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Vector network equilibrium problems and nonlinear scalarization methods

MATHEMATICAL methods OF OPERATIONS RESEARCH 1999年第2期49卷 239-253页

作者： Chen, GY Goh, CJ Yang, XQ Hong Kong Polytech Univ Dept Appl Math Kowloon Peoples R China Univ Western Australia Dept Math Nedlands WA 6907 Australia Acad Sinica Inst Syst Sci Beijing 100080 Peoples R China

The conventional equilibrium problem found in many economics and network models is based on a scalar cost, or a single objective. Recently, equilibrium problems based on a vector cost, or multicriteria, have received considerable attention. In this paper, we study a scalarization method for analyzing network equilibrium problems with vector-valued cost function. The method is based on a strictly monotone function originally proposed by Gerstewitz. Conditions that are both necessary and sufficient for weak vector equilibrium are derived, with the prominent feature that no convexity assumptions are needed, in contrast to other existing scalarization methods.

关键词： multicriteria network equilibrium scalarization methods strictly monotone functions variational inequalities

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Pareto-optimal security strategies as minimax strategies of a standard matrix game

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1999年第1期102卷 203-210页

作者： Voorneveld, M Tilburg Univ Dept Econometr & Ctr NL-5000 LE Tilburg Netherlands

We prove that, in a multicriteria matrix game, a strategy is a Pareto-optimal security strategy if and only if it is a minimax strategy of a standard matrix game that is obtained by a suitable weighting of the criteria.

关键词： multicriteria games matrix games scalarization methods Pareto-optimal security strategies

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A NECESSARY AND SUFFICIENT CONDITION FOR PARETO-OPTIMAL SECURITY STRATEGIES IN MULTICRITERIA MATRIX GAMES

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1991年第3期68卷 463-481页

作者： GHOSE, D INDIAN INST SCI DEPT ELECT ENGN BANGALORE 560012 KARNATAKA INDIA

In this paper, a scalar game is derived from a zero-sum multicriteria matrix game, and it is proved that the solution of the new game with strictly positive scalarization is a necessary and sufficient condition for a strategy to be a Pareto-optimal security strategy (POSS) for one of the players in the original game. This is done by proving that a certain set, which is the extension of the set of security level vectors in the criterion function space, is convex and polyhedral. It is also established that only a finite number of scalarizations are necessary to obtain all the POSS for a player. An example is included to illustrate the main steps in the proof.

关键词： GAME THEORY MULTICRITERIA GAMES GAMES WITH VECTOR PAYOFFS PARETO-OPTIMAL SECURITY STRATEGIES MULTICRITERIA OPTIMIZATION scalarization methods

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scalarization and Kuhn-Tucker-like conditions for weak vector generalized quasivariational inequalities

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2006年第2期130卷 309-316页

作者： Morgan, J. Romaniello, M. Univ Naples Federico II Dipartimento Matemat & Stat Naples Italy Univ Calabria Dipartimento Sci Aziendali I-87036 Cosenza Italy

We consider a weak vector generalized quasivariational inequality. By introducing a method of scalarization which does not require any assumption on the data and by using previous results of the authors concerning scalar generalized quasivariational inequalities, we present Kuhn-Tucker-like conditions for this problem in the case in which the set-valued operator of the constraints is defined by a finite number of inequalities.

关键词： weak vector generalized quasivariational inequalities set-valued operators vector optimization scalarization methods generalized Kuhn-Tucker conditions

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Vector linear programming in zero-sum multicriteria matrix games

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1996年第1期89卷 115-127页

作者： Fernandez, FR Puerto, J Departmento de Estadística e Investigación Operativa Universidad de Sevilla Sevilla España

In this paper, a multiple-objective linear problem is derived from a zero-sum multicriteria matrix game. It is shown that the set of efficient solutions of this problem coincides with the set of Pareto-optimal security strategies (POSS) for one of the players in the original game. This approach emphasizes the existing similarities between the scalar and multicriteria matrix games, because in both cases linear programming can be used to solve the problems. It also leads to different scalarizations which are alternative ways to obtain the set of all POSS. The concept of ideal strategy for a player is introduced, and it is established that a pair of Pareto saddle-point strategies exists if both players have ideal strategies. Several examples are included to illustrate the results in the paper.

关键词： game theory multicriteria games Pareto-optimal security strategies vector linear programming scalarization methods

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Self-adaptive memetic algorithms for multi-objective single machine learning-effect scheduling problems with release times

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FLEXIBLE SERVICES AND MANUFACTURING JOURNAL 2022年第3期34卷 748-784页

作者： Deliktas, Derya Kutahya Dumlupynar Univ Fac Engn Dept Ind Engn Kutahya Turkey Univ Nottingham Sch Comp Sci Computat Optimisat & Learning COL Lab Nottingham NG8 1BB England

This paper proposes a single machine scheduling problem with learning-effect and release times by considering two objectives requiring minimization of makespan and total tardiness, simultaneously. Due to the NP-hardness of this problem, two memetic algorithms with meme variants are presented for solving the bi-objective problem and applied by combining three different scalarization methods, including weighted sum, conic, and tchebycheff. The performance of all memetic algorithms with the meme is investigated across randomly generated twenty-seven test problems ranging from 'small' to 'large' size. The experimental results indicate that the Multimeme Memetic Algorithm using the tchebycheff outperforms the other algorithms producing the best-known results for almost all bi-objective single machine scheduling instances with learning-effects. All algorithms perform effectively in solving large-sized problems with up to 200 jobs.

关键词： Memetic algorithms Hyper-heuristics Single machine scheduling scalarization methods Learning effect

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Pareto solutions in multicriteria optimization under uncertainty

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2020年第2期281卷 357-368页

作者： Engau, Alexander Sigler, Devon Univ Lancaster Management Sch Lancaster LA1 4YX England Dalhousie Univ Rowe Sch Business Kenneth C Rowe Management Bldg6100 Univ Ave Halifax NS B3H 4R2 Canada Univ Colorado Dept Math & Stat Sci Campus Box 170POB 173364 Denver CO 80217 USA Natl Renewable Energy Lab 15013 Denver West Pkwy Golden CO 80401 USA

We present and analyze several definitions of Pareto optimality for multicriteria optimization or decision problems with uncertainty primarily in their objective function values. In comparison to related notions of Pareto robustness, we first provide a full characterization of an alternative efficient set hierarchy that is based on six different ordering relations both with respect to the multiple objectives and a possibly finite, countably infinite or uncountable number of scenarios. We then establish several scalarization results for the generation of the corresponding efficient points using generalized weighted-sum and epsilon-constraint techniques. Finally, we leverage these scalarization results to also derive more general conditions for the existence of efficient points in each of the corresponding optimality classes, under suitable assumptions. (C) 2019 Elsevier B.V. All rights reserved.

关键词： Multiple objective programming Pareto solutions Optimization under uncertainty Efficient set hierarchy scalarization methods

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Regularized gap functions and error bounds for generalized mixed strong vector quasiequilibrium problems

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COMPUTATIONAL & APPLIED MATHEMATICS 2018年第5期37卷 5935-5950页

作者： Lam Quoc Anh Nguyen Van Hung Vo Minh Tam Can Tho Univ Dept Math Teacher Coll Can Tho Vietnam Ton Duc Thang Univ Dept Management Sci & Technol Dev Ho Chi Minh City Vietnam Ton Duc Thang Univ Fac Math & Stat Ho Chi Minh City Vietnam Dong Thap Univ Dept Math 783 Pham Huu Lau StWard 6 Cao Lanh City Vietnam

In this paper, we consider generalized mixed strong vector quasiequilibrium problems in infinite-dimensional spaces. Using the nonlinear scalarization method, we first propose gap functions for such problems. Then, based on the Moreau-Yosida regularization of these functions, we introduce regularized gap functions for the reference problems. Next, we study sufficient conditions for the mentioned functions to be continuous. Finally, error bounds for the concerning problems in terms of regularized gap functions are investigated. Some examples are provided to illustrate our results.

关键词： Strong vector quasiequilibrium Strong monotonicity Error bounds Regularized gap function scalarization methods 49J53 58E35 90C25 90C33

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Multi-objective genetic algorithm based on the fuzzy MULTIMOORA method for solving the cardinality constrained portfolio optimization

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APPLIED INTELLIGENCE 2023年第12期53卷 14717-14743页

作者： Deliktas, Derya Ustun, Ozden Kutahya Dumlupinar Univ Fac Engn Dept Ind Engn Kutahya Turkey

Over the recent decades, Markowitz mean-variance models with additional constraints and objectives have still attracted the attention of many researchers. Stock evaluation and portfolio optimization have a decision-making process that contains tangible and intangible factors under vague parameters. This study presents an integrated approach including the fuzzy MULTIMOORA based on Correlation Coefficient and Standard Deviation (CCSD) method and the mean-variance-ranking cardinality constrained portfolio optimization (MVRCCPO), which is the extension of classical mean-variance cardinality constrained portfolio optimization model. The ranking obtained from the fuzzy MULTIMOORA based on the CCSD method is handled as one of the objective functions in the proposed model. The MVRCCPO problem is an NP-Complete problem when taking cardinality constraints into account. In this case, existing exact algorithms such as quadratic programming may not be efficient for solving the problem. Due to the complexity of the problem and the nature of the multiple objectives, we used the multi-objective genetic algorithm based on three different scalarization techniques, involving conic, tchebycheff, and weighted-sum. The performance of the multi-objective genetic algorithm with scalarization variants is investigated across forty-nine benchmark instances consisting of seven different objective weight combinations and seven different instances from 'small' to 'large' size derived from a large real-life problem. The computational results indicate that the multi-objective genetic algorithm utilizing weighted-sum in accordance with the weighted distance to the ideal point outperforms the rest in point of the obtained results in most cases. Moreover, the multi-objective genetic algorithm using conic in accordance with the weighted distance to the reference point is better performance than the multi-objective genetic algorithm using tchebycheff.

关键词： Portfolio optimization Fuzzy MULTIMOORA scalarization methods Markowitz mean-variance model Cardinality constraints

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Efficient scalarization in multiobjective optimal control of a nonsmooth PDE

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2022年第2期83卷 435-464页

作者： Bernreuther, Marco Mueller, Georg Volkwein, Stefan Univ Konstanz Dept Math & Stat D-78457 Constance Germany Heidelberg Univ Interdisciplinary Ctr Sci Comp IWR D-69120 Heidelberg Germany

This work deals with the efficient numerical characterization of Pareto stationary fronts for multiobjective optimal control problems with a moderate number of cost functionals and a mildly nonsmooth, elliptic, semilinear PDE-constraint. When "ample" controls are considered, strong stationarity conditions that can be used to numerically characterize the Pareto stationary fronts are known for our problem. We show that for finite dimensional controls, a sufficient adjoint-based stationarity system remains obtainable. It turns out that these stationarity conditions remain useful when numerically characterizing the fronts, because they correspond to strong stationarity systems for problems obtained by application of weighted-sum and reference point techniques to the multiobjective problem. We compare the performance of both scalarization techniques using quantifiable measures for the approximation quality. The subproblems of either method are solved with a line-search globalized pseudo-semismooth Newton method that appears to remove the degenerate behavior of the local version of the method employed previously. We apply a matrix-free, iterative approach to deal with the memory and complexity requirements when solving the subproblems of the reference point method and compare several preconditioning approaches.

关键词： Multiobjective optimal control Nonsmooth optimization Stationarity conditions Pareto optimality scalarization methods Pseudo-semismooth Newton method

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