Integral Global Optimality Conditions and an Algorithm for Multiobjective Problems

Abstract

In this work, we propose integral global optimality conditions for multiobjective problems not necessarily differentiable. The integral characterization, already known for single objective problems, are extended to multiobjective problems by weighted sum and Chebyshev weighted scalarizations. Using this last scalarization, we propose an algorithm for obtaining an approximation of the weak Pareto front whose effectiveness is illustrated by solving a collection of multiobjective test problems.

Keywords:

• Chebyshev weighted scalarization
• integral global optimality conditions
• multiobjective optimization
• Pareto front
• weighted sum scalarization

AMS CLASSIFICATION:

• 90C29
• 65K05
• 49M37

Acknowledgments

The authors are thankful to Fernanda Maria Pereira, Valeriano Antunes de Oliveira and to the anonymous referees whose suggestions led to improvements in the article. The first author was partially supported by CAPES - Brazil and Fundação para a Ciência e a Tecnologia (FCT) through the projects PTDC/MAT-APL/28400/2017, UIDB/00297/2020, UI/BD/151246/2021, and UIDP/00297/2020 (CMA), Portugal. The third author was partially supported by the European Regional Development Fund (ERDF) and by the Ministry of Economy, Knowledge, Business and University, of the Junta de Andalucía – Spain, within the framework of the FEDER Andalucía 2014-2020 operational program (UPO-1381297).

Notes

1 The efficient set is uniformly dominant if for every non-efficient point $x\prime$ there exists an efficient point $x^{\ast }$ such that $f_{\mathcal{\ell }}(x\prime )>f_{\mathcal{\ell }}(x^{\ast })$ for all $\mathcal{\ell }=1,\dots ,r.$