Proximal Point Method for Locally Lipschitz Functions in Multiobjective Optimization of Hadamard Manifolds

• Published in: Journal of optimization theory and applications Vol. 179; no. 1; pp. 37 - 52
• Main Authors: Bento, Glaydston de C., da Cruz Neto, João Xavier, de Meireles, Lucas V.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-10-2018; Springer Nature B.V
• Subjects: Applications of Mathematics; Calculus of Variations and Optimal Control; Optimization; Convergence; Convexity; Critical point; Engineering; Goal programming; Manifolds; Mathematics; Mathematics and Statistics; Multiple objective analysis; Operations Research/Decision Theory; Optimization; Pareto optimum; Theory of Computation; Proximal method; 90C29; 49M30; 90C30; Multiobjective optimization; Hadamard manifold
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-018-1330-5

A proximal point method for nonsmooth multiobjective optimization in the Riemannian context is proposed, and an optimality condition for multiobjective problems is introduced. This allowed replacing the classic approach, via “scalarization,” by a purely vectorial and considering the method without any assumption of convexity over the constraint sets that determine the vectorial improvement steps. The main convergence result ensures that each cluster point (if any) of any sequence generated by the method is a Pareto critical point. Moreover, when the problem is convex on a Hadamard manifold, full convergence of the method for a weak Pareto optimal is obtained.