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Use of substitute scalarizing functions to guide a local search based heuristic: The case of moTSP

JOURNAL OF HEURISTICS 2000年第3期6卷 419-431页

作者： Hansen, MP Pilegaard Planning DK-2100 Copenhagen Denmark

Solving the Tchebycheff program means optimizing a particular scalarizing function. When dealing with combinatorial problems, however, it is due to computational intractability often necessary to apply heuristics and settle for approximations to the optimal solution. The experiments in this paper suggest that for the multiobjective traveling salesman problem (moTSP) instances considered, heuristic optimization of the Tchebycheff program gives better results when using a substitute scalarizing function instead of the Tchebycheff based one to guide the local search path. Two families of substitute scalarizing functions are considered.

关键词： multiple objective combinatorial optimization multiple criteria decision making scalarizing functions tabu search local search

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Analysis of inverted PBI and comparison with other scalarizing functions in decomposition based MOEAs

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JOURNAL OF HEURISTICS 2015年第6期21卷 819-849页

作者： Sato, Hiroyuki Univ Electrocommun Fac Informat & Engn 1-5-1 Chofugaoka Chofu Tokyo 1828585 Japan

MOEA/D is one of the promising evolutionary approaches for solving multi and many-objective optimization problems. MOEA/D decomposes a multi-objective optimization problem into a number of single objective optimization problems. Each single objective optimization problem is defined by a scalarizing function using a weight vector. InMOEA/D, there are several scalarizing approaches such as weighted Tchebycheff, reciprocal weighted Tchebycheff, weighted sum (WS) and penalty-based boundary intersection (PBI). Each scalarizing function has a characteristic effect on the search performance of MOEA/D and provides a scenario of multi-objective solution search. To improve the availability of MOEA/D framework for solving various kinds of problems, it is important to provide a new scalarizing function which has different characteristics from the conventional scalarizing functions. In particular, the conventional scalarizing approaches face a difficulty to approximate a widely spread Pareto front in some problems. To approximate the entire Pareto front by improving the spread of solutions in the objective space and enhance the search performance of MOEA/D in multi and many-objective optimization problems, in this work we propose the inverted PBI scalarizing approach which is an extension of the conventional PBI and WS. In this work, we analyze differences between inverted PBI and other scalarizing functions, and compare the search performances of NSGA-III and five MOEA/Ds using weighted Tchebycheff, reciprocal weighted Tchebycheff, WS, PBI and inverted PBI in many-objective knapsack problems and WFG4 problems with 2-8 objectives. As results, we show that the inverted PBI based MOEA/D achieves higher search performance than other algorithms in problems with many-objectives and the difficulty to approximate a widely spread Pareto front in the objective space. Also, we show the robustness of the inverted PBI on Pareto front geometry by using problems with four representative conca

关键词： Evolutionary multi-objective optimization MOEA/D scalarizing functions

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Generation of New scalarizing functions Using Genetic Programming 16th

Generation of New Scalarizing Functions Using Genetic Progra...

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16th International Conference on Parallel Problem Solving from Nature (PPSN)

作者： Rodriguez, Amin V. Bernabe Coello, Carlos A. Coello CINVESTAV IPN Evolutionary Computat Grp Av IPN 2508 Mexico City 07360 San Pedro Zacat Mexico

ISBN: (纸本)9783030581145;9783030581152

In recent years, there has been a growing interest in multi-objective evolutionary algorithms (MOEAs) with a selection mechanism different from Pareto dominance. This interest has been mainly motivated by the poor performance of Pareto-based selection mechanisms when dealing with problems having more than three objectives (the so-called many-objective optimization problems). Two viable alternatives for solving many-objective optimization problems are decomposition-based and indicator-based MOEAs. However, it is well-known that the performance of decomposition-based MOEAs (and also of indicator-based MOEAs designed around R2) heavily relies on the scalarizing function adopted. In this paper, we propose an approach for generating novel scalarizing functions using genetic programming. Using our proposed approach, we were able to generate two new scalarizing functions (called AGSF1 and AGSF2), which were validated using an indicator-based MOEA designed around R2 (MOMBI-II). This validation was conducted using a set of standard test problems and two performance indicators (hypervolume and s-energy). Our results indicate that AGSF1 has a similar performance to that obtained when using the well-known Achievement scalarizing Function (ASF). However, AGSF2 provided a better performance than ASF in most of the test problems adopted. Nevertheless, our most remarkable finding is that genetic programming can indeed generate novel (and possible more competitive) scalarizing functions.

关键词： Multi-objective optimization Genetic programming scalarizing functions

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Designing scalarizing functions Using Grammatical Evolution

Designing Scalarizing Functions Using Grammatical Evolution

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Genetic and Evolutionary Computation Conference (GECCO)

作者： Bernabe Rodriguez, Amin V. Coello Coello, Carlos A. CINVESTAV IPN Comp Sci Dept Mexico City DF Mexico

ISBN: (纸本)9798400701207

In this paper, we present a grammatical evolution-based framework to produce new scalarizing functions, which are known to have a significant impact on the performance of both decomposition-based multi-objective evolutionary algorithms (MOEAs) and indicator-based MOEAs which use R2. We perform two series of experiments using this framework. First, we produce many scalarizing functions using different benchmark problems to explore the behavior of the resulting functions according to the geometry of the problem adopted to generate them. Then, we perform a second round of experiments adopting two combinations of problems which yield better results in some test instances. We present the experimental validation of these new functions compared against the Achievement scalarizing Function (ASF), which is known to provide a very good performance. For this comparative study, we adopt several benchmark problems and we are able to show that our proposal is able to generate new scalarizing functions that can outperform ASF in different problems.

关键词： multi-objective optimization genetic programming grammatical evolution scalarizing functions

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An Ensemble of scalarizing functions and Weight Vectors for Evolutionary Multi-Objective Optimization

An Ensemble of Scalarizing Functions and Weight Vectors for ...

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IEEE Congress on Evolutionary Computation (IEEE CEC)

作者： Cristina Valencia-Rodriguez, Diana Coello Coello, Carlos A. IPN Comp Sci Dept CINVESTAV Mexico City DF Mexico

ISBN: (纸本)9781728183923

Ensembles have been used in the evolutionary computation literature to evolve several populations in an independent manner, using different search approaches. Moreover, each population's parents compete with their offspring and the other population's offspring to improve diversity. It has been shown that ensemble algorithms improve the performance of the techniques embedded within them, when considered independently. Furthermore, scalarizing functions have been successfully used in decomposition-based and some indicator-based Multi-objective Evolutionary Algorithms (MOEAs). However, it has been shown that the performance of scalarizing function tends to be tied to the geometrical shape of the Pareto front. In this work, we propose a new ensemble algorithm that adopts different scalarizing functions and weight vectors using Hungarian Differential Evolution as the baseline multi-objective optimizer. Our experimental study shows that our proposed approach outperforms the original HDE, and it is competitive with respect to modern MOEAs.

关键词： ensemble scalarizing functions weight vectors multi-objective optimization

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Using objective values to start multiple objective linear programming algorithms

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2001年第3期128卷 587-596页

作者： Arbel, A Korhonen, P Helsinki Sch Econ Helsinki 00100 Finland Tel Aviv Univ Dept Ind Engn IL-69978 Tel Aviv Israel

We introduce in this paper a new starting mechanism for multiple-objective linear programming (MOLP) algorithms. This makes it possible to start an algorithm from any solution in object:ive space. The original problem is first augmented in such a way that a given starting solution is feasible. The augmentation is explicitly or implicitly controlled by one parameter during the search process, which verifies the feasibility (efficiency) of the final solution. This starting mechanism can be applied either to traditional algorithms, which search the exterior of the constraint polytope, or to algorithms moving through the interior of the constraints. We provide recommendations on the suitability of an algorithm for the various locations of a starting point in objective space. Numerical considerations illustrate these ideas. (C) 2001 Elsevier Science B.V. All rights reserved.

关键词： multiple-objective linear programming interactive starting solution aspiration levels scalarizing functions reference direction interior-point algorithms

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Using aspiration levels in an interactive interior multiobjective linear programming algorithm

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 1996年第1期89卷 193-201页

作者： Abel, A Korhonen, P HELSINKI SCH ECON & BUSINESS ADM SF-00100 HELSINKIFINLAND TEL AVIV UNIV DEPT IND ENGNIL-69978 TEL AVIVISRAEL

In this paper, we propose the use of an interior-point linear programming algorithm for multiple objective linear programming (MOLP) problems. At each iteration, a Decision Maker (DM) is asked to specify aspiration levels for the various objectives, and an achievement scalarizing function is applied to project aspiration levels onto the nondominated set. The interior-point algorithm is used to find an interior solution path from a starting solution to a nondominated solution corresponding to the optimum of the achievement scalarizing function. The proposed approach allows the DM to re-specify aspiration levels during the solution process and thus steer the interior solution path toward different areas in objective space. We illustrate the use of the approach with a numerical example.

关键词： multiple objective linear programming aspiration levels scalarizing functions interior point algorithms affine-scaling primal algorithm

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Reference Point Specification in MOEA/D for Multi-Objective and Many-Objective Problems

Reference Point Specification in MOEA/D for Multi-Objective ...

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IEEE International Conference on Systems, Man, and Cybernetics (SMC)

作者： Ishibuchi, Hisao Doi, Ken Nojima, Yusuke Osaka Prefecture Univ Grad Sch Engn Dept Comp Sci & Intelligent Syst Sakai Osaka 5998531 Japan

ISBN: (纸本)9781509018970

Recently a number of evolutionary multi-objective optimization (EMO) algorithms have been proposed using the framework of MOEA/D (multi-objective evolutionary algorithm based on decomposition). Those algorithms are characterized by the use of uniformly distributed normalized weight vectors from which a set of uniformly distributed reference lines is generated. Their basic idea is to search for a Pareto optimal solution along each reference line. While they are different in various aspects such as fitness evaluation, solution assignment to each reference line, and solution replacement, they share the same basic idea (i.e., to search for a Pareto optimal solution along each reference line). The importance of weight vector specification has been emphasized in the literature. However, the specification of a reference point has not been examined in detail whereas it plays an important role as a starting point of all reference lines. The reference point usually consists of the best value of each objective over the examined solutions, which is an approximation of the ideal point. However, this approximation is not good in early generations where the true ideal point may be much better than the best value of each objective over the examined solutions (even if it is very good in later generations). Based on these discussions, we propose a reference point specification method.

关键词： Evolutionary multi-objective optimization (EMO) many-objective optimization MOEA/D scalarizing functions

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Inverted PBI in MOEA/D and its Impact on the Search Performance on Multi and Many-Objective Optimization 14

Inverted PBI in MOEA/D and its Impact on the Search Performa...

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16th Genetic and Evolutionary Computation Conference (GECCO)

作者： Sato, Hiroyuki Univ Electrocommun Chofu Tokyo 1828585 Japan

ISBN: (纸本)9781450326629

MOEA/D decomposes a multi-objective optimization problem into a number of single objective optimization problems. Each single objective optimization problem is defined by a scalarizing function using a weight vector. In MOEA/D, there are several scalarizing approaches such as the weighted Tchebycheff, the weighted sum, and the PBI (penalty-based boundary intersection). However, these conventional scalarizing approaches face a difficulty to approximate a widely spread Pareto front in some problems. To enhance the spread of Pareto optimal solutions in the objective space and improve the search performance of MOEA/D especially in many-objective optimization problems, in this work we propose the inverted PBI scalarizing approach which is an extension of the conventional PBI. We use many-objective knapsack problems and WFG4 problems with 2-8 objectives, and compare the search performance of NSGA-III and four MOEA/Ds using the weighted Tchebycheff, the weighted sum, the PBI and the inverted PBI. As results, we show that MOEA/D using the inverted PBI achieves higher search performance than other algorithms in problems with many-objectives and the difficulty to obtain a widely spread Pareto front in the objective space.

关键词： Algorithms Design Performance multi-objective optimization MOEA/D scalarizing functions

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Constraint Handling within MOEA/D Through an Additional scalarizing Function

Constraint Handling within MOEA/D Through an Additional Scal...

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Genetic and Evolutionary Computation Conference (GECCO)

作者： Zapotecas-Martinez, Saul Ponsich, Antonin UAM Cuajimalpa Mexico City DF Mexico UAM Azcapotzalco Mexico City DF Mexico

ISBN: (纸本)9781450371285

The Multi-Objective Evolutionary Algorithm based on Decomposition (MOEA/D) has shown high-performance levels when solving complicated multi-objective optimization problems. However, its adaptation for dealing with constrained multi-objective optimization problems (cMOPs) keeps being under the scope of recent investigations. This paper introduces a novel selection mechanism inspired by the epsilon-constraint method, which builds a bi-objective problem considering the scalarizing function (used into the decomposition approach of MOEA/D) and the constraint violation degree as an objective function. During the selection step of MOEA/D, the scalarizing function is considered to choose the best solutions to the cMOP. Preliminary results obtained over a set of complicated test problems drawn from the CF test suite indicate that the proposed algorithm is highly competitive regarding state-of-the-art MOEAs adopted in our comparative study.

关键词： Constrained handling techniques multi-objective evolutionary algorithms decomposition approach scalarizing functions

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