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A Hybrid Metaheuristic for Multiobjective Unconstrained Binary Quadratic Programming Arnaud Liefooghe, Sébastien Verel, Jin-Kao Hao

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A Hybrid Metaheuristic for Multiobjective Unconstrained Binary Quadratic Programming Arnaud Liefooghe, Sébastien Verel, Jin-Kao Hao A Hybrid Metaheuristic for Multiobjective Unconstrained Binary Quadratic Programming Arnaud Liefooghe, Sébastien Verel, Jin-Kao Hao To cite this version: Arnaud Liefooghe, Sébastien Verel, Jin-Kao Hao. A Hybrid Metaheuristic for Multiobjective Uncon- strained Binary Quadratic Programming. 2013. hal-00801793v2 HAL Id: hal-00801793 https://hal.archives-ouvertes.fr/hal-00801793v2 Submitted on 3 Jun 2013 (v2), last revised 15 Nov 2013 (v3) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. A Hybrid Metaheuristic for Multiobjective Unconstrained Binary Quadratic Programming Arnaud Liefooghe∗,a,b, S´ebastien Verelb,c, Jin-Kao Haod aLIFL, Universit´eLille 1, UMR CNRS 8022, Cit´escientifique, Bˆat. M3, 59655 Villeneuve d’Ascq cedex, France bInria Lille-Nord Europe, Parc Scientifique de la Haute Borne, 40 avenue Halley, 59650 Villeneuve d’Ascq, France cI3S, Universit´eNice Sophia Antipolis, UMR CNRS 6070, 2000 route des Lucioles, BP 121, 06903 Sophia Antipolis cedex, France dLERIA, Universit´ed’Angers, 2 bd. Lavoisier, 49045 Angers, France Abstract The conventional Unconstrained Binary Quadratic Programming (UBQP) problem is known to be a unified modeling and solution framework for many combinatorial optimization problems. This paper extends the single-objective UBQP to the multiobjective case (mUBQP) where multiple objectives are to be optimized simultaneously. We propose a hybrid metaheuristic which combines an elitist evolutionary multiobjective optimization algorithm and a state-of-the-art single-objective tabu search procedure by using an achievement scalarizing function. Finally, we define a formal model to generate mUBQP instances and validate the performance of the proposed approach in obtaining competitive results on large-size mUBQP instances with two and three objectives. Key words: Unconstrained binary quadratic programming, Multiobjective combinatorial optimization, Hybrid Metaheuristic, Evolutionary Multiobjective Optimization, Tabu search, Scalarizing function 1. Introduction the UBQP itself is clearly a NP-hard problem [7]. During the past few decades, a large number of algorithms and approaches Given a collection of n items such that each pair is associated have been proposed for the single-objective UBQP in the liter- with a profit value that can be positive, negative or zero, uncon- ature. This includes several exact methods based on branch and strained binary quadratic programming (UBQP) seeks a subset bound or branch and cut [8, 9, 10] and a number of heuristic of items that maximizes the sum of their paired values. The and metaheuristic methods like simulated annealing [11], tabu value of a pair is accumulated in the sum only if the two corre- search [12, 13, 14, 15, 16], path-relinking [17], evolutionary sponding items are selected. A candidate solution to a UBQP and memetic algorithms [18, 19, 20, 21]. instance can be specified by a binary string of size n, such that each variable indicates whether the corresponding item is in- In this paper, we extend this conventional single-objective cluded in the selection or not. More formally, the conventional UBQP problem to the multiobjective case, denoted by mUBQP, and single-objective UBQP problem is to maximize the follow- where multiple objectives are to be optimized simultaneously. ing objective function. Such an extension naturally increases the expressive ability of the UBQP and provides a convenient formulation to fit situa- n n tions where the single-objective UBQP cannot accommodate. ′ f (x) = x Qx = qi j xi x j (1) For instance, UBQP can recast the vertex coloring problem Xi=1 Xj=1 (of determining the chromatic number of a graph) [5] and the sum coloring problem (of determining the chromatic sum of a where Q = (qi j) is an n by n matrix of constant values and x graph) [22]. Still, UBQP is not convenient to formulate the is a vector of n binary (zero-one) variables, i.e., xi ∈ {0, 1}, i ∈ {1,..., n}. bi-objective coloring problem which requires to determine a The UBQP is known to be a general model able to repre- legal vertex coloring of a graph while minimizing simultane- sent a wide range of important problems, including those from ously the number of colors used and the sum of colors. For this financial analysis [1], social psychology [2], computer aided bi-objective coloring problem, the mUBQP formulation can be design [3] and cellular radio channel allocation [4]. More- employed in a straightforward way. over, a number of NP-hard problems can be conveniently trans- In addition of introducing the mUBQP problem, the paper formed into the UBQP, such as graph coloring problems, max- has two additional contributions. First, given that the single- cut problem, set packing problem, set partitioning problem, objective UBQP is NP-hard, its generalized mUBQP formula- maximum clique problem and so on [5, 6]. As a consequence, tion is also a difficult problem to solve in the general case. For the purpose of approximating the Pareto set of a given mUBQP instance, heuristic approaches are then compulsory. Follow- ∗ Corresponding author, Tel.: +33 3 59 35 86 30. ing the studies on memetic algorithms for the UBQP and many Email addresses: [email protected] (Arnaud Liefooghe), [email protected] (S´ebastien Verel), other problems, we adopt as our solution approach the memetic [email protected] (Jin-Kao Hao) framework and propose a hybrid metaheuristic which combines 1 an elitist evolutionary multiobjective optimization algorithm the Pareto set, denoted by XPS , and its mapping in the objective with a state-of-the-art single-objective tabu search procedure space is called the Pareto front. One of the most challenging based on an achievement scalarizing function. The last con- issue in multiobjective combinatorial optimization is to identify tribution of this work is to define a formal and flexible model a minimal complete Pareto set, i.e., one Pareto optimal solution to generate hard mUBQP instances. An experimental analy- mapping to each point from the Pareto front. Note that such a sis validates the interest of the proposed hybrid metaheuristic set may not be unique, since multiple solutions can map to the by achieving a clear improvement over non-hybrid and conven- same non-dominated vector. tional algorithms on large-size mUBQP instances with two and three objectives. 2.3. Properties The paper is organized as follows. Section 2 introduces the For many multiobjective combinatorial optimization prob- multiobjective formulation of the UBQP problem (mUBQP). lems, computing the Pareto set is computationally prohibitive Section 3 presents the hybrid metaheuristic (HM) proposed for for two main reasons. First, the question of deciding if the mUBQP and its main ingredients, including the scalarizing a candidate solution is dominated is known to be NP-hard evaluation function, the tabu search procedure, the initialization for numerous multiobjective combinatorial optimization prob- phase and the variation operators. Section 4 gives an experi- lems [23, 24]. This is also the case for the mUBQP problem mental analysis of the HM algorithm on a large set of mUBQP since its single-objective counterpart is NP-hard [7]. Second, instances of different structure and size. The last section con- the number of Pareto optimal solutions typically grows expo- cludes and suggests further research lines. nentially with the size of the problem instance [24]. In that sense, most multiobjective combinatorial optimization prob- 2. Multiobjective Unconstrained Binary Quadratic Pro- lems are said to be intractable. In the following, we prove that gramming the mUBQP problem is intractable. This section first introduces the multiobjective unconstrained Proposition 1. The multiobjective unconstrained binary binary quadratic programming problem. Some definitions re- quadratic programming problem (2) is intractable, even for lated to multiobjective combinatorial optimization are then re- m = 2. called, followed by problem complexity-related properties and Proof. Consider the following bi-objective mUBQP instance. a link with similar problem formulations. Last, the construc- n(i−1)− i(i−1) + j−1 tion of problem instances, together with an experimental study 2 2 if i > j q1 = i, j ∈ {1,..., n} on the objective values correlation and the cardinality of the i j ( 0 if i < j Pareto set, are presented. 2 1 Let qi j = −qi j for all i, j ∈ {1,..., n}. As illustrated in Figure 1 = 2.1. Problem Formulation for n 3, it is obvious that all solutions are mutually non- dominated. Therefore, all feasible solutions are Pareto optimal, The multiobjective unconstrained binary quadratic program- n and |XPS | = |X| = 2 . ming (mUBQP) problem can be stated as follows. n n In order to cope with NP-hard and intractable multiobjective k combinatorial optimization problems, researchers have devel- max fk(x) = qi j xi x j k ∈ {1,..., m} (2) Xi=1 Xj=1 oped approximate algorithms that identify a Pareto set approx- imation having both good convergence and distribution proper- subject to xi ∈ {0, 1} i ∈ {1,..., n} ties [25, 26]. To this end, metaheuristics in general, and evolu- where f = ( f1, f2,..., fm) is an objective vector function with tionary algorithms in particular, have received a growing inter- k k m > 2, n is the problem size, and we have m matrices Q = (qi j) est since the late eighties [27]. of size n by n with constant values, k ∈ {1,..., m}. The decision space X is defined on binary strings of size n. 2.4. Links with Existing Problem Formulations The single-objective UBQP problem is of high interest in 2.2.
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