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Partial Convexification of General MIPs by Dantzig-Wolfe Reformulation

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15th international conference on integer programming and combinatorial optimization (ipco)

作者： Bergner, Martin Caprara, Alberto Furini, Fabio Luebbecke, Marco E. Malaguti, Enrico Traversi, Emiliano Rhein Westfal TH Aachen Chair Operat Res Templergraben 64 D-52056 Aachen Germany Univ Bologna DEIS Viale Risorgimento 2 I-40136 Bologna Italy

ISBN: (纸本)9783642208072

Dantzig-Wolfe decomposition is well-known to provide strong dual bounds for specially structured mixed integer programs (MIPs) in practice. However, the method is not implemented in any state-of-the-art MIP solver: it needs tailoring to the particular problem;the decomposition must be determined from the typical bordered block-diagonal matrix structure;the resulting column generation subproblems must be solved efficiently;etc. We provide a computational proof-of-concept that the process can be automated in principle, and that strong dual bounds can be obtained on general MIPs for which a solution by a decomposition has not been the first choice. We perform an extensive computational study on the 0-1 dynamic knapsack problem (without block-diagonal structure) and on general MIPLIB2003 instances. Our results support that Dantzig-Wolfe reformulation may hold more promise as a general-purpose tool than previously acknowledged by the research community.

关键词： integer programming

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Lift-and-Project Cuts for Mixed integer Convex Programs

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15th international conference on integer programming and combinatorial optimization (ipco)

作者： Bonami, Pierre Aix Marseille Univ LIF CNRS 163 Ave Luminy Case 901 F-13288 Marseille 9 France

ISBN: (纸本)9783642208072

this paper addresses the problem of generating cuts for mixed integer nonlinear programs where the objective is linear and the relations between the decision variables are described by convex functions defining a convex feasible region. We propose a new method for strengthening the continuous relaxations of such problems using cutting planes. Our method can be seen as a practical implementation of the lift-and-project technique in the nonlinear case. To derive each cut we use a combination of a nonlinear programming subproblem and a linear outer approximation. One of the main features of the approach is that the subproblems solved to generate cuts are typically not more complicated than the original continuous relaxation. In particular they do not require the introduction of additional variables or nonlinearities. We propose several strategies for using the technique and present preliminary computational evidence of its practical interest. In particular, the cuts allow us to improve over the state of the art branch-and-bound of the solver Bonmin, solving more problems in faster computing times on average.

关键词： Mixed integer Nonlinear programming Disjunctive programming Lift-and-Project Cutting Planes

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Design and Verify: A New Scheme for Generating Cutting-Planes

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15th international conference on integer programming and combinatorial optimization (ipco)

作者： Dey, Santanu S. Pokutta, Sebastian Georgia Inst Technol H Milton Stewart Sch Ind & Syst Engn Atlanta GA 30332 USA MIT Sloan Sch Management Cambridge MA 02139 USA

ISBN: (纸本)9783642208072

A cutting-plane procedure for integer programming (IP) problems usually involves invoking a black-box procedure (such as the Gomory-Chvatal (GC) procedure) to compute a cutting-plane. In this paper, we describe an alternative paradigm of using the same cutting-plane black-box. this involves two steps. In the first step, we design an inequality cx <= d, independent of the cutting-plane black-box. In the second step, we verify that the designed inequality is a valid inequality by verifying that the set P boolean AND {x is an element of R-n : cx >= d + 1} boolean AND Z(n) is empty using cutting-planes from the black-box. Here P is the feasible region of the linear-programming relaxation of the IP. We refer to the closure of all cutting-planes that can be verified to be valid using a specific cutting-plane black-box as the verification closure of the considered cutting-plane black-box. this paper conducts a systematic study of properties of verification closures of various cutting-plane black-box procedures.

关键词： Verification Scheme Cutting-Planes integer programming

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Integrality Gaps of Linear and Semi-Definite programming Relaxations for Knapsack

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15th international conference on integer programming and combinatorial optimization (ipco)

作者： Karlin, Anna R. Mathieu, Claire Nguyen, C. thach Univ Washington Seattle WA 98195 USA Brown Univ Providence RI 02912 USA

ISBN: (纸本)9783642208072

In this paper, we study the integrality gap of the Knapsack linear program in the Sherali-Adams and Lasserre hierarchies. First, we show that an integrality gap of 2 - epsilon persists up to a linear number of rounds of Sherali-Adams, despite the fact that Knapsack admits a fully polynomial time approximation scheme [24, 30]. Second, we show that the Lasserre hierarchy closes the gap quickly. Specifically, after t rounds of Lasserre, the integrality gap decreases to t/(t - 1). this answers the open question in [9]. Also, to the best of our knowledge, this is the first positive result that uses more than a small number of rounds in the Lasserre hierarchy. Our proof uses a decomposition theorem for the Lasserre hierarchy, which may be of independent interest.

关键词： Polynomial approximation

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An Iterative Scheme for Valid Polynomial Inequality Generation in Binary Polynomial programming

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15th international conference on integer programming and combinatorial optimization (ipco)

作者： Ghaddar, Bissan Vera, Juan C. Anjos, Miguel F. Univ Waterloo Dept Management Sci Waterloo ON N2L 3G1 Canada Tilburg Univ Tilburg Sch Econ & Management Tilburg Netherlands Ecole Polytech Dept Math & Genie Ind Montreal PQ H3T 1J4 Canada Ecole Polytech Gerad Montreal PQ H3T 1J4 Canada

ISBN: (纸本)9783642208072

Semidefinite programming has been used successfully to build hierarchies of convex relaxations to approximate polynomial programs. this approach rapidly becomes computationally expensive and is often tractable only for problems of small sizes. We propose an iterative scheme that improves the semidefinite relaxations without incurring exponential growth in their size. the key ingredient is a dynamic scheme for generating valid polynomial inequalities for general polynomial programs. these valid inequalities are then used to construct better approximations of the original problem. As a result, the proposed scheme is in principle scalable to large general combinatorial optimization problems. For binary polynomial programs, we prove that the proposed scheme converges to the global optimal solution for interesting cases of the initial approximation of the problem. We also present examples illustrating the computational behaviour of the scheme and compare it to other methods in the literature.

关键词： Binary Polynomial programming Binary Quadratic programming Valid Inequality Generation Semidefinite programming

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Backdoor Branching

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15th international conference on integer programming and combinatorial optimization (ipco)

作者： Fischetti, Matteo Monaci, Michele Univ Padua DEI Via Gradenigo 6-A I-35131 Padua Italy

ISBN: (纸本)9783642208072

Which is the minimum number of variables that need branching for a given MIP instance? Can this information be effective in producing compact branching trees, hence improving the performance of a state-of-the-art solver? In this paper we present a restart exact MIP solution scheme where a set covering model is used to find a small set of variables (a "backdoor", in the terminology of [8]) to be used as first-choice variables for branching. In a preliminary "sampling" phase, our method quickly collects a number of relevant low-cost fractional solutions that qualify as obstacles for LP bound improvement. then a set covering model is solved to detect a small subset of variables (the backdoor) that "cover the fractionality" of the collected fractional solutions. these backdoor variables are put in a priority branching list, and a black-box MIP solver is eventually run-in its default mode-by taking this list into account, thus avoiding any other interference with its highly-optimized internal mechanisms. Computational results on a large set of instances from MIPLIB 2010 are presented, showing that some speedup can be achieved even with respect to a state-of-the-art solver such as IBM ILOG Cplex 12.2.

关键词： integer programming

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Fixed-Charge Transportation on a Path: Linear programming Formulations

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15th international conference on integer programming and combinatorial optimization (ipco)

作者： Van Vyve, Mathieu Ctr Operat Res & Econometr Voie Roman Pays 34 B-1348 Louvain La Neuve Belgium Louvain Sch Management Voie Roman Pays 34 B-1348 Louvain La Neuve Belgium

ISBN: (纸本)9783642208072

the fixed-charge transportation problem is a fixed-charge network flow problem on a bipartite graph. this problem appears as a subproblem in many hard transportation problems, and has also strong links with the challenging big-bucket multi-item lot-sizing problem. We provide a polyhedral analysis of the polynomially solvable special case in which the associated bipartite graph is a path. We describe a new class of inequalities that we call "path-modular" inequalities. We give two distinct proofs of their validity. the first one is direct and crucially relies on sub-and super-modularity of an associated set function, thereby providing an interesting link with flow-cover type inequalities. the second proof is by projecting a tight extended formulation, therefore also showing that these inequalities suffice to describe the convex hull of the feasible solutions to this problem. We finally show how to solve the separation problem associated to the path-modular inequalities in O(n(3)) time.

关键词： mixed-integer programming lot-sizing transportation convex hull extended formulation

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Complexity Analyses of Bienstock-Zuckerberg and Lasserre Relaxations on the Matching and Stable Set Polytopes

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15th international conference on integer programming and combinatorial optimization (ipco)

作者： Au, Yu Hin Tuncel, Levent Univ Waterloo Dept Combinator & Optimizat Fac Math Waterloo ON N2L 3G1 Canada

ISBN: (纸本)9783642208072

Many hierarchies of lift-and-project relaxations for 0,1 integer programs have been proposed, two of the most recent and strongest being those by Lasserre in 2001, and Bienstock and Zuckerberg in 2004. We prove that, on the LP relaxation of the matching polytope of the complete graph on (2n+1) vertices defined by the nonnegativity and degree constraints, the Bienstock-Zuckerberg operator (even with positive semidefiniteness constraints) requires theta(root n) rounds to reach the integral polytope, while the Lasserre operator requires theta(n) rounds. We also prove that Bienstock-Zuckerberg operator, without the positive semidefiniteness constraint requires approximately n/2 rounds to reach the stable set polytope of the n-clique, if we start with the fractional stable set polytope. As a by-product of our work, we consider a significantly strengthened version of Sherali-Adams operator and a strengthened version of Bienstock-Zuckerberg operator. Most of our results also apply to these stronger operators.

关键词： matching polytope lift-and-project methods integer programming semidefinite programming convex relaxations

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An Excluded Minor Characterization of Seymour Graphs

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15th international conference on integer programming and combinatorial optimization (ipco)

作者： Ageev, Alexander Benchetrit, Yohann Sebo, Andras Szigeti, Zoltan Sobolev Inst Math Novosibirsk Russia Lab G SCOP Grenoble France

ISBN: (纸本)9783642208072

A graph G is said to be a Seymour graph if for any edge set F there exist vertical bar F vertical bar pairwise disjoint cuts each containing exactly one element of F, provided for every circuit C of G the necessary condition vertical bar C boolean AND F vertical bar = vertical bar C\F vertical bar is satisfied. Seymour graphs behave well with respect to some integer programs including multiflow problems, or more generally odd cut packings, and are closely related to matching theory. A first coNP characterization of Seymour graphs has been shown by Ageev, Kostochka and Szigeti [1], the recognition problem has been solved in a particular case by Gerards [2], and the related cut packing problem has been solved in the corresponding special cases. In this article we show a new, minor-producing operation that keeps this property, and prove excluded minor characterizations of Seymour graphs: the operation is the contraction of full stars, or of odd circuits. this sharpens the previous results, providing at the same time a simpler and self-contained algorithmic proof of the existing characterizations as well, still using methods of matching theory and its generalizations.

关键词： integer programming

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integer programming and combinatorial optimization 2011

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丛书名： Lecture Notes in Computer Science

2011年

作者： Oktay Günlük Gerhard J. Woeginger

ISBN: (数字)9783642208072

ISBN: (纸本)9783642208065

this book constitutes the proceedings of the 15th international conference on integer programming and combinatorial optimization, ipco 2011, held in New York, USA in June 2011. the 33 papers presented were carefully reviewed and selected from 110 submissions. the conference is a forum for researchers and practitioners working on various aspects of integer programming and combinatorial optimization with the aim to present recent developments in theory, computation, and applications. the scope of ipco is viewed in a broad sense, to include algorithmic and structural results in integer programming and combinatorial optimization as well as revealing computational studies and novel applications of discrete optimization to practical problems.

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