The matching relaxation for a class of generalized set partitioning problems

This paper introduces a discrete relaxation for the class of combinatorial optimization problems which can be described by a set partitioning formulation under packing constraints. We present two combinatorial relaxations based on computing maximum weighted matchings in suitable graphs. Besides providing dual bounds, the relaxations are also used on a variable reduction technique and a matheuristic. We show how that general method can be tailored to sample applications, and also perform a successful computational evaluation with benchmark instances of a problem in maritime logistics.Comment: 33 pages. A preliminary (4-page) version of this paper was presented at CTW 2016 (Cologne-Twente Workshop on Graphs and Combinatorial Optimization), with proceedings on Electronic Notes in Discrete Mathematic

Algoritmo relax-and-cut para o problema do conjunto independente máximo

Heuristic and exact algorithms are proposed for the Maximum Independent Set Problem. Instead of the standard formulation of the problem, which usually leads to very weak linear relaxation bounds, we use a much stronger reformulation, based on an edge clique cover. The reformulation is new and is introduced here. To solve it, we developed a Non Delayed Relax-and-Cut algorithm, a Lagrangian analog to a cutting plane algorithm. The algorithm generates primal and dual bounds for the problem. It does that while dynamically dualizing valid inequalities, as they become violated by solutions to Lagrangian Subproblems. Next, in a warm start procedure, clique inequalities (that were dynamically dualized throughout the Relax-and-Cut algorithm) are appended to the reformulation, which is then submitted, in that reinforced form, to solver CPLEX. The Relax-and-Cut algorithm, just by itself and restricted to dualizing only clique inequalities, was able to attain optimal primal or dual bounds for 107 of the 119 instances tested. On the other hand, the exact Relax-and-Cut-CPLEX algorithm obtained optimality certificates for 64 instances, three of them previously open for thirty years. Additionally, it also attained optimal primal or dual bounds for 110 instances.Propomos algoritmos exatos e heurísticos para o Problema do Conjunto Independente Máximo. Ao invés da formulação padrão do problema, que normalmente leva a limites de relaxação linear muito fracos, utilizamos uma reformulação muito mais forte, baseada numa cobertura de arestas por cliques. A reformulação é original e é aqui introduzida. Para resolvê-la, desenvolvemos um algoritmo do tipo Non Delayed Relax-and-Cut, um análogo Lagrangeano de um algoritmo de planos de corte. O algoritmo gera limites primais e duais para o problema. Isso á feito dualizando desigualdades válidas dinamicamente, à medida em que ão violadas por soluções de Subproblemas Lagrangeanos. A seguir, num procedimento de warm start, desigualdades de clique (dualizadas dinamicamente durante a aplicação do algoritmo Relax-and-Cut) são agregadas à reformulação, que é então submetida, nessa forma reforçada, ao software CPLEX. O algoritmo Relax-and-Cut, atuando isoladamente e restrito apenas à dualização dinâmica de desigualdades de clique, conseguiu obter limites primais ou duais ótimos para 107 das 119 instâncias testadas. Por sua vez, o algoritmo exato, Relax-and-Cut-CPLEX, conseguiu obter certificados de otimalidade para 64 instâncias, três delas previamente em aberto há trinta anos. Além disso, conseguiu obter limites primais ou duais ótimos para 110 instâncias

A Relax-and-Cut algorithm for the set partitioning problem

Relax-and-Cut algorithms offer an alternative to strengthen Lagrangian relaxation bounds. The main idea behind these algorithms is to dynamically select and dualize inequalities (cuts) within a Lagrangian relaxation framework. This paper proposes a Relax-and-Cut algorithm for the Set Partitioning Problem. Computational tests are reported for benchmark instances from the literature. For Set Partitioning instances with integrality gaps, a variant of the classical Lagrangian relaxation is often used in the literature. It introduces a knapsack constraint to the standard formulation of the problem. Our results indicate that the proposed Relax-and-Cut algorithm outperforms the latter approach in terms of lower bound quality. Furthermore, it turns out to be very competitive in terms of CPU time. Decisive in achieving that performance was the implementation of dominance rules to manage inequalities in the cut pool. The Relax-and-Cut framework proposed here could also be used as a preprocessing tool for Linear Integer Programming solvers. Computational experiments demonstrated that the combined use of our framework and XPRESS improved the performance of that Linear Integer Programming solver for the test sets used in this study. (c) 2006 Elsevier Ltd. All rights reserved.3561963198

Exact rotamer optimization for computational protein design

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2008.Includes bibliographical references (leaves 235-244).The search for the global minimum energy conformation (GMEC) of protein side chains is an important computational challenge in protein structure prediction and design. Using rotamer models, the problem is formulated as a NP-hard optimization problem. Dead-end elimination (DEE) methods combined with systematic A* search (DEE/A*) have proven useful, but may not be strong enough as we attempt to solve protein design problems where a large number of similar rotamers is eligible and the network of interactions between residues is dense. In this thesis, we present an exact solution method, named BroMAP (branch-and-bound rotamer optimization using MAP estimation), for such protein design problems. The design goal of BroMAP is to be able to expand smaller search trees than conventional branch-and-bound methods while performing only a moderate amount of computation in each node, thereby reducing the total running time. To achieve that, BroMAP attempts reduction of the problem size within each node through DEE and elimination by energy lower bounds from approximate maximurn-a-posteriori (MAP) estimation. The lower bounds are also exploited in branching and subproblem selection for fast discovery of strong upper bounds. Our computational results show that BroMAP tends to be faster than DEE/A* for large protein design cases. BroMAP also solved cases that were not solvable by DEE/A* within the maximum allowed time, and did not incur significant disadvantage for cases where DEE/A* performed well. In the second part of the thesis, we explore several ways of improving the energy lower bounds by using Lagrangian relaxation. Through computational experiments, solving the dual problem derived from cyclic subgraphs, such as triplets, is shown to produce stronger lower bounds than using the tree-reweighted max-product algorithm.(cont.) In the second approach, the Lagrangian relaxation is tightened through addition of violated valid inequalities. Finally, we suggest a way of computing individual lower bounds using the dual method. The preliminary results from evaluating BroMAP employing the dual bounds suggest that the use of the strengthened bounds does not in general improve the running time of BroMAP due to the longer running time of the dual method.by Eun-Jong Hong.Ph.D