by the study of Balas and Saxena, and the results that so far can be realized by generating such inequalities from optimal basic solutions, led

5 Cornuejols to suggest that one should look for deep split cuts that can be separated efficiently. In our work we suggest a heuristic way of generating GMI/split inequalities that is based on information from the structure of the underlying lattice. We present examples and some computational indications. This talk is based on joint work with Frederik von Heymann, Andrea Lodi, Andrea Tramontani, and Laurence Wolsey. 11:15a-12:00p: Sanjeeb Dash, IBM Research, New York, USA Title: On two-branch split cuts Abstract: In this talk we present some properties of two-branch split cuts, which generalize the split cuts of Cook, Kannan and Schrijver, and were studied by Li and Richard (2008). In particular, we show that the closure of a polyhedral set with respect to two-branch split cuts is a polyhedron. Furthermore, we use this result to show that the quadrilateral closure of the two-row continuous group relaxation the set of points satisfying all cutting planes obtained from maximal lattice-free quadrilaterals is a polyhedron. We also discuss computational results with split cuts and two-branch split cuts in a few different settings. Joint work with Oktay Gunluk and Diego Moran. 2:00p-2:30p: Marco Molinaro, Georgia Institute of Technology, USA Title: How Good Are Sparse Cutting-Planes? Abstract: Sparse cutting-planes are often the ones used in mixed-integer programing (MIP) solvers, since they help in solving the linear programs encountered during branch-&-bound more efficiently. However, how well can we approximate the integer hull by just using sparse cutting-planes? In order to understand this question better, given a polyope P (e.g. the integer hull of a MIP), let P^k be its best approximation using cuts with at most k non-zero coefficients. We consider d(p, P^k) = max_{x in P^k} (min_{y in P} x - y ) as a measure of the quality of sparse cuts. In our first result, we present general upper bounds on d(p, P^k) which depend on the number of vertices in the polytope and exhibits three phases as k increases. Our bounds imply that if P has polynomially many vertices, using half sparsity already approximates it very well. Second, we present a lower bound on d(p, P^k) for random polytopes that show that the upper bounds are quite tight. Third, we show that for a class of hard packing IPs, sparse cutting-planes do not approximate the integer hull well. Finally, we show that using sparse cuttingplanes in extended formulations is at least as good as using them in the original polyhedron, and give an example where the former is actually much better. Joint work with Santanu Dey and Qianyi Wang.

6 2:30p-3:00p: Minjiao Zhang, University of Alabama, USA Title: On Knapsack-Constrained Continuous Mixing Set Abstract: We study the knapsack-constrained continuous mixing set, which arises in the joint chance-constrained program. We characterize the extreme points and rays of the convex hull of the knapsack-constrained continuous mixing set, and develop two extended formulations and linear descriptions. In addition, for the cardinality-constrained continuous mixing set, which is a special case of the knapsack-constrained continuous mixing set, we propose the sufficient and necessary conditions for the valid inequalities, and present a set of valid inequalities in an explicit form. In particular, we show that the proposed valid inequalities are enough to describe a particular case of the cardinality-constrained continuous mixing set. 3:30p-4:00p: Pierre Bonami, IBM CPLEX, Spain Title: Cut generation through binarization Abstract: For a mixed integer linear program with bounded general integer variables, we study the combination of a reformulation introduced by Roy that maps general integer variables to a collection of binary variables and simple split cuts. We show that a pure integer problem with two bounded integer variables is solved by doing this reformulation and computing the rank-2 simple split closure. We show that this result does not generalize to problems in higher dimensions. We compare in particular rank-2 simple split cuts from the reformulated problem to intersection cuts from two dimensional lattice free sets. Finally, we present an algorithm to approximate the rank-2 simple split cut closure and report empirical results on 22 benchmark instances. We show that the bounds obtained compare favorably with those obtained with other approximate methods to compute the split closure or lattice-free cut closure. Joint work with François Margot. 4:00p-4:45p: Alper Atamtürk, UC Berkeley, USA Title: Network design with uncertain capacities Abstract: We consider a network design problem with uncertain edge capacities, which, under various assumptions, can be modeled using conic quadratic constraints on binary variables. Conic quadratic constraints on binary variables lead to supermodular or submodular knapsack problems. We will discuss cover and pack inequalities, their extensions, reductions, lifting and separation for uncorrelated and correlated cases. Joint work with Avinash Bhardwaj and Vishnu Narayanan. 4:45p-5:15p: Business meeting

7 WEDNESDAY, JULY 23: 9:30a-10:15a: Sebastian Pokutta, Georgia Institute of Technology, USA Title: The information-theoretic method in optimization Abstract: Recently problems in extended formulations, convex black-box optimization, and compressed sensing have been shown to be especially tractable via an information-theoretic approach. We provide an overview of the informationtheoretic method, the underlying paradigms, and present three applications demonstrating the power of the method, both for obtaining lower bounds as well as upper bounds. 10:45a-11:15a: Laurent Poirrier, University of Waterloo, Canada Title: Permutations in the factorization of simplex bases Abstract: The basis matrices corresponding to consecutive iterations of the simplex method only differ in a single column. This fact is commonly exploited in current LP solvers to avoid the computation of a fresh factorization at every iteration. Instead, a previous factorization is updated to reflect the modified column. We present a variant of this process for the special case where the update can be performed purely by permuting rows and columns of the factors. Joint work with Ricardo Fukasawa. 11:15a-12:00p: Zonghao Gu, Gurobi, USA Title: Solving LP and MIP Models with Piecewise Linear Objective Functions Abstract: LP and MIP models often contain piecewise linear structure; this structure may capture a true piecewise linear function, or it may be used to approximate a non-linear function. Current solvers use one optimization variable for each piece of the piecewise function, which can dramatically increase the model size. This talk will discuss extensions of the simplex and MIP B&B algorithms that allow them to solve piecewise linear models without requiring additional variables. 2:00p-2:45p: Andrea Lodi, University of Bologna, Italy Title: Indicator Constraints in Mixed-Integer Programming Abstract: Mixed Integer Linear Programming (MILP) models are commonly used to model indicator constraints, which either hold or are relaxed depending on the value of a binary variable. Classification problems with Ramp Loss functions are an important application of such models. Mixed Integer Nonlinar Programming (MINLP) models are usually dismissed because they cannot be solved as efficiently. However, we show here that a subset of classification problems can be solved much more efficiently by a MINLP model with nonconvex constraints. This calls for a

8 reconsideration of the modeling of these indicator constraints, and we present several new results and interpretations obtained by digging into the relationship between MILP and MINLP. 2:45p-3:15p: Timo Berthold, FICO Xpress Optimization, Germany Title: Cloud branching: How to exploit dual degeneracy in global search Abstract: Branch-and-bound methods for MIP are traditionally based on solving an LP relaxation and branching on a variable which takes a fractional value in the (single) computed relaxation optimum. We study branching strategies for mixedinteger programs that exploit the knowledge of *multiple* alternative optimal solutions (a cloud) of the current LP relaxation. These strategies naturally extend state-of-the-art methods like strong branching, pseudocost branching, and their hybrids. We show that by exploiting dual degeneracy, and thus multiple alternative optimal solutions, it is possible to enhance traditional methods. We present preliminary computational results, applying the newly proposed strategy to full strong branching, which is known to be the MIP branching rule leading to the fewest number of search nodes, and pseudo cost branching, which is the basis of most state-of-the-art branching algorithms. Experiments are carried out in the state-ofthe-art MIP solvers SCIP and FICO Xpress Optimizer. Joint work with Domenico Salvagnin and Gerald Gamrath. 3:45p-4:15p: Michele Monaci, University of Padova, Italy Title: Proximity Search for 0-1 Mixed-Integer Convex Programming Abstract: In this talk we investigate the effects of replacing the objective function of a 0-1 Mixed-Integer Convex Program with a ``proximity'' one, with the aim of enhancing the heuristic behavior of a black-box solver. The relationship of this approach with primal integer methods is also addressed. Promising computational results on different proof-of-concept implementations are presented, suggesting that proximity search can be very effective in quickly improving the incumbent in the early part of the search. This is particularly true when a sequence of similar MIPs has to be solved as, e.g., in a column-generation setting, or for problems that can be easily decomposed, e.g., according to a Benders' scheme. Joint work with Matteo Fischetti. 4:15p-4:45p: Jeff Linderoth, University of Wisconsin-Madison, USA Title: Orbital Conflict-When Worlds Collide Abstract: In this talk, we smash together ideas that have proven to be effective at exploiting symmetry with strong cutting planes for mixed integer programs. We

9 hope that the computational results will be earth-shattering. Joint work with Jim Ostrowski, University of Tennessee and Fabrizio Rossi and Stefano Smriglio, Universita di L Aquila. 4:45p-5:15p: Domenico Salvagnin, University of Padova, Italy Title: Detecting and exploiting permutation structures in MIPs Abstract: Many combinatorial optimization problems can be formulated as the search for the best possible permutation of a given set of objects, according to a given objective function. The corresponding MIP formulation is thus typically made of an assignment substructure, plus additional constraints and variables (as needed) to express the objective function. Unfortunately, the permutation structure is generally lost when the model is flattened out as a mixed integer program, and state-of-the-art MIP solvers do not take full advantage of it. In the present paper we propose a heuristic procedure to detect permutation problems from their MIP formulation, and show how we can take advantage of this knowledge to speedup the solution process. Computational results on quadratic assignment and single machine scheduling problems show that the technique, when embedded in a stateof-the-art MIP solver, can indeed improve performance. THURSDAY, JULY 24: 9:30a-10:00a: Raymond Hemmecke, TU Munich, Germany Title: Augmentation Algorithms for Linear and Integer Linear Programming Abstract: Separable convex IPs can be solved via polynomially many augmentation steps if best augmenting steps along Graver basis directions are performed. Using instead augmentation along directions with best ratio of cost improvement/unit length, we show that for linear objectives the number of augmentation steps is bounded by the number of elements in the Graver basis of the problem matrix, giving strongly polynomial-time algorithms for the solution of N-fold LPs and ILPs. 10:30a-11:00a: Robert Hildebrand, ETH Zurich, Switzerland Title: Convex Set Operators and Polynomial Integer Minimization in Fixed Dimension Abstract: We propose a new approach to minimizing a polynomial over the integer points in a polyhedron based on a convex set operator. This problem is known to be NP-Hard in dimension two even when the feasible region is bounded and the objective is a polynomial of degree four. We solve the feasibility problem by dividing the plane into regions where a sub-level set is convex or its complement

10 is convex. The operator then applies to solving the feasibility problem on convex sets or polyhedra after removing a convex set. We use this approach to show that cubic and homogeneous polynomials can be minimized over a polytope in dimension two in polynomial time. In particular, this completes a complexity classification by degree of the minimization problem in dimension two. 11:00a-11:45a: Warren Adams, Clemson University, USA Title: Modeling Polynomial Functions of Two Discrete Variables Abstract: Given two discrete variables x and y, the challenge is to construct efficient polyhedral representations of a general polynomial function of these variables. To obtain such representations, we recast the function in an extendedvariable space by defining a new continuous variable for each distinct nonlinear term. Then we construct, in an encompassing space, a polytope that possesses two key properties. The first property is a one-to-one correspondence between the extreme points and the number of possible pairwise-realizations of x and y. Second, at every extreme point, each variable x and y realizes one of its permissible values, and each auxiliary variable equals to its intended product. In this manner, optimization of any such polynomial function reduces to a linear program. The methodology uses Lagrange Interpolating Polynomials, as in the reformulationlinearization-technique for general discrete variables, to compute the extended spaces, and a projection operation to re-express the polytopes in more tractable lower-dimensional regions. Complete characterizations of the projections are explicitly available when one of the discrete variables is binary, extending the classic Fortet - McCormick inequalities for two binary variables. Joint work with Stephen M. Henry.