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Evolutionary multiobjective optimization assisted by scalarization function approximation for high-dimensional expensive problems

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SWARM AND EVOLUTIONARY COMPUTATION 2024年 86卷

作者： Horaguchi, Yuma Nishihara, Kei Nakata, Masaya Yokohama Natl Univ Fac Engn Tokiwadai 79-5 Yokohama Kanagawa 2408501 Japan

Surrogate -assisted evolutionary algorithms (SAEAs) are a promising approach for solving expensive multiobjective optimization problems, but they often cannot address high -dimensional problems. Although one common approach to overcoming this challenge is to construct reliable surrogates, their accuracy inevitably deteriorates in a high -dimensional search space. Thus, this paper presents a novel SAEA based on scalarization function approximation, which is designed to strengthen its robustness against this deterioration. The proposed algorithm constructs an approximation model for each scalarization function defined in a decompositionbased framework. Each decomposed problem is then solved using multiple independent models trained for its neighbor problems. The intent is to decrease the risk of search performance degradations caused by unreliable approximations and retain the redundancy of the surrogate -assisted search to hedge the risk of over -fitting. Furthermore, each approximation model is adapted to a promising region of its corresponding decomposed problem to reduce the complexity of model fitting given a limited number of training samples. Experimental results show that the proposed algorithm is competitive with state-of-the-art SAEAs adapted for high -dimensional problems.

关键词： Surrogate-assisted evolutionary algorithm Multiobjective optimization scalarization function

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Lipschitz properties of the scalarization function and applications

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OPTIMIZATION 2010年第2期59卷 305-319页

作者： Tammer, Christiane Zalinescu, Constantin Univ Halle Wittenberg Inst Math Halle Saale Germany Alexandru Ioan Cuza Univ Fac Math Iasi Romania Acad Romana Octav Mayer Inst Math Iasi Romania

The scalarization functions were used in vector optimization for a long period. Similar functions were introduced and used in economics under the name of shortage function or in mathematical finance under the name of (convex or coherent) measures of risk. The main aim of this article is to study Lipschitz continuity properties of such functions and to give some applications for deriving necessary optimality conditions for vector optimization problems using the Mordukhovich subdifferential.

关键词： scalarization function Lipschitz continuity epi-Lipschitz set vector optimization

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Data transmission optimization in edge computing using multi-objective reinforcement learning

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JOURNAL OF SUPERCOMPUTING 2024年第14期80卷 21179-21206页

作者： Li, Xiaole Liu, Haitao Wang, Haifeng Linyi Univ Sch Informat Sci & Engn Linyi 276000 Shandong Peoples R China Shandong Univ Sch Software Jinan 250000 Shandong Peoples R China

Reducing network energy consumption and balancing workload are two key optimization goals for data transmission in edge computing field. However, these two goals are likely to be conflicting in some cases and fail to achieve the optimum simultaneously. In this paper, we design a new data transmission optimization algorithm using multi-objective reinforcement learning. We design the vector of rewards for the two objectives, and update Pareto approximate set by multiple state steps to approach the optimal solution. In every step, we classify the candidate links into four different levels for path selection. We aggregate network traffic to construct minimum topology subset, minimizing the number of occupied device to reduce energy consumption. We optimize the load distribution on those selected links, minimizing maximum congestion factor to balance workload. For action selection, we leverage roulette-based Chebyshev scalarization function to solve the weight selection problem for multi-objectives and enforce exploration to avoid falling into local optimum. To improve the convergence rate, we design heuristic factor to control the search of solution space and enhance the guiding effect of the existing optimal solution. Simulation result shows that the proposed algorithm achieves good performance in energy-saving and load balance at the same time.

关键词： Energy-saving Load balance Multi-objective reinforcement learning scalarization function Edge computing

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Connectedness of solution sets for generalized vector equilibrium problems via free-disposal sets in complete metric space

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OPTIMIZATION LETTERS 2022年第7期16卷 2073-2093页

作者： Shao, Chong-Yang Peng, Zai-Yun Xiao, Yi-Bin Zhao, Yong Chongqing JiaoTong Univ Coll Math & Stat Chongqing 400074 Peoples R China Univ Elect Sci & Technol China Sch Math Sci Chengdu 610000 Peoples R China

In this paper, the connectedness of solution sets for generalized vector equilibrium problems via free-disposal sets (GVEPVF) in complete metric space is discussed. Firstly, by virtue of Gerstewitz scalarization functions and oriented distance functions, a new scalarization function omega is constructed and some properties of it are given. Secondly, with the help of omega, the existence of solutions for scalarization problems (GVEPVF)(omega) and the relationship between the solution sets of (GVEPVF)(omega) and (GVEPVF) are obtained. Then, under some suitable assumptions, sufficient conditions of (path) connectedness of solution sets for (GVEPVF) are established. Finally, as an application, the connectedness results of E-efficient solution set for a class of vector programming problems are derived. The obtained results are new, and some examples are given to illustrate the main results.

关键词： Generalized vector equilibrium problems Solution sets scalarization function Connectedness Free-disposal set

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Lipschitz properties of nonsmooth functions and set-valued mappings via generalized differentiation

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NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2013年 89卷 110-120页

作者： Nguyen Mau Nam Lafferriere, Gerardo Portland State Univ Faribom Maseeh Dept Math & Stat Portland OR 97207 USA

In this paper, we revisit the Mordukhovich subdifferential criterion for Lipschitz continuity of nonsmooth functions and the coderivative criterion for the Aubin/Lipschitz-like property of set-valued mappings in finite dimensions. The criteria are useful and beautiful results in modern variational analysis showing the state of the art of the field. As an application, we establish necessary and sufficient conditions for Lipschitz continuity of the minimal time function and the scalarization function, which play an important role in many aspects of nonsmooth analysis and optimization. (C) 2013 Elsevier Ltd. All rights reserved.

关键词： Lipschitz property Generalized differentiation Subdifferential Coderivative Minimal time function scalarization function

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On Generalized Quasi-Vector Equilibrium Problems via scalarization Method

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2016年第2期168卷 584-599页

作者： Farajzadeh, Ali Lee, Byung Soo Plubteing, Somyot Razi Univ Dept Math Kermanshah 67149 Iran Kyungsung Univ Dept Math Busan 608736 South Korea Naresuan Univ Fac Sci Dept Math Phitsanulok 65000 Thailand

In this paper, we consider the nonlinear scalarization function in the setting of topological vector spaces and present some properties of it. Moreover, using the nonlinear scalarization function and Fan-Glicksberg-Kakutani's fixed point theorem, we obtain an existence result of a solution for a generalized vector quasi-equilibrium problem without using any monotonicity and upper semi-continuity (or continuity) on the given maps. Our result can be considered as an improvement of the known corresponding result. After that, we introduce a system of generalized vector quasi-equilibrium problem which contains Nash equilibrium and Debreu-type equilibrium problem as well as the system of vector equilibrium problem posed previously. We provide two existence theorems for a solution of a system of generalized vector quasi-equilibrium problem. In the first one, our multi-valued maps have closed graphs and the maps are continuous, while in the second one, we do not use any continuity on the maps. Moreover, the method used for the existence theorem of a solution of a system of generalized vector quasi-equilibrium problem is not based upon a maximal element theorem. Finally, as an application, we apply the main results to study a system of vector optimization problem and vector variational inequality problem.

关键词： scalarization function Generalized vector quasi-equilibrium System of generalized vector quasi-equilibrium Topological vector space

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On scalarization and well-posedness in set optimization with a partial set order relation

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POSITIVITY 2024年第1期28卷 1-15页

作者： Gupta, Sakshi Gupta, Rekha Srivastava, Manjari Univ Delhi Dept Math Delhi 110007 India Univ Delhi Dept Math Miranda House Delhi 110007 India

In this paper, a new scalarization function is introduced with respect to a partial set order relation established by Karaman et al. (Positivity 22(3):783-802, 2018). A few properties of this function are studied. scalarization results and some characterizations for set of minimal and weak minimal solutions of a set optimization problem (SOP) in terms of the optimal solution set of the scalar optimization problem (P) are obtained using the newly defined scalarization function. Further, two types of well-posedness for (SOP) are introduced. Equivalence between the well-posedness of (SOP) with (P) is established and a few necessary conditions are obtained for the two well-posedness defined above.

关键词： Set optimization Partial set order relation scalarization function Well-posedness

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EKELAND'S VARIATIONAL PRINCIPLE WITH A scalarization TYPE WEIGHTED SET ORDER RELATION

JOURNAL OF NONLINEAR AND VARIATIONAL ANALYSIS

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JOURNAL OF NONLINEAR AND VARIATIONAL ANALYSIS 2023年第3期7卷 381-396页

作者： Han, Wenyan Yu, Guolin Ningxia Univ Sch Math & Stat Yinchuan 750021 Peoples R China North Minzu Univ Sch Math & Informat Sci Yinchuan 750021 Peoples R China

In this paper, a weighted set order relation given by the oriented distance function is introduced, which does not need any convexity and is applicable even the ordering cone has an empty interior. The Ekeland's variational principle, Caristi's fixed point theorem, and Takahashi's minimization theorem associated with the introduced weighted set order relation are constructed, and the equivalences among them are deduced. As an application, the existence of solutions to a set optimization problem is examined to verify the validity of the results obtained.

关键词： Ekeland's variational principle Set order scalarization function Set optimization

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New Results on Henig Proper Generalized Vector Quasiequilibrium Problems

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NUMERICAL functionAL ANALYSIS AND OPTIMIZATION 2015年第3期36卷 387-403页

作者： Pham Huu Sach Nguyen Ba Minh Vietnam Acad Sci & Technol Hanoi Inst Math Hanoi 10307 Vietnam Hanoi Univ Commerce Hanoi Vietnam

New sufficient conditions are given for the existence of solutions of a Henig proper generalized vector quasiequilibrium problem with moving cones. They are established by a new scalarizing approach, which is based on a suitable nonlinear scalarization function, proposed recently for set-valued maps in Sach and Tuan (J. Optim. Theory Appl. 157:347-364 (2013)). Examples are given to illustrate our main results.

关键词： 49K40 90C31 Set-valued map scalarization function Proper efficiency Cone-preconcavity Existence result Quasiequilibrium problem

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A Calculation Approach to scalarization for Polyhedral Sets by Means of Set Relations

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TAIWANESE JOURNAL OF MATHEMATICS 2019年第1期23卷 255-267页

作者： Yu, Hui Ike, Koichiro Ogata, Yuto Tanaka, Tamaki Niigata Univ Grad Sch Sci & Technol Niigata 9502181 Japan

In this paper, we focus on certain functions as scalarization for six types of set relations and discuss calculation algorithms for them between polyhedral sets, while those between polytopes have been already investigated. A major difference between polyhedral sets and polytopes is in boundedness. Polyhedral sets are no longer necessarily bounded. Methods for calculating types (1), (2), (4), (6) are easily available by a similar way to existing ideas. However, those for types (3) and (5), which are actually the most famous and long-standing types, require some technical ways approaching to the value of them by using the fact that finitely generatedness and polyhedrality coincide and can be algorithmically switched in finite-dimensional spaces. As a result, we show all types are reduced to a finite number of linear programming problems. Also, we demonstrate our methods through an example and give detailed calculation process.

关键词： set relation scalarization function polyhedral set finitely generated set

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