A Linear Scalarization Proximal Point Method for Quasiconvex Multiobjective Minimization

Erik Alex Papa Quiroz; Hellena Christina Fernandes Apolinário; Kely Diana Villacorta; Paulo Roberto Oliveira

doi:10.1007/s10957-019-01582-z

A Linear Scalarization Proximal Point Method for Quasiconvex Multiobjective Minimization

Erik Alex Papa Quiroz
, Hellena Christina Fernandes Apolinário
, Kely Diana Villacorta
, Paulo Roberto Oliveira

Universidad Nacional Mayor de San Marcos
Universidade Federal do Tocantins
Universidade Federal da Paraíba
Universidade Federal do Rio de Janeiro

Research output: Contribution to journal › Article › peer-review

20 Scopus citations

Abstract

In this paper, we propose a linear scalarization proximal point algorithm for solving lower semicontinuous quasiconvex multiobjective minimization problems. Under some natural assumptions and, using the condition that the proximal parameters are bounded, we prove the convergence of the sequence generated by the algorithm and, when the objective functions are continuous, we prove the convergence to a generalized critical point of the problem. Furthermore, for the continuously differentiable case we introduce an inexact algorithm, which converges to a Pareto critical point.

Original language	English
Pages (from-to)	1028-1052
Number of pages	25
Journal	Journal of Optimization Theory and Applications
Volume	183
Issue number	3
DOIs	https://doi.org/10.1007/s10957-019-01582-z
State	Published - 1 Dec 2019
Externally published	Yes

Keywords

Fejér convergence
Lower semicontinuous quasiconvex functions
Multiobjective minimization
Pareto–Clarke critical point
Proximal point methods

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10.1007/s10957-019-01582-z

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title = "A Linear Scalarization Proximal Point Method for Quasiconvex Multiobjective Minimization",

abstract = "In this paper, we propose a linear scalarization proximal point algorithm for solving lower semicontinuous quasiconvex multiobjective minimization problems. Under some natural assumptions and, using the condition that the proximal parameters are bounded, we prove the convergence of the sequence generated by the algorithm and, when the objective functions are continuous, we prove the convergence to a generalized critical point of the problem. Furthermore, for the continuously differentiable case we introduce an inexact algorithm, which converges to a Pareto critical point.",

keywords = "Fej{\'e}r convergence, Lower semicontinuous quasiconvex functions, Multiobjective minimization, Pareto–Clarke critical point, Proximal point methods",

author = "Papa Quiroz, \{Erik Alex\} and Apolin{\'a}rio, \{Hellena Christina Fernandes\} and Villacorta, \{Kely Diana\} and Oliveira, \{Paulo Roberto\}",

note = "Publisher Copyright: {\textcopyright} 2019, Springer Science+Business Media, LLC, part of Springer Nature.",

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AU - Papa Quiroz, Erik Alex

AU - Apolinário, Hellena Christina Fernandes

AU - Villacorta, Kely Diana

AU - Oliveira, Paulo Roberto

PY - 2019/12/1

Y1 - 2019/12/1

N2 - In this paper, we propose a linear scalarization proximal point algorithm for solving lower semicontinuous quasiconvex multiobjective minimization problems. Under some natural assumptions and, using the condition that the proximal parameters are bounded, we prove the convergence of the sequence generated by the algorithm and, when the objective functions are continuous, we prove the convergence to a generalized critical point of the problem. Furthermore, for the continuously differentiable case we introduce an inexact algorithm, which converges to a Pareto critical point.

AB - In this paper, we propose a linear scalarization proximal point algorithm for solving lower semicontinuous quasiconvex multiobjective minimization problems. Under some natural assumptions and, using the condition that the proximal parameters are bounded, we prove the convergence of the sequence generated by the algorithm and, when the objective functions are continuous, we prove the convergence to a generalized critical point of the problem. Furthermore, for the continuously differentiable case we introduce an inexact algorithm, which converges to a Pareto critical point.

KW - Fejér convergence

KW - Lower semicontinuous quasiconvex functions

KW - Multiobjective minimization

KW - Pareto–Clarke critical point

KW - Proximal point methods

UR - https://www.scopus.com/pages/publications/85073969071

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JO - Journal of Optimization Theory and Applications

JF - Journal of Optimization Theory and Applications

IS - 3

ER -

A Linear Scalarization Proximal Point Method for Quasiconvex Multiobjective Minimization

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Keywords

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