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achievement scalarizing function sorting for strength Pareto evolutionary algorithm in many-objective optimization

NEURAL COMPUTING & APPLICATIONS 2021年第11期33卷 6369-6388页

作者： Li, Xin Li, Xiaoli Wang, Kang Yang, Shengxiang Li, Yang Beijing Univ Technol Fac Informat Technol Beijing 100124 Peoples R China Minist Educ Beijing Key Lab Computat Intelligence & Intellige Engn Res Ctr Digital Commun Beijing 100124 Peoples R China De Montfort Univ Sch Comp Sci & Informat Ctr Computat Intelligence Leicester Leics England Commun Univ China Beijing 100024 Peoples R China

Multi-objective evolutionary algorithms (MOEAs) have proven their effectiveness in solving two or three objective problems. However, recent research shows that Pareto-based MOEAs encounter selection difficulties facing many similar non-dominated solutions in dealing with many-objective problems. In order to reduce the selection pressure and improve the diversity, we propose achievement scalarizing function sorting strategy to make strength Pareto evolutionary algorithm suitable for many-objective optimization. In the proposed algorithm, we adopt density estimation strategy to redefine a new fitness value of a solution, which can select solution with good convergence and distribution. In addition, a clustering method is used to classify the non-dominated solutions, and then, an achievement scalarizing function ranking method is designed to layer different frontiers and eliminate redundant solutions in the environment selection stage, thus ensuring the convergence and diversity of non-dominant solutions. The performance of the proposed algorithm is validated and compared with some state-of-the-art algorithms on a number of test problems with 3, 5, 8, 10 objectives. Experimental studies demonstrate that the proposed algorithm shows very competitive performance.

关键词： Evolutionary algorithm Many-objective optimization achievement scalarizing function Convergence Diversity

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A preference-based evolutionary algorithm for multiobjective optimization: the weighting achievement scalarizing function genetic algorithm

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JOURNAL OF GLOBAL OPTIMIZATION 2015年第1期62卷 101-129页

作者： Ruiz, Ana Belen Saborido, Ruben Luque, Mariano Univ Malaga Dept Appl Econ Math E-29071 Malaga Spain

When solving multiobjective optimization problems, preference-based evolutionary multiobjective optimization (EMO) algorithms introduce preference information into an evolutionary algorithm in order to focus the search for objective vectors towards the region of interest of the Pareto optimal front. In this paper, we suggest a preference-based EMO algorithm called weighting achievement scalarizing function genetic algorithm (WASF-GA), which considers the preferences of the decision maker (DM) expressed by means of a reference point. The main purpose of WASF-GA is to approximate the region of interest of the Pareto optimal front determined by the reference point, which contains the Pareto optimal objective vectors that obey the preferences expressed by the DM in the best possible way. The proposed approach is based on the use of an achievement scalarizing function (ASF) and on the classification of the individuals into several fronts. At each generation of WASF-GA, this classification is done according to the values that each solution takes on the ASF for the reference point and using different weight vectors. These vectors of weights are selected so that the vectors formed by their inverse components constitute a well-distributed representation of the weight vectors space. The efficiency and usefulness of WASF-GA is shown in several test problems in comparison to other preference-based EMO algorithms. Regarding a metric based on the hypervolume, we can say that WASF-GA has outperformed the other algorithms considered in most of the problems.

关键词： Multiobjective optimization Pareto optimal solutions Reference point approach achievement scalarizing function Evolutionary algorithm

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On computing objective function values in multiple objective quadratic-linear programming

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 1998年第1期106卷 184-190页

作者： Korhonen, P Yu, GY Helsinki Sch Econ & Business Adm Dept Econ & Management Sci Helsinki 00100 Finland

In this paper, we will consider the computation of objective function values when a nondominated frontier is searched in multiple objective quadratic-linear programming (MOQLP). Reference directions and weighted-sums constitute a methodological basis for the search. This idea leads to a parametric linear complementarity model formulation. A critical task of making a search procedure efficient, is to compute the changes in quadratic and linear objective functions efficiently when a search direction is changed or a basis change is performed, Those changes in objective functions can be computed by a so-called direct or indirect method. The direct method is a straightforward one and based on the use of unit changes in basic decision variables. Instead, the indirect method utilizes some other basic variables of the model. We will introduce the indirect method and make theoretical and empirical comparisons between the methods. Based on the comparisons, we point out that the indirect method is clearly much more efficient than the direct one. (C) 1998 Elsevier Science B.V.

关键词： multiple objective programming nondominated frontier quadratic programming achievement scalarizing function reference direction linear complementarity model parametric programming

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Multi-objective optimization for balancing CO 2 emissions and thermal comfort in a central heat pump system: A case study of a Danish office building

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Energy and Buildings 2025年 342卷

作者： Yujie Yang Muhyiddine Jradi Center for Energy Informatics The Maersk Mc-Kinney Moller Institute University of Southern Denmark Campusvej 55 5230 Odense Denmark

Buildings consume a large portion of global energy and contribute significantly to CO 2 emissions, with HVAC systems being major energy users. Improving HVAC efficiency is crucial for sustainability and climate goals. Model Predictive Control (MPC) has shown promise in optimizing HVAC operations, enhancing energy savings while maintaining comfort. However, full-scale implementation of MPC in real-world buildings remains limited. This study bridges that gap by developing a high-fidelity energy model of an office building at the University of Southern Denmark to evaluate MPC performance in practice. Unlike traditional methods focusing on terminal components, this research targets the central heat source, a heat pump, and uses MPC to balance CO 2 emissions and thermal comfort. A multi-objective optimization framework is used, weighing CO 2 emissions at 0.6 and thermal comfort at 0.4 when selecting control strategies at each prediction step. Results show that MPC reduced average indoor temperature by 0.74 °C, cut total energy use by 27.96 %, and lowered CO 2 emissions by 26.63 %. Additionally, the optimized heat pump operation aligns energy use with periods of higher renewable energy availability, further reducing dependence on fossil fuels. By shifting the optimization focus to the central heat source and integrating grid-responsive strategies, this study offers a scalable and intelligent solution for decarbonizing HVAC systems. The findings underscore MPC’s potential to enhance building energy performance and contribute to a more sustainable and resilient energy future.

关键词： Model predictive control CO 2 emissions Thermal comfort Energy Heat pump ASF achievement scalarizing function AWHP Air-to-water heat pump COP Coefficient of performance EU European Union HVAC Heating, Ventilation, and Air Conditioning IQR Interquartile range MCDM Multi-Criteria Decision Making MIMO Multiple-input, multiple-output MPC Model Predictive NSGA-II Non-dominated genetic algorithm-II PMV Predicted Mean Vote PPD Predicted Percentage Dissatisfied RBC Rule-based control SHGC Solar heat gain coefficient WWR Window-to-wall ratio

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Set-based many-objective optimization guided by a preferred region

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NEUROCOMPUTING 2017年 228卷 241-255页

作者： Gong, Dunwei Sun, Fenglin Sun, Jing Sun, Xiaoyan China Univ Min & Technol Sch Informat & Elect Engn Xuzhou 221116 Jiangsu Peoples R China Huai Hai Inst Technol Coll Sci Lianyungang 222005 Jiangsu Peoples R China

Set-based evolutionary optimization based on performance indicators is one of effective methods to solve many objective optimization problems. However, preference information of a high-dimensional objective space has not yet been fully used to guide the evolution of a population. In this paper, we propose a set-based many objective evolutionary algorithm guided by a preferred region. In the set-based evolution, the preferred region of a high-dimensional objective space is dynamically determined, a selection Strategy on sets by combining the Pareto dominance on sets with the above preferred region is designed, and the crossover operators on sets guided by the above preferred region are developed to produce a Pareto front with superior performances. The proposed method is applied to four benchmark many-objective optimization problems and a real-world engineering design optimization problem, and the experimental results empirically demonstrate its effectiveness.

关键词： Many-objective optimization Set-based evolution achievement scalarizing function Preferred region Pareto dominance on sets

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A reference direction approach to multiple objective quadratic-linear programming

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 1997年第3期102卷 601-610页

作者： Korhonen, P Yu, GY Helsinki School of Economics and Business Administration Department of Economics and Management Science Runeberginkatu 14-16 00100 Helsinki Finland

In this paper, we propose an interactive procedure for solving multiple criteria problems with one quadratic objective, several linear objectives, and a set of linear constraints. The procedure is based on the use of reference directions and weighted sums. Reference directions for the linear functions, and weighted sums for combining the quadratic function with the linear ones are used as parameters to implement the free search of nondominated solutions. The idea leads to the parametric linear complementarity problem formulation. An approach to deal with this type of problems is given as well. The approach is illustrated with a numerical example. (C) 1997 Elsevier Science B.V.

关键词： multiple objective programming quadratic programming achievement scalarizing function reference direction linear complementarity model nondominated criterion values

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On the sizing of a solar thermal electricity plant for multiple objectives using evolutionary optimization

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APPLIED SOFT COMPUTING 2012年第10期12卷 3300-3311页

作者： Deb, Kalyanmoy Ruiz, Francisco Luque, Mariano Tewari, Rahul Cabello, Jose M. Cejudo, Jose M. Indian Inst Technol Dept Mech Engn Kanpur 208016 Uttar Pradesh India Michigan State Univ Dept Elect & Comp Engn E Lansing MI 48824 USA Aalto Univ Sch Econ Dept Informat & Serv Econ Helsinki 00100 Finland Univ Malaga E-29071 Malaga Spain Deutsch Bank Grp Bombay Maharashtra India

Design, implementation and operation of solar thermal electricity plants are no more an academic task, rather they have become a necessity. In this paper, we work with power industries to formulate a multi-objective optimization model and attempt to solve the resulting problem using classical as well as evolutionary optimization techniques. On a set of four objectives having complex trade-offs, our proposed procedure first finds a set of trade-off solutions showing the entire range of optimal solutions. Thereafter, the evolutionary optimization procedure is combined with a multiple criterion decision making (MCDM) approach to focus on preferred regions of the trade-off frontier. Obtained solutions are compared with a classical generating method. Eventually, a decision-maker is involved in the process and a single preferred solution is obtained in a systematic manner. Starting with generating a wide spectrum of trade-off solutions to have a global understanding of feasible solutions, then concentrating on specific preferred regions for having a more detailed understanding of preferred solutions, and then zeroing on a single preferred solution with the help of a decision-maker demonstrates the use of multi-objective optimization and decision making methodologies in practice. As a by-product, useful properties among decision variables that are common to the obtained solutions are gathered as vital knowledge for the problem. The procedures used in this paper are ready to be used to other similar real-world problem solving tasks. (C) 2012 Elsevier B.V. All rights reserved.

关键词： Solar thermal power plant Multi-objective optimization Pareto-optimal solutions Reference point approach achievement scalarizing function Preferred solution KKT optimality condition

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Equivalent reference points in multiobjective programming

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EXPERT SYSTEMS WITH APPLICATIONS 2015年第4期42卷 2205-2212页

作者： Luque, M. Lopez-Agudo, L. A. Marcenaro-Gutierrez, O. D. Univ Malaga Dept Appl Econ Math E-29071 Malaga Spain Univ Malaga Dept Appl Econ Stat & Econometr E-29071 Malaga Spain

In this paper, we concentrate on reference point based methods in multiobjective programming to demonstrate, as main contribution, that the solution to a multiobjective optimization problem stays unchanged if the reference point is changed to any point on a set defined by means of the original reference point, the nondominated objective solution and some parameters of the ASF. Concretely, this new set of "equivalent reference points" is the convex linear combination of two straight lines, one containing the original reference point and the other a nondominated objective solution, where the slope of both straight lines is given by the inverses of the weights of the ASF. An illustrative example is used to show the results obtained and an empirical model (application with real data) allows us to highlight possible implications. (C) 2014 Elsevier Ltd. All rights reserved.

关键词： Multiobjective programming Reference point achievement scalarizing function Interactive methods

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Global WASF-GA: An Evolutionary Algorithm in Multiobjective Optimization to Approximate the Whole Pareto Optimal Front

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EVOLUTIONARY COMPUTATION 2017年第2期25卷 309-349页

作者： Saborido, Ruben Ruiz, Ana B. Luque, Mariano Ecole Polytech Montreal Polytech Montreal Researchers Software Engn Montreal PQ Canada Univ Malaga Dept Appl Econ Math Calle Ejido 6 E-29071 Malaga Spain

In this article, we propose a new evolutionary algorithm for multiobjective optimization called Global WASF-GA (global weighting achievement scalarizing function genetic algorithm), which falls within the aggregation-based evolutionary algorithms. The main purpose of Global WASF-GA is to approximate the whole Pareto optimal front. Its fitness function is defined by an achievement scalarizing function (ASF) based on the Tchebychev distance, in which two reference points are considered (both utopian and nadir objective vectors) and the weight vector used is taken from a set of weight vectors whose inverses are well-distributed. At each iteration, all individuals are classified into different fronts. Each front is formed by the solutions with the lowest values of the ASF for the different weight vectors in the set, using the utopian vector and the nadir vector as reference points simultaneously. Varying the weight vector in the ASF while considering the utopian and the nadir vectors at the same time enables the algorithm to obtain a final set of nondominated solutions that approximate the whole Pareto optimal front. We compared Global WASF-GA to MOEA/D (different versions) and NSGA-II in two-, three-, and five-objective problems. The computational results obtained permit us to conclude that Global WASF-GA gets better performance, regarding the hyper-volume metric and the epsilon indicator, than the other two algorithms in many cases, especially in three-and five-objective problems.

关键词： Multiobjective optimization Pareto optimal solutions achievement scalarizing function Evolutionary algorithm

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On the estimation of pareto front and dimensional similarity in many-objective evolutionary algorithm

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INFORMATION SCIENCES 2021年 563卷 375-400页

作者： Li, Li Yen, Gary G. Sahoo, Avimanyu Chang, Liang Gu, Tianlong Guilin Univ Elect Technol Guangxi Key Lab Trusted Software Guilin 541004 Peoples R China Oklahoma State Univ Sch Elect & Comp Engn Stillwater OK 74078 USA Oklahoma State Univ Div Engn Technol Stillwater OK 74078 USA

Evolutionary algorithms have been proven to be effective in solving multi-objective optimization problems. However, their performance deteriorates progressively in handling many-objective optimization problems due to the sensitivity upon the curvature of Pareto front, as well as the implicit evaluation on similarity in high dimensionality. This paper proposes an on-line Pareto front curvature estimator for an adaptive selection, in which the achievement scalarizing function is used to identify the pivotal solution to extrapolate the geometric information. Then an adaptive scalarizing function based fitness assessment, which guarantees the Pareto optimality, is presented. The diversity of the Pareto optimal solutions is also ensured by introducing a novel similarity metric. Finally, an extensive experimental analysis is presented to corroborate the analytical result by evaluating problems with various types of Pareto fronts. The experimental results substantiate the efficacy of the results with competitive performance. (c) 2021 Elsevier Inc. All rights reserved.

关键词： achievement scalarizing function Angle-based selection Decomposition Evolutionary algorithm Many-objective optimization Reference vector Pareto front curvature Similarity

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