Accelerated Diagonal Steepest Descent Method for Unconstrained Multiobjective Optimization
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Accelerated Diagonal Steepest Descent Method for Unconstrained Multiobjective Optimization Mustapha El Moudden1
· Abdelkrim El Mouatasim2
Received: 12 March 2020 / Accepted: 7 November 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, we propose two methods for solving unconstrained multiobjective optimization problems. First, we present a diagonal steepest descent method, in which, at each iteration, a common diagonal matrix is used to approximate the Hessian of every objective function. This method works directly with the objective functions, without using any kind of a priori chosen parameters. It is proved that accumulation points of the sequence generated by the method are Pareto-critical points under standard assumptions. Based on this approach and on the Nesterov step strategy, an improved version of the method is proposed and its convergence rate is analyzed. Finally, computational experiments are presented in order to analyze the performance of the proposed methods. Keywords Multiobjective optimization · Diagonal steepest descent methods · Pareto critical · Unconstrained problems · Nesterov step Mathematics Subject Classification 90C29 · 90C30 · 90C53
1 Introduction Multiobjective optimization refers to the general problem of simultaneously minimizing (or maximizing) several objective functions. Naturally, the objective functions are
Communicated by Xiaoqi Yang.
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Mustapha El Moudden [email protected]; [email protected] Abdelkrim El Mouatasim [email protected]
1
Modeling, Simulation and Data Analysis (MSDA), Mohammed VI Polytechnic University, Benguerir, Morocco
2
Polydisciplinary Faculty of Ouarzazate, Ibn Zohr University, B.P. 284, Ouarzazate 45800, Morocco
123
Journal of Optimization Theory and Applications
conflicting, and since there is no single point which can optimize all objective functions simultaneously, we have a large number of Pareto-optimal points. One should recall that a Pareto-optimal point is a solution, in which the improvement of any objective function is not possible without impairing at least one other objective function. Multiobjective optimization problems are encountered in many application areas, for instance, in engineering [1], environmental analysis [2] and management science [3]. The literature on unconstrained multiobjective problems is vast, and several methods have been developed to solve them. The scalarization approach is one of the most popular strategies for solving these problems. Using some parameters, this approach consists in converting the multiobjective problem into one or more scalar-valued problems. This enables the use of the theory and methods of standard optimization. Several major well-known drawbacks occur in scalarization approach. First, almost all choices of the parameters may lead to unbounded scalar-valued optimization problems; see [4]. Second, the parameters are not known a priori and have to be prespecified, just as shown in [5]. Third, in the non-convex case, we may
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