The Proximal Point Method for Locally Lipschitz Functions in Multiobjective Optimization with Application to the Compromise Problem

G. Bento (1) , J. Cruz Neto (2) , G. López (3) , Antoine Soubeyran (4) , J. Souza (2, 5)

1 UFG - Universidade Federal de Goiás [Goiânia]
2 UFPI - Universidade Federal do Piauí
3 Universidad de Sevilla = University of Seville
4 AMSE - Aix-Marseille Sciences Economiques
5 UFRJ - Universidade Federal do Rio de Janeiro [Brasil] = Federal University of Rio de Janeiro [Brazil] = Université fédérale de Rio de Janeiro [Brésil]

This paper studies the constrained multiobjective optimization problem of finding Pareto critical points of vector-valued functions. The proximal point method considered by Bonnel, Iusem, and Svaiter [SIAM J. Optim., 15 (2005), pp. 953--970] is extended to locally Lipschitz functions in the finite dimensional multiobjective setting. To this end, a new (scalarization-free) approach for convergence analysis of the method is proposed where the first-order optimality condition of the scalarized problem is replaced by a necessary condition for weak Pareto points of a multiobjective problem. As a consequence, this has allowed us to consider the method without any assumption of convexity over the constraint sets that determine the vectorial improvement steps. This is very important for applications; for example, to extend to a dynamic setting the famous compromise problem in management sciences and game theory.

Domaines: Economies et finances

G. Bento, J. Cruz Neto, G. López, Antoine Soubeyran, J. Souza. The Proximal Point Method for Locally Lipschitz Functions in Multiobjective Optimization with Application to the Compromise Problem. SIAM Journal on Optimization, 2018, 28 (2), pp.1104-1120. ⟨10.1137/16M107534X⟩. ⟨hal-01985333⟩