Inexact Proximal Point Methods for Multiobjective Quasiconvex Minimization on Hadamard Manifolds
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Inexact Proximal Point Methods for Multiobjective Quasiconvex Minimization on Hadamard Manifolds Erik Alex Papa Quiroz1
· Nancy Baygorrea Cusihuallpa2 · Nelson Maculan3
Received: 5 July 2019 / Accepted: 18 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, we present two inexact scalarization proximal point methods to solve quasiconvex multiobjective minimization problems on Hadamard manifolds. Under standard assumptions on the problem, we prove that the two sequences generated by the algorithms converge to a Pareto critical point of the problem and, for the convex case, the sequences converge to a weak Pareto solution. Finally, we explore an application of the method to demand theory in economy, which can be dealt with using the proposed algorithm. Keywords Proximal point methods · Quasiconvex function · Hadamard manifolds · Multiobjective optimization · Pareto optimality Mathematics Subject Classification 49M37 · 65K05 · 65K10 · 90C26 · 90C29
1 Introduction Minimization of multiobjective functions is very important in the applications of science and engineering, because decision makers need to consider multiple, conflicting
Communicated by Alexandru Kristály.
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Erik Alex Papa Quiroz [email protected] Nancy Baygorrea Cusihuallpa [email protected] Nelson Maculan [email protected]
1
Universidad Nacional Mayor de San Marcos and Universidad Privada del Norte, Lima, Peru
2
Universidade Federal do Rio de Janeiro and Centro de Tecnologia Mineral-CETEM, Rio de Janeiro, Brazil
3
Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil
123
Journal of Optimization Theory and Applications
objectives in their decision process; see for example the books of Ehrgott [1] and Miettinen [2]. In finite-dimensional linear spaces, multiobjective optimization models have many significant applications in decision-making problems such as economic theory, management science and engineering design. On the other hand, the class of quasiconvex functions has applications, for instance, in economy because utility functions are quasiconcave; see [3,4]. Several methods have been proposed to solve multiobjective optimization problems. We may reference, for instance, the steepest descent method for multiobjective optimization of Fliege [5], the projective gradient method to the convex constrained case proposed by Graña Drummond and Iusem [6], the interior-point method for solving convex multiobjective problem proposed by Fliege and Svaiter [7], the proximal point scalarization methods for multiobjective optimization of Gregório and Oliveira [8] and Rocha et al. [9], the derivative-free methodology of Custódio et al. [10], the proximal point method, based on the results of Rockafellar [11], of Bonnel et al. [12] and its approximate case of Ceng and Yao [13]. For multiobjective quasiconvex minimization, several known methods in Euclidean spaces have been extended: the proximal point scalarization method, see Apolinario et al. [14], the subgradient type method for no
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