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检索条件"任意字段=7th International Conference on Integer Programming and Combinatorial Optimization, IPCO 1999"

共 85 条记录，以下是21-30 订阅

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Verifying integer programming Results 1

引用

19th international conference on integer programming and combinatorial optimization (ipco)

作者： Cheung, Kevin K. H. Gleixner, Ambros Steffy, Daniel E. Carleton Univ Sch Math & Stat Ottawa ON Canada Zuse Inst Berlin Dept Math Optimizat Takustr 7 D-14195 Berlin Germany Oakland Univ Dept Math & Stat Rochester MI 48063 USA

ISBN: (数字)9783319592503

ISBN: (纸本)9783319592503;9783319592497

Software for mixed-integer linear programming can return incorrect results for a number of reasons, one being the use of inexact floating-point arithmetic. Even solvers that employ exact arithmetic may suffer from programming or algorithmic errors, motivating the desire for a way to produce independently verifiable certificates of claimed results. Due to the complex nature of state-of-the-art MIP solution algorithms, the ideal form of such a certificate is not entirely clear. this paper proposes such a certificate format designed with simplicity in mind, which is composed of a list of statements that can be sequentially verified using a limited number of inference rules. We present a supplementary verification tool for compressing and checking these certificates independently of how they were created. We report computational results on a selection of MIP instances from the literature. To this end, we have extended the exact rational version of the MIP solver SCIP to produce such certificates.

关键词： Correctness Verification Proof Certificate Optimality Infeasibility Mixed-integer linear programming

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Market split and basis reduction: Towards a solution of the cornuéjols-dawande instances 1

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7th international conference on integer programming and combinatorial optimization, ipco 1999

作者： Aardal, Karen Bixby, Robert E. Hurkens, Cor A. J. Lenstra, Arjen K. Smeltink, Job W. Department of Computer Science Utrecht University Netherlands Department of Computational and Applied Mathematics Rice University United States Department of Mathematics and Computing Science Eindhoven University of Technology Netherlands Emerging Technology Citibank N.A United States

ISBN: (数字)9783540487777

ISBN: (纸本)3540660194

At the ipco VI conference Cornuéjols and Dawande proposed a set of 0-1 linear programming instances that proved to be very hard to solve by traditional methods, and in particular by linear programming based branch-and-bound. they offered these market spliti nstances as a challenge to the integer programming community. the market split problem can be formulated as a system of linear diophantine equations in 0-1 variables. In our study we use the algorithm of Aardal, Hurkens, and Lenstra (1998) based on lattice basis reduction. this algorithm is not restricted to deal with market split instances only but is a general method for solving systems of linear diophantine equations with bounds on the variables. We show computational results from solving both feasibility and optimization versions of the market split instances with up to 7 equations and 60 variables, and discuss various branching strategies and their effect on the number of nodes enumerated. To our knowledge, the largest feasibility and optimization instances solved before have 6 equations and 50 variables, and 4 equations and 30 variables respectively. We also present a probabilistic analysis describing how to compute the probability of generating infeasible market split instances. the formula used by Cornuéjols and Dawande tends to produce relatively many feasible instances for sizes larger than 5 equations and 40 variables. © Springer-Verlag Berlin Heidelberg 1999.

关键词： integer programming

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Linear programming Using Limited-Precision Oracles 20th

Linear Programming Using Limited-Precision Oracles

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20th international conference on integer programming and combinatorial optimization (ipco)

作者： Gleixner, Ambros Steffy, Daniel E. Konrad Zuse Zentrum Informat Tech Berlin Takustr 7 D-14195 Berlin Germany Oakland Univ Math & Stat Rochester MI 48063 USA

ISBN: (纸本)9783030179533;9783030179526

Linear programming is a foundational tool for many aspects of integer and combinatorial optimization. this work studies the complexity of solving linear programs exactly over the rational numbers through use of an oracle capable of returning limited-precision LP solutions. Under mild assumptions, it is shown that a polynomial number of calls to such an oracle and a polynomial number of bit operations, is sufficient to compute an exact solution to an LP. Previous work has often considered oracles that provide solutions of an arbitrary specified precision. While this leads to polynomial-time algorithms, the level of precision required is often unrealistic for practical computation. In contrast, our work provides a foundation for understanding and analyzing the behavior of the methods that are currently most effective in practice for solving LPs exactly.

关键词： Linear programming Oracle complexity Diophantine approximation Exact solutions Symbolic computation Rational arithmetic Extended-precision arithmetic Iterative refinement

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Two-Stage Stochastic Stable Matching 25th

Two-Stage Stochastic Stable Matching

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25th international conference on integer programming and combinatorial optimization (ipco)

作者： Faenza, Yuri Foussoul, Ayoub He, Chengyue Columbia Univ IEOR New York NY 10027 USA

ISBN: (纸本)9783031598340;9783031598357

We introduce and study a two-stage stochastic stable matching problem between students and schools. A decision maker chooses a stable matching in a marriage instance;then, after some agents enter or leave the market following a probability distribution D, chooses a stable matching in the new instance. the goal is, roughly speaking, to maximize the expected quality of the matchings across the two stages and minimize the expected students' discontent for being downgraded to a less preferred school in the second-stage. We consider both the case when D is given explicitly and when it is accessed via a sampling oracle. In the former case, we give a polynomial time algorithm. In the latter case, we show that, unless P = NP, no algorithm can find the optimal value or the optimal solution of the problem in polynomial-time. On the positive side, we give a pseudopolynomial algorithm that computes a solution of arbitrarily small additive error. Our techniques include the use of a newly defined poset of stable pairs, which may be of independent interest.

关键词： Stable Matching Two-stage stochastic programming Poset

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Lower Bounds on the Complexity of Mixed-integer Programs for Stable Set and Knapsack 25th

Lower Bounds on the Complexity of Mixed-Integer Programs for...

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25th international conference on integer programming and combinatorial optimization (ipco)

作者： Schade, Jamico Sinha, Makrand Weltge, Stefan Tech Univ Munich Munich Germany Univ Illinois Champaign IL 61820 USA

ISBN: (纸本)9783031598340;9783031598357

Standard mixed-integer programming formulations for the stable set problem on n-node graphs require n integer variables. We prove that this is almost optimal: We give a family of n-node graphs for which every polynomial-sizeMIP formulation requires Omega(n/ log(2) n) integer variables. By a polyhedral reduction we obtain an analogous result for n-item knapsack problems. In both cases, this improves the previously known bounds of Omega(root n/ log n) by Cevallos, Weltge & Zenklusen (SODA 2018). To this end, we show that there exists a family of n-node graphs whose stable set polytopes satisfy the following: any (1 + epsilon/n)-approximate extended formulation for these polytopes, for some constant epsilon > 0, has size 2(Omega(n / log n)). Our proof extends and simplifies the information-theoretic methods due to Goos, Jain & Watson (FOCS 2016, SIAM J. Comput. 2018) who showed the same result for the case of exact extended formulations (i.e. epsilon = 0).

关键词： mixed-integer programming stable set problem knapsack problem extended formulations

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Mixed-integer Linear Representability, Disjunctions, and Variable Elimination 1

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19th international conference on integer programming and combinatorial optimization (ipco)

作者： Basu, Amitabh Martin, Kipp Ryan, Christopher thomas Wang, Guanyi Johns Hopkins Univ Dept Appl Math & Stat Baltimore MD 21218 USA Univ Chicago Booth Sch Business Chicago IL 60637 USA Georgia Inst Technol Ind & Syst Engn Atlanta GA 30332 USA

ISBN: (数字)9783319592503

ISBN: (纸本)9783319592503;9783319592497

Jeroslow and Lowe gave an exact geometric characterization of subsets of R-n that are projections of mixed-integer linear sets, a.k.a MILP-representable sets. We give an alternate algebraic characterization by showing that a set is MILP-representable if and only if the set can be described as the intersection of finitely many affine Chvatal inequalities. these inequalities are a modification of a concept introduced by Blair and Jeroslow. this gives a sequential variable elimination scheme that, when applied to the MILP representation of a set, explicitly gives the affine Chvatal inequalities characterizing the set. this is related to the elimination scheme of Wiliams and Williams-Hooker, who describe projections of integer sets using disjunctions of affine Chvatal systems. Our scheme extends their work in two ways. First, we show that disjunctions are unnecessary, by showing how to find the affine Chvatal inequalities that cannot be discovered by the Williams-Hooker scheme. Second, disjunctions of Chvatal systems can give sets that are not projections of mixed-integer linear sets;so the Williams-Hooker approach does not give an exact characterization of MILP representability.

关键词： integer programming

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Strong Mixed-integer programming Formulations for Trained Neural Networks 1

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20th international conference on integer programming and combinatorial optimization (ipco)

作者： Anderson, Ross Huchette, Joey Tjandraatmadja, Christian Vielma, Juan Pablo Google Res Cambridge MA 02142 USA MIT Cambridge MA 02139 USA

ISBN: (数字)9783030179533

ISBN: (纸本)9783030179533;9783030179526

We present an ideal mixed-integer programming (MIP) formulation for a rectified linear unit (ReLU) appearing in a trained neural network. Our formulation requires a single binary variable and no additional continuous variables beyond the input and output variables of the ReLU. We contrast it with an ideal "extended" formulation with a linear number of additional continuous variables, derived through standard techniques. An apparent drawback of our formulation is that it requires an exponential number of inequality constraints, but we provide a routine to separate the inequalities in linear time. We also prove that these exponentially-many constraints are facet-defining under mild conditions. Finally, we study network verification problems and observe that dynamically separating from the exponential inequalities (1) is much more computationally efficient and scalable than the extended formulation, (2) decreases the solve time of a state-of-the-art MIP solver by a factor of 7 on smaller instances, and (3) nearly matches the dual bounds of a state-of-the-art MIP solver on harder instances, after just a few rounds of separation and in orders of magnitude less time.

关键词： Mixed-integer programming Formulations Deep learning

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Assortment optimization with Visibility Constraints 25th

Assortment Optimization with Visibility Constraints

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25th international conference on integer programming and combinatorial optimization (ipco)

作者： Barre, theo El Housni, Omar Lodi, Andrea Univ Calif Berkeley IEOR Berkeley CA USA Cornell Univ Cornell Tech ORIE New York NY 10021 USA

ISBN: (纸本)9783031598340;9783031598357

Motivated by applications in e-retail and online advertising, we study the problem of assortment optimization under visibility constraints, referred to as APV. We are given a universe of substitutable products and a stream of T customers. the objective is to determine the optimal assortment of products to offer to each customer in order to maximize the total expected revenue, subject to the constraint that each product is required to be shown to a minimum number of customers. the minimum display requirement for each product is given exogenously and we refer to these constraints as visibility constraints. We assume that customer choices follow a Multinomial Logit model. We provide a characterization of the structure of the optimal assortments and present an efficient polynomial time algorithm for solving APV. To accomplish this, we introduce a novel function called the "expanded revenue" of an assortment and establish its supermodularity. Our algorithm takes advantage of this structural property. Additionally, we demonstrate that APV can be formulated as a compact linear program. Finally, we propose a novel, fair strategy for pricing the revenue loss due to the enforcement of visibility constraints.

关键词： Assortment optimization Supermodularity Algorithm Design Multinomial Logit model Visibility Constraints

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Sensitivity Analysis for Mixed Binary Quadratic programming 25th

Sensitivity Analysis for Mixed Binary Quadratic Programming

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25th international conference on integer programming and combinatorial optimization (ipco)

作者： Cifuentes, Diego Dey, Santanu S. Xu, Jingye Georgia Inst Technol H Milton Stewart Sch Ind & Syst Engn Atlanta GA 30332 USA

ISBN: (纸本)9783031598340;9783031598357

We consider sensitivity analysis for Mixed Binary Quadratic Programs (MBQPs) with respect to changing right-hand-sides (rhs). We show that even if the optimal solution of a given MBQP is known, it is NP-hard to approximate the change in objective function value with respect to changes in rhs. Next, we study algorithmic approaches to obtaining dual bounds for MBQP with changing rhs. We leverage Burer's completely-positive (CPP) reformulation of MBQPs. Its dual is an instance of co-positive programming (COP), and can be used to obtain sensitivity bounds. We prove that strong duality between the CPP and COP problems holds if the feasible region is bounded or if the objective function is convex, while the duality gap can be strictly positive if neither condition is met. We also show that the COP dual has multiple optimal solutions, and the choice of the dual solution affects the quality of the bounds with rhs changes. We finally provide a method for finding good nearly optimal dual solutions, and we present preliminary computational results on sensitivity analysis for MBQPs.

关键词： Sensitivity Analysis Mixed Binary Quadratic programming Copositive programming Duality theory

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On the Partial Convexification of the Low-Rank Spectral optimization: Rank Bounds and Algorithms 25th

On the Partial Convexification of the Low-Rank Spectral Opti...

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25th international conference on integer programming and combinatorial optimization (ipco)

作者： Li, Yongchun Xie, Weijun Georgia Inst Technol H Milton Stewart Sch Ind & Syst Engn Atlanta GA 30332 USA

ISBN: (纸本)9783031598340;9783031598357

A Low-rank Spectral optimization Problem (LSOP) minimizes a linear objective function subject to multiple two-sided linear inequalities intersected with a low-rank and spectral constrained domain. Although solving LSOP is, in general, NP-hard, its partial convexification (i.e., replacing the domain by its convex hull) termed "LSOP-R", is often tractable and yields a high-quality solution. this motivates us to study the strength of LSOP-R. Specifically, we derive rank bounds for any extreme point of the feasible set of LSOP-R with different matrix spaces and prove their tightness. the proposed rank bounds recover two well-known results in the literature from a fresh angle and allow us to derive sufficient conditions under which the relaxation LSOP-R is equivalent to LSOP. To effectively solve LSOP-R, we develop a column generation algorithm with a vector-based convex pricing oracle, coupled with a rank-reduction algorithm, which ensures that the output solution always satisfies the theoretical rank bound. Finally, we numerically verify the strength of LSOP-R and the efficacy of the proposed algorithms.

关键词： Linear programming

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