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Combining convergence and diversity in evolutionary multiobjective optimization
 Evolutionary Computation
, 2002
"... Over the past few years, the research on evolutionary algorithms has demonstrated their niche in solving multiobjective optimization problems, where the goal is to �nd a number of Paretooptimal solutions in a single simulation run. Many studies have depicted different ways evolutionary algorithms c ..."
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Cited by 118 (11 self)
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Over the past few years, the research on evolutionary algorithms has demonstrated their niche in solving multiobjective optimization problems, where the goal is to �nd a number of Paretooptimal solutions in a single simulation run. Many studies have depicted different ways evolutionary algorithms can progress towards the Paretooptimal set with a widely spread distribution of solutions. However, none of the multiobjective evolutionary algorithms (MOEAs) has a proof of convergence to the true Paretooptimal solutions with a wide diversity among the solutions. In this paper, we discuss why a number of earlier MOEAs do not have such properties. Based on the concept ofdominance, new archiving strategies are proposed that overcome this fundamental problem and provably lead to MOEAs that have both the desired convergence and distribution properties. A number of modi�cations to the baseline algorithm are also suggested. The concept ofdominance introduced in this paper is practical and should make the proposed algorithms useful to researchers and practitioners alike.
Approximation Algorithms for Combinatorial Multicriteria Optimization Problems
 International Transactions in Operational Research
, 2000
"... The computational complexity of combinatorial multiple objective programming problems is investigated. INP completeness and #P completeness results are presented. Using two definitions of approximability, general results are presented, which outline limits for approximation algorithms. The perform ..."
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Cited by 18 (1 self)
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The computational complexity of combinatorial multiple objective programming problems is investigated. INP completeness and #P completeness results are presented. Using two definitions of approximability, general results are presented, which outline limits for approximation algorithms. The performance of the well known tree and Christofides' heuristics for the TSP is investigated in the multicriteria case with respect to the two definitions of approximability. Keywords: Multicriteria Optimization, Combinatorial Optimization, INP completeness, Approximation Algorithms AMS subject classification: 90C29, 90C27 1 Introduction In this paper we will study combinatorial optimization problems with multiple criteria. Interest in multicriteria optimization (or multiple objective programming, MOP) with respect to theory and applications has been growing in recent years, as can be seen from the literature reviews in [32, 39]. The reason for this is certainly that in real world problems almost...
Archiving with Guaranteed Convergence and Diversity in MultiObjective Optimization
 In Proceedings of the Genetic and Evolutionary Computation Conference
, 2002
"... Over the past few years, the research on evolutionary algorithms has demonstrated their niche in solving multiobjective optimization problems, where the goal is to find a number of Paretooptimal solutions in a single simulation run. However, none of the multiobjective evolutionary algorithm ..."
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Cited by 17 (4 self)
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Over the past few years, the research on evolutionary algorithms has demonstrated their niche in solving multiobjective optimization problems, where the goal is to find a number of Paretooptimal solutions in a single simulation run. However, none of the multiobjective evolutionary algorithms (MOEAs) has a proof of convergence to the true Paretooptimal solutions with a wide diversity among the solutions. In this paper we discuss why a number of earlier MOEAs do not have such properties. A new archiving strategy is proposed that maintains a subset of the generated solutions. It guarantees convergence and diversity according to welldefined criteria, i.e. #dominance and #Pareto optimality.
Fast approximation schemes for multicriteria combinatorial Optimization
, 1994
"... The solution to an instance of the standard Shortest Path problem is a single shortest route in a directed graph. Suppose, however, that each arc has both a distance and a cost, and that one would like to find a route that is both short and inexpensive. In general, no single route will be both short ..."
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Cited by 8 (0 self)
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The solution to an instance of the standard Shortest Path problem is a single shortest route in a directed graph. Suppose, however, that each arc has both a distance and a cost, and that one would like to find a route that is both short and inexpensive. In general, no single route will be both shortest and cheapest; rather, the solution to an instance of this multicriteria problem will be a set of efficient or Pareto optimal routes. The (distance, cost) pairs associated with the efficient routes define an efficient frontier or tradeoff curve. An efficient set for a multicriteria problem can be exponentially large, even when the underlying singlecriterion;oblem is in P. This work therefore considers approximate solutions to rlulticriteria discrete optimization problems and investigates when they can be found quickly. This requires generalizing the notion of a fully polynomial time approximatiofi scheme to multicriteria problems. In this paper, necessary and sufficient conditions are developed for the existence of such a fast approximation scheme for a problem. Although the focus is multicriteria problems, the conditions are of interest even in the single criterion case. In addition, an appropriate form of problem reduction is introduced to facilitate the application of these conditions to a variety of problems. A companion paper uses the results of this paper to study the existence of fast approximation schemes for several interesting network flow, knapsack, and
Computing a Finite Size Representation of the Set of Approximate Solutions of an MOP
, 2008
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Generating epsilonefficient solutions in multiobjective programming
 DEPARTMENT OF MATHEMATICAL SCIENCES, CLEMSON UNIVERSITY
, 2005
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Stochastic convergence of random search to fixed size Pareto set approximations. Arxiv preprint arXiv:0711.2949
, 2007
"... This paper presents the first convergence result for random search algorithms to a subset of the Pareto set of given maximum size k with bounds on the approximation quality ǫ. The core of the algorithm is a new selection criterion based on a hypothetical multilevel grid on the objective space. It is ..."
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Cited by 2 (1 self)
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This paper presents the first convergence result for random search algorithms to a subset of the Pareto set of given maximum size k with bounds on the approximation quality ǫ. The core of the algorithm is a new selection criterion based on a hypothetical multilevel grid on the objective space. It is shown that, when using this criterion for accepting new search points, the sequence of solution archives converges with probability one to a subset of the Pareto set that ǫdominates the entire Pareto set. The obtained approximation quality ǫ is equal to the size of the grid cells on the finest level of resolution that allows an approximation with at most k points in the family of grids considered. While the convergence result is of general theoretical interest, the archiving algorithm might be of high practical value for any type iterative multiobjective optimization method, such as evolutionary algorithms or other metaheuristics, which all rely on the usage of a finite online memory to store the best solutions found so far as the current approximation of the Pareto set. 1
On local optima in multiobjective combinatorial optimization problems
, 2004
"... Abstract. In this article, local optimality in multiobjective combinatorial optimization is used as a baseline for the design and analysis of two iterative improvement algorithms. Both algorithms search in a neighborhood that is defined on a collection of sets of feasible solutions and their accepta ..."
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Cited by 2 (1 self)
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Abstract. In this article, local optimality in multiobjective combinatorial optimization is used as a baseline for the design and analysis of two iterative improvement algorithms. Both algorithms search in a neighborhood that is defined on a collection of sets of feasible solutions and their acceptance criterion is based on outperformance relations. Proofs of the soundness and completeness of these algorithms are given. 1.
Multiple Objective Minimum Cost Flow Problems: A Review
, 2005
"... In this paper, theory and algorithms for solving the multiple objective minimum cost flow problem are reviewed. For both the continuous and integer case exact and approximation algorithms are presented. In addition, a section on compromise solutions summarizes corresponding results. The reference li ..."
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In this paper, theory and algorithms for solving the multiple objective minimum cost flow problem are reviewed. For both the continuous and integer case exact and approximation algorithms are presented. In addition, a section on compromise solutions summarizes corresponding results. The reference list consists of all papers known to the authors which deal with the multiple objective minimum cost flow problem.