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Structure and Weak Sharp Minimum of the Pareto Solution Set for Piecewise Linear multiobjective optimization

JOURNAL OF optimization THEORY AND APPLICATIONS 2010年第1期147卷 113-124页

作者： Yang, X. Q. Yen, N. D. Hong Kong Polytech Univ Dept Appl Math Hong Kong Hong Kong Peoples R China Vietnamese Acad Sci & Technol Inst Math Hanoi Vietnam

In this paper, the Pareto solution set of a piecewise linear multiobjective optimization problem in a normed space is shown to be the union of finitely many semiclosed polyhedra. If the problem is further assumed to b... 详细信息

关键词： Piecewise linear functions multiobjective optimization problems Pareto solution sets Global weak sharp minimum Image space analysis

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A Multi-Objective Evolutionary Concept Learner

A Multi-Objective Evolutionary Concept Learner

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2010 IEEE World Congress on Computational Intelligence

作者： Dandois, Celine Divina, Federico Vanhoof, Wim Univ Namur Fac Comp Sci Rue Grandgagnage 21 B-5000 Namur Belgium Pablo Olavide Univ BIGS E-41013 Seville Spain

ISBN: (纸本)9781424481262

Learning concept descriptions from data is a challenging, and inherently multi-objective, optimization problem. The model induced by the learner has to be complete, consistent and easily interpretable, and producing it should take few computational resources. These objectives are often conflicting and require thus some heuristics to be balanced. The classical approach is to combine all the objectives into a single scoring function that is used for guiding the search in the hypothesis space. However, we believe that a multi-objective approach is more appropriate for this problem. Evolutionary Algorithms (EAs) are particularly suited for solving multi-objective optimization problems. In this paper, we propose an improved version of the Evolutionary Concept Learner (ECL) system, called Multi-Objective ECL (MOECL). While the search performed by ECL is based on a single-objective EA, MOECL adopts a multi-objective search strategy. It uses two objectives independent of class distribution, the true positive rate and the true negative rate, and is inspired by the simple and effective Pareto-based optimization algorithm SPEA2 for fitness assignement. Experiments show that the results of the MOECL system are globally more accurate than those of ECL and of the other state-of-the-art systems, on propositional datasets and on one relational dataset. As such, we believe the MOECL system to constitute a promising approach towards a multi-objective and effective concept learner.

关键词： Pareto optimisation Pareto-based optimization algorithm evolutionary algorithms evolutionary computation fitness assignment learning (artificial intelligence) multiobjective evolutionary concept learner multiobjective optimization problems multiobjective search strategy scoring function search problems

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Automated Discovery of Vital Knowledge from Pareto-optimal Solutions: First Results from Engineering Design

Automated Discovery of Vital Knowledge from Pareto-optimal S...

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2010 IEEE World Congress on Computational Intelligence

作者： Bandaru, Sunith Deb, Kalyanmoy Indian Inst Technol Dept Mech Engn Kanpur 208016 Uttar Pradesh India

ISBN: (纸本)9781424481262

Real world multi-objective optimization problems are often solved with the only intention of selecting a single trade-off solution by taking up a decision-making task. The computational effort and time spent on obtaining the entire Pareto front is thus not justifiable. The Pareto solutions as a whole contain within them a lot more information than that is used. Extracting this knowledge would not only give designers a better understanding of the system, but also bring worth to the resources spent. The obtained knowledge acts as governing principles which can help solve other similar systems easily. We propose a genetic algorithm based unsupervised approach for learning these principles from the Pareto-optimal dataset of the base problem. The methodology is capable of discovering analytical relationships of a certain type between different problem entities.

关键词： Pareto front Pareto optimisation Pareto-optimal solutions data mining genetic algorithm genetic algorithms knowledge discovery multiobjective optimization problems unsupervised learning unsupervised learning approach

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Many-Objective Test problems to Visually Examine the Behavior of multiobjective Evolution in a Decision Space

Many-Objective Test Problems to Visually Examine the Behavio...

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11th International Conference on Parallel Problem Solving from Nature

作者： Ishibuchi, Hisao Hitotsuyanagi, Yasuhiro Tsukamoto, Noritaka Nojima, Yusuke Osaka Prefecture Univ Dept Comp Sci & Intelligent Syst Grad Sch Engn Naka Ku Osaka 5998531 Japan

ISBN: (纸本)9783642158704

Many-objective optimization is a hot issue in the EMO (evolutionary multiobjective optimization) community. Since almost all solutions in the current population are non-dominated with each other in many-objective EMO algorithms, we may need a different fitness evaluation scheme from the case of two and three objectives. One difficulty in the design of many-objective EMO algorithms is that we cannot visually observe the behavior of multiobjective evolution in the objective space with four or more objectives. In this paper, we propose the use of many-objective test problems in a two- or three-dimensional decision space to visually examine the behavior of multiobjective evolution. Such a visual examination helps us to understand the characteristic features of EMO algorithms for many-objective optimization. Good understanding of existing EMO algorithms may facilitates their modification and the development of new EMO algorithms for many-objective optimization.

关键词： Evolutionary multiobjective optimization (EMO) many-objective optimization multiobjective optimization problems test problems

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Generalized (F,ρ)-convexity and duality in nonsmooth problems of multiobjective optimization

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JOURNAL OF optimization THEORY AND APPLICATIONS 2008年第1期136卷 61-68页

作者： Nobakhtian, S. Univ Isfahan Dept Math Esfahan Iran

A mixed-type dual for a nonsmooth multiobjective optimization problem with inequality and equality constraints is formulated. We obtain weak and strong duality theorems for a mixed-type dual without requiring the regularity assumptions and the nonnegativeness of the Lagrange multipliers associated to the equality constraints. We apply also a nonsmooth constraint qualification for multiobjective programming to establish strong duality results. In this case, our constraint qualification assures the existence of positive Lagrange multipliers associated with the vector-valued objective function.

关键词： multiobjective optimization problems efficient solutions generalized (F,rho)-convexity duality nonsmooth analysis

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Computational Approach to Essential and Nonessential Objective Functions in Linear Multicriteria optimization

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JOURNAL OF optimization THEORY AND APPLICATIONS 2008年第3期139卷 577-590页

作者： Malinowska, A. B. Torres, D. F. M. Bialystok Tech Univ Fac Comp Sci Bialystok Poland Univ Aveiro Dept Math P-3800 Aveiro Portugal

The question of obtaining well-defined criteria for multiple-criteria decision making problems is well known. One of the approaches dealing with this question is the concept of nonessential objective functions. A certain objective function is called nonessential if the set of efficient solutions is the same with or without that objective function. We present two methods for determining nonessential objective functions. A computational implementation is done using a computer algebra system.

关键词： multiobjective optimization problems Efficient (Pareto optimal) solutions Essential nonessential objective functions

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Changes of the solution set by adding objectives to multicriteria optimization problems

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8th International Conference on Fuzzy Logic and Intelligent Technologies in Nuclear Science

作者： Malinowska, A. B. Bialystok Tech Univ Fac Comp Sci PL-15351 Bialystok Poland

ISBN: (纸本)9789812799463

In this work we are interested in characterizing objective functions which do not change the set of efficient solutions (weakly efficient solutions, properly efficient solutions). Necessary and sufficient conditions for an objective function to be nonessential (weakly nonessential, properly nonessential) are presented. We establish relations between: weakly nonessential, properly nonessential and nonessential functions.

关键词： multiobjective optimization problems efficient (Pareto optimal) solutions essential/nonessential objective functions

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Fuzzy set based multiobjective allocation of resources and its applications

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2006年第1-2期52卷 197-210页

作者： Ekel, P. Ya. Martins, C. A. P. S. Pereira, J. G., Jr. Palhares, R. M. Canha, L. N. Pontif Catholic Univ Minas Gerais Grad Program Elect Engn BR-30535610 Belo Horizonte MG Brazil Univ Fed Minas Gerais Dept Elect Engn BR-31270010 Belo Horizonte MG Brazil Univ Fed Santa Maria Grad Program Elect Engn BR-97105900 Santa Maria RS Brazil

This paper presents results of research into the use of the Bellman-Zadeh approach to decision making in a fuzzy environment for solving multiobjective optimization problems. Its application conforms to the principle of guaranteed result and provides constructive lines in obtaining harmonious solutions on the basis of analyzing associated maxmin problems. The use of the Bellman-Zadeh approach has served as a basis for solving a problem of multiobjective allocation of resources (or their shortages) and developing a corresponding adaptive interactive decision-making system (AIDMS1). Its calculating kernel permits one to solve maxmin problems using an algorithm based on a nonlocal search (modification of the Gelfand's and Tsetlin's "long valley" method). The AIDMS1 includes procedures for considering linguistic variables to reflect conditions that are difficult to formalize as well as procedures for constructing and correcting vectors of importance factors for goals. The use of these procedures permits one to realize an adaptive approach to processing information of a decision maker to provide successive improving of the solution quality. C++ windows of the AIDMS1 are presented for input, output, and special possibilities related to considering linguistic variables and constructing and correcting vectors of importance factors. The results of the paper are universally applicable and are already being used to solve power engineering problems. (c) 2006 Elsevier Ltd. All rights reserved.

关键词： multiobjective optimization problems Bellman-Zadeh approach Maximin problems allocation of resources

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A path following method for box-constrained multiobjective optimization with applications to goal programming problems

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MATHEMATICAL METHODS OF OPERATIONS RESEARCH 2003年第1期58卷 69-85页

作者： Recchioni, MC Univ Ancona Ist Teoria Decis & Finanza Innovat I-60121 Ancona Italy

We propose a path following method to find the Pareto optimal solutions of a box-constrained multiobjective optimization problem. Under the assumption that the objective functions are Lipschitz continuously differentiable we prove some necessary conditions for Pareto optimal points and we give a necessary condition for the existence of a feasible point that minimizes all given objective functions at once. We develop a method that looks for the Pareto optimal points as limit points of the trajectories solutions of suitable initial value problems for a system of ordinary differential equations. These trajectories belong to the feasible region and their computation is well suited for a parallel implementation. Moreover the method does not use any scalarization of the multiobjective optimization problem and does not require any ordering information for the components of the vector objective function. We show a numerical experience on some test problems and we apply the method to solve a goal programming problem.

关键词： multiobjective optimization problems Pareto-optimal fronts path following methods dynamical systems goal programming

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Optimality conditions in directionally differentiable Pareto problems with a set constraint via tangent cones

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NUMERICAL FUNCTIONAL ANALYSIS AND optimization 2003年第5-6期24卷 557-574页

作者： Jiménez, B Novo, V Univ Salamanca Dept Econ & Hist Econ Fac Econ & Empresa Salamanca 37007 Spain Univ Nacl Educ Distancia ETSI Ind Dept Matemat Aplicada Madrid Spain

We study a multiobjective optimization problem in R-n with a feasible set defined by inequality and equality constraints and a convex set constraint. All the involved functions are, at least, directionally differentiable. Quasiconvex inequalities are considered. Given these assumptions, an expression for the contingent cone to the feasible set using an extended Mangasarian-Fromovitz constraint qualification is provided. As application, necessary and sufficient optimality conditions of Kuhn-Tucker type are established for a local Pareto minimal point.

关键词： multiobjective optimization problems tangent cone optimality conditions for a Pareto minimum Lagrange multipliers quasiconvexity dini differentiable functions Hadamard differentiable functions

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