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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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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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Particle Swarm optimization for Constrained and multiobjective problems:A Brief Review

Particle Swarm Optimization for Constrained and Multiobjecti...

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2011 International Conference on Management and Artificial Intelligence（ICMAI 2011）

作者： Nor Azlina Ab.Aziz Mohamad Yusoff Alias Ammar W.Mohemmed Kamarulzaman Ab.Aziz Faculty of Engineering and Technology Multimedia University Malaysia Faculty of Engineering Multimedia University Malaysia Knowledge Engineering & Discovery Research Institute Auckland University of Technology New Zealand Faculty of Management Multimedia University Malaysia

Particle swarm optimization (PSO) is an optimization method that belongs to the swarm intelligence *** was initially introduced as continuous problem optimization *** has evolved to being applied to more complex multiobjective and constrained *** paper presents a systematic literature review of PSO for constrained and multiobjective optimization problems.

关键词： Constrained optimization problems multiobjective optimization problems particle swarm optimization

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Bare-Bones multiobjective Particle Swarm optimization Based on Parallel Cell Balanceable Fitness Estimation

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IEEE ACCESS 2018年 6卷 32493-32506页

作者： Qiao, Junfei Zhou, Hongbiao Yang, Cuili Beijing Univ Technol Fac Informat Technol Beijing 100124 Peoples R China Beijing Key Lab Computat Intelligence & Intellige Beijing 100124 Peoples R China

The convergence and diversity of the Pareto optimal solutions is of great importance for multiobjective evolutionary algorithms. Based on parallel cell balanceable fitness estimation (PCBFE), a novel bare-bones multiobjective particle swarm optimization (NBBMOPSO) algorithm is proposed in this paper. First, the PCBFE strategy, which is based on the parallel cell mapping approach, is developed to retain the balance between the proximity and the diversity. After that, the PCB1-E strategy is adopted to maintain external archive and update leaders. Second, an adaptive update strategy for crossover probability is designed to repair the weakness of particle search. Finally, an elitism learning strategy is performed to exchange useful information among solutions in the external archive, which can enhance the capability of dropping out of the local Pareto front. To demonstrate the merits of NBBMOPSO for multiobjective optimization, Zitzler-Deb-Thiele (ZDT) and Deb-Thiele-Laumanns-Zitzler (DTLZ) test suits are examined with comparisons against the other seven state-of-the-art competitors. Experimental results show that the proposed NBBMOPSO outperforms all the other methods in terms of the chosen performance metrics.

关键词： multiobjective optimization problems bare-bones particle swarm optimization parallel cell balanceable fitness estimation adaptive crossover probability elitism learning strateg

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A note on approximate Karush-Kuhn-Tucker conditions in locally Lipschitz multiobjective optimization

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optimization LETTERS 2019年第1期13卷 163-174页

作者： Nguyen Van Tuyen Yao, Jen-Chih Wen, Ching-Feng Univ Elect Sci & Technol China Sch Math Sci Chengdu Sichuan Peoples R China Hanoi Pedag Univ 2 Dept Math Phuc Yen Vinh Phuc Vietnam China Med Univ Ctr Gen Educ Taichung 40402 Taiwan Kaohsiung Med Univ Ctr Fundamental Sci Kaohsiung 80708 Taiwan Kaohsiung Med Univ Res Ctr Nonlinear Anal & Optimizat Kaohsiung 80708 Taiwan Kaohsiung Med Univ Hosp Dept Med Res Kaohsiung 80708 Taiwan

In the recent paper of Giorgi et al. (J Optim Theory Appl 171:70-89, 2016), the authors introduced the so-called approximate Karush-Kuhn-Tucker (AKKT) condition for smooth multiobjective optimization problems and obtained some AKKT-type necessary optimality conditions and sufficient optimality conditions for weak efficient solutions of such a problem. In this note, we extend these optimality conditions to locally Lipschitz multiobjective optimization problems using Mordukhovich subdifferentials. Furthermore, we prove that, under some suitable additional conditions, an AKKT condition is also a KKT one.

关键词： Approximate optimality conditions multiobjective optimization problems Locally Lipschitz functions Mordukhovich subdifferential

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Optimistic NAUTILUS navigator for multiobjective optimization with costly function evaluations

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JOURNAL OF GLOBAL optimization 2022年第4期83卷 865-889页

作者： Saini, Bhupinder Singh Emmerich, Michael Mazumdar, Atanu Afsar, Bekir Shavazipour, Babooshka Miettinen, Kaisa Univ Jyvaskyla Fac Informat Technol POB 35 Agora FI-40014 Jyvaskyla Finland Leiden Inst Adv Comp Sci Fac Sci Niels Bohrweg 1 NL-2333 CA Leiden Netherlands

We introduce novel concepts to solve multiobjective optimization problems involving (computationally) expensive function evaluations and propose a new interactive method called O-NAUTILUS. It combines ideas of trade-off free search and navigation (where a decision maker sees changes in objective function values in real time) and extends the NAUTILUS Navigator method to surrogate-assisted optimization. Importantly, it utilizes uncertainty quantification from surrogate models like Kriging or properties like Lipschitz continuity to approximate a so-called optimistic Pareto optimal set. This enables the decision maker to search in unexplored parts of the Pareto optimal set and requires a small amount of expensive function evaluations. We share the implementation of O-NAUTILUS as open source code. Thanks to its graphical user interface, a decision maker can see in real time how the preferences provided affect the direction of the search. We demonstrate the potential and benefits of O-NAUTILUS with a problem related to the design of vehicles.

关键词： Interactive methods multiobjective optimization problems Decision makers Preference information Computational cost Kriging

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A decomposition-based multiobjective evolutionary algorithm with weights updated adaptively

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INFORMATION SCIENCES 2021年 572卷 343-377页

作者： Liu, Yuan Hu, Yikun Zhu, Ningbo Li, Kenli Zou, Juan Li, Miqing Hunan Univ Coll Comp Sci & Elect Engn Changsha Hunan Peoples R China Xiangtan Univ Sch Comp Sci Xiangtan Hunan Peoples R China Univ Birmingham Sch Comp Sci CERCIA Birmingham B15 2TT W Midlands England

Recently, decomposition-based multiobjective evolutionary algorithms (DMEAs) have become more prevalent than other patterns (e.g., Pareto-based algorithms and indicator based algorithms) for solving multiobjective optimization problems (MOPs). They utilize a scalarizing method to decompose an MOP into several subproblems based on the weights provided, resulting in the performances of the algorithms being highly dependent on the uniformity between the problem's optimal Pareto front and the distribution of the specified weights. However, weight generation is generally based on a simplex lattice design, which is suitable for "regular" Pareto fronts (i.e., simplex-like fronts) but not for other "irregular" Pareto fronts. To improve the efficiency of this type of algorithm, we develop a DMEA with weights updated adaptively (named DMEA-WUA) for the problems regarding various Pareto fronts. Specifically,the DMEA-WUA introduces a novel exploration versus exploitation model for environmental *** exploration process finds appropriate weights for a given problem in four steps: weight generation, weight deletion, weight addition and weight replacement. Exploitation means using these weights from the exploration step to guide the evolution of the population. Moreover, exploration is carried out when the exploitation process is stagnant;this is different from the existing method of periodically updating weights. Experimental results show that our algorithm is suitable for solving problems with various Pareto fronts, including those with "regular" and "irregular" shapes. (c) 2021 Elsevier Inc. All rights reserved.

关键词： multiobjective optimization problems Exploration Exploitation Weights updated adaptively The decomposition-based multiobjective evolutionary algorithm

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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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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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A decomposition-based binary ACO algorithm for the multiobjective UBQP

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NEUROCOMPUTING 2017年 246卷 58-68页

作者： Zangari, Murilo Pozo, Aurora Santana, Roberto Mendiburu, Alexander Univ Fed Parana UFPR Comp Sci Dept PO 19081 BR-81531970 Curitiba Parana Brazil Univ Basque Country UPV EHU Dept Comp Sci & Artificial Intelligence Intelligent Syst Grp Paseo Manuel Lardizabal 1 San Sebastian 20080 Guipuzcoa Spain Univ Basque Country UPV EHU Dept Comp Architecture & Technol Intelligent Syst Grp Paseo Manuel Lardizabal 1 San Sebastian 20080 Guipuzcoa Spain

The multiobjective unconstrained binary quadratic programming (mUBQP) is a combinatorial optimization problem which is able to represent several multiobjective optimization problems (MOPs). The problem can be characterized by the number of variables, the number of objectives and the objective correlation strength. multiobjective evolutionary algorithms (MOEAs) are known as an efficient technique for solving MOPs. Moreover, several recent studies have shown the effectiveness of the MOEA/D framework applied to different MOPs. Previously, we have presented a preliminary study on an algorithm based on MOEA/D framework and the bio-inspired metaheuristic called binary ant colony optimization (BACO). The metaheuristic uses a positive feedback mechanism according to the best solutions found so far to update a probabilistic model which maintains the learned information. This paper presents the improved MOEA/D-BACO framework for solving the mUBQP. The components (i) mutation-like effect, and (ii) diversity preserving method are incorporated into the framework to enhance its search ability avoiding the premature convergence of the model and consequently maintaining a more diverse population of solutions. Experimental studies were conducted on a set of mUBQP instances. The results have shown that the proposed MOEA/D-BACO has outperformed MOEA/D, which uses genetic operators, in most of the test instances.. Moreover, the algorithm has produced competitive results in comparison to the best approximated Pareto fronts from the literature. (C) 2017 Elsevier B.V. All rights reserved.

关键词： multiobjective optimization problems Probabilistic modeling Binary Ant Colony optimization MOEA/D mUBQP

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