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Variable Bound Tightening and Valid Constraints for Multiperiod Blending

INFORMS JOURNAL ON COMPUTING 2022年第4期34卷 2073-2090页

作者： Chen, Yifu Maravelias, Christos T. Univ Wisconsin Dept Chem & Biol Engn Madison WI 53706 USA Princeton Univ Dept Chem & Biol Engn Princeton NJ 08544 USA Princeton Univ Andlinger Ctr Energy & Environm Princeton NJ 08544 USA

Multiperiod blending has a number of important applications in a range of industrial sectors. It is typically formulated as a nonconvex mixed integer nonlinear program (MINLP), which involves binary variables and bilinear terms. In this study, we first propose a reformulation of the constraints involving bilinear terms using lifting. We introduce a method for calculating tight bounds on the lifted variables calculated by aggregating multiple constraints. We propose valid constraints derived from the reformulation-linearization technique (RLT) that use the bounds on the lifted variables to further tighten the formulation. Computational results indicate our method can substantially reduce the solution time and optimality gap.

关键词： preprocessing reformulation-linearization technique variable lifting bilinear terms

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Efficient Separation of RLT Cuts for Implicit and Explicit Bilinear Products 1

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24th International Conference on Integer Programming and Combinatorial Optimization (IPCO)

作者： Bestuzheva, Ksenia Gleixner, Ambros Achterberg, Tobias Zuse Inst Berlin Berlin Germany HTW Berlin Berlin Germany Gurobi GmbH Frankfurt Germany

ISBN: (数字)9783031327261

ISBN: (纸本)9783031327254;9783031327261

The reformulation-linearization technique (RLT) is a prominent approach to constructing tight linear relaxations of non-convex continuous and mixed-integer optimization problems. The goal of this paper is to extend the applicability and improve the performance of RLT for bilinear product relations. First, a method for detecting bilinear product relations implicitly contained in mixed-integer linear programs is developed based on analyzing linear constraints with binary variables, thus enabling the application of bilinear RLT to a new class of problems. Our second contribution addresses the high computational cost of RLT cut separation, which presents one of the major difficulties in applying RLT efficiently in practice. We propose a new RLT cutting plane separation algorithm which identifies combinations of linear constraints and bound factors that are expected to yield an inequality that is violated by the current relaxation solution. A detailed computational study based on implementations in two solvers evaluates the performance impact of the proposed methods.

关键词： reformulation-linearization technique Bilinear products Cutting planes Mixed-integer programming

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An LP-based characterization of solvable QAP instances with chess-board and graded structures

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JOURNAL OF COMBINATORIAL OPTIMIZATION 2023年第5期45卷 114-114页

作者： Waddell, Lucas A. Phillips, Jerry L. Liu, Tianzhu Dhar, Swarup Bucknell Univ Lewisburg PA 17837 USA Francis Mar Univ Florence SC 29502 USA

The quadratic assignment problem (QAP) is perhaps the most widely studied nonlinear combinatorial optimization problem. It has many applications in various fields, yet has proven to be extremely difficult to solve. This difficulty has motivated researchers to identify special objective function structures that permit an optimal solution to be found efficiently. Previous work has shown that certain such structures can be explained in terms of a mixed 0-1 linear reformulation of the QAP known as the level-1 reformulation-linearization-technique (RLT) form. Specifically, the objective function structures were shown to ensure that a binary optimal extreme point solution exists to the continuous relaxation. This paper extends that work by considering classes of solvable cases in which the objective function coefficients have special chess-board and graded structures, and similarly characterizing them in terms of the level-1 RLT form. As part of this characterization, we develop a new relaxed version of the level-1 RLT form, the structure of which can be readily exploited to study the special instances under consideration.

关键词： Quadratic assignment linearization reformulation-linearization technique

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Tight lower bounds for the Traveling Salesman Problem with draft limits

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COMPUTERS & OPERATIONS RESEARCH 2023年 154卷

作者： Balma, Ali Mrad, Mehdi Ladhari, Talel Univ Tunis Tunis Business Sch Business Analyt & Decis Making POB 65 Tunis 2059 Tunisia Univ Tunis Ecole Natl Super Ingenieurs Tunis Ave Taha Hussein Tunis 1008 Tunisia Univ Tunis Ecole Super Sci Econ & Commerciales Tunis 4 Rue Abou Zakaria El Hafsi Montfleury 1089 Tunisia Umm Al Qura Univ Coll Business Adm Mecca Saudi Arabia King Saud Univ Coll Engn Dept Ind Engn Riyadh 11421 Saudi Arabia

In the present paper, we consider the Traveling Salesman Problem with Draft Limits (TSPDL), which is a variant of the Traveling Salesman Problem (TSP) arising in maritime transportation. We provide compact formulations of polynomial size developed by using the reformulation-linearization technique (RLT) with one and two levels. In doing so, we recover time-dependent formulations of TSP and provide their factorization into products involving the decision variables and their complements. This factorization affords further RLT-lifting to the model of TSPDL by combining time-dependent and flow variables that lead to very tight linear relaxation. Computational results conducted on benchmark instances confirm the tightness of the lower bounds.

关键词： Traveling Salesman Time-dependent model Multi-commodity flow reformulation-linearization technique Draft limits

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Second order cone constrained convex relaxations for nonconvex quadratically constrained quadratic programming

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JOURNAL OF GLOBAL OPTIMIZATION 2019年第2期75卷 461-494页

作者： Jiang, Rujun Li, Duan Fudan Univ Sch Data Sci Shanghai Peoples R China City Univ Hong Kong Sch Data Sci Hong Kong Peoples R China

In this paper, we present new convex relaxations for nonconvex quadratically constrained quadratic programming (QCQP) problems. While recent research has focused on strengthening convex relaxations of QCQP using the reformulation-linearization technique (RLT), the state-of-the-art methods lose their effectiveness when dealing with (multiple) nonconvex quadratic constraints in QCQP, except for direct lifting and linearization. In this research, we decompose and relax each nonconvex constraint to two second order cone (SOC) constraints and then linearize the products of the SOC constraints and linear constraints to construct some new effective valid constraints. Moreover, we extend the reach of the RLT-like techniques for almost all different types of constraint-pairs (including valid inequalities by linearizing the product of a pair of SOC constraints, and the Hadamard product or the Kronecker product of two respective valid linear matrix inequalities), examine dominance relationships among different valid inequalities, and explore almost all possibilities of gaining benefits from generating valid constraints. We also successfully demonstrate that applying RLT-like techniques to additional redundant linear constraints could reduce the relaxation gap significantly. We demonstrate the efficiency of our results with numerical experiments.

关键词： Nonconvex quadratically constrained quadratic programming Convex relaxations reformulation-linearization technique SOC-RLT

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ALLOCATING NODES TO HUBS FOR MINIMIZING THE HUBS PROCESSING RESOURCES: A CASE STUDY

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RAIRO-OPERATIONS RESEARCH 2019年第3期53卷 807-827页

作者： Balma, Ali Mrad, Mehdi Univ Tunis Ecole Natl Super Ingenieurs Tunis 5 Av Taha Hussein Tunis 1008 Tunisia Univ Tunis Tunis Business Sch BADEM POB 65 Bir El Kassaa 2059 Tunisia King Saud Univ Dept Ind Engn Coll Engn POB 800 Riyadh 11421 Saudi Arabia

This paper addresses the problem of allocating the terminal nodes to the hub nodes in a telecommunication network. Since the flow processing induces some undesirable delay, the objective is to minimize the total flow processed by the hubs. This study focuses on a real life network of the tunisian operator Tunisie Telecom whose operations managers are concerned by the quality of service. We provide three compact formulations that give optimal solutions for networks of large size. In particular, the last two are obtained by applying the reformulation-linearization technique to a nonlinear formulation of the problem. The latter formulation derived within this approach is the most computationally effective, as pointed out by the computational experiments conducted on the real life network of Tunisie Telecom with 110 nodes and 5 hubs. Finally, we discuss and compare between the single allocation and double allocation configurations.

关键词： Hub allocation Non-linear programming Combinatorial optimization Graphs and Networks reformulation-linearization technique Telecommunications

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SOLVING QUADRATIC PROGRAMMING BY CUTTING PLANES

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SIAM JOURNAL ON OPTIMIZATION 2019年第2期29卷 1076-1105页

作者： Bonami, Pierre Lodi, Andrea Schweiger, Jonas Tramontani, Andrea IBM Spain CPLEX Optimizat Madrid Spain Polytech Montreal Canada Excellence Res Chair CP 6079Succ Ctr Ville Montreal PQ H3C 3A7 Canada Zuse Inst Berlin Berlin Germany

We propose new cutting planes for strengthening the linear relaxations that appear in the solution of nonconvex quadratic problems with linear constraints. By a famous result of Motzkin and Straus, these problems are connected to the clique number of a graph. Our cuts are derived in the context of a spatial branch-and-bound algorithm where linearization variables are introduced to represent products. Their validity is based on the result of Motzkin and Straus, in that it depends on the clique number of certain graphs. For convenience, we derive our cuts for the special case of the so-called standard quadratic programs, where the only (linear) constraint imposes that variables must belong to a simplex. We specifically consider cuts that correspond to carefully constructed complete bipartite graphs. We study the relation between these cuts and the classical ones obtained at the first level of the reformulation-linearization technique. By studying this relation, we derive a new type of valid inequalities that generalizes both types of cuts, and thus leading to a large family of cutting planes to strengthen the linear relaxation. We present extensive computational results using the different cutting planes we propose within the spatial branch and bound implemented by the commercial solver CPLEX. We show that our cuts allow us to obtain a significantly better bound than reformulation-linearization cuts and reduce computing times for global optimality. Finally, we formally establish the scaling necessary for using the proposed cuts to solve nonconvex quadratic problems with linear constraints, e.g., quadratic knapsack problems.

关键词： nonconvex programming standard quadratic programming global optimization cutting planes reformulation-linearization technique

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Strong multi-commodity flow formulations for the asymmetric traveling salesman problem

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2018年第1期271卷 72-79页

作者： Balma, Ali Ben Salem, Safa Mrad, Mehdi Ladhari, Talel Univ Tunis Ecole Natl Super Ingenieurs Tunis Ave Taha Hussein Tunis 1008 Tunisia King Saud Univ RCSE Adv Mfg Inst Riyadh 11421 Saudi Arabia Univ Tunis Tunis Business Sch Business Analyt & Decis Making POB 65 Bir El Kassaa 2059 Tunisia

We provide new compact formulations of polynomial size for the asymmetric traveling salesman problem obtained through the reformulation-linearization technique. The first one is obtained directly by this latter approach while the two others are derived by performing projections of this formulation on the variables of the existing models. We show that the devised formulations are stronger than the state-of-the-art models. Computational experiments conducted on benchmark instances for the classical variant and with precedence constraints confirm the better quality of the relaxations provided by our proposed formulations. (C) 2018 Elsevier B.V. All rights reserved.

关键词： Traveling salesman reformulation-linearization technique Compact formulations Sequential ordering problem

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A practical solution approach for the green vehicle routing problem

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TRANSPORTATION RESEARCH PART E-LOGISTICS AND TRANSPORTATION REVIEW 2017年 104卷 97-112页

作者： Leggieri, Valeria Haouari, Mohamed Free Univ Bozen Bolzano Fac Sci & Technol Bolzano Italy Qatar Univ Mech & Ind Engn Dept Doha Qatar

Green vehicle routing (GVRP) is an active research area that is concerned with the development and analysis of distribution activities with eco-friendly vehicles. We propose a practical solution approach for the GVRP that uses a mixed-integer linear formulation and a reduction procedure. The newly formulation offers two significant advantages: compactness and flexibility. We provide empirical evidence that the formulation and the reduction procedure enable to derive optimal solutions for medium-sized instances using a general-purpose solver. We show that the proposed exact approach consistently outperforms a state-of-the-art branch-and-cut algorithm and constitutes an appealing and practical alternative for optimally solving GVRPs. (C) 2017 Elsevier Ltd. All rights reserved.

关键词： Green vehicle routing problem Compact mixed integer linear programming formulation reformulation-linearization technique Preprocessing procedures

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Lifted polynomial size formulations for the homogeneous and heterogeneous vehicle routing problems

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2017年第3期263卷 755-767页

作者： Leggieri, Valeria Haouari, Mohamed Free Univ Bozen Bolzano Fac Sci & Technol I-39100 Bolzano Italy Qatar Univ Coll Engn Dept Mech & Ind Engn Doha 2713 Qatar

We propose compact formulations for the symmetric and asymmetric capacitated vehicle routing problems. These formulations are obtained by lifting, using the reformulation-linearization technique, a novel polynomial size ordered path-based formulation. We show that the strongest proposed formulation is equivalent to the strongest multi-commodity flow formulation presented so far. In addition, we propose polynomial size valid inequalities that aim at further tightening the proposed formulations. Also, the tightest formulation is extended to model the fleet size and mix vehicle routing problem with fixed costs. We show that the new derived model is not comparable with a state-of-the-art polynomial length formulation. We present the results of computational experiments that demonstrate the tightness of the proposed formulations and the impact of the valid inequalities. (C) 2017 Elsevier B.V. All rights reserved.

关键词： Transportation Capacitated vehicle routing problem reformulation-linearization technique Heterogeneous VRP Compact formulations

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