Abstract
The aims of this paper are twofold. First, it is shown, for the first time, which types of nonsmooth functions are characterized by all vector critical points as being efficient or weakly efficient solutions of vector optimization problems in constrained and unconstrained scenarios on Hadamard manifolds. This implies the need to extend different concepts, such as the Karush–Kuhn–Tucker vector critical points and generalized invexity functions, to Hadamard manifolds. The relationships between these quantities are clarified through a great number of explanatory examples. Second, we present an economic application proving that Nash’s critical and equilibrium points coincide in the case of invex payoff functions. This is done on Hadamard manifolds, a particular case of noncompact Riemannian symmetric spaces.
Highlights
Our area of interest is the Hadamard manifolds
This paper is concerned with the pursuit of solutions of optimization problems defined on Hadamard manifolds through critical points, where the objective function may be nonsmooth
We have introduced a great number of explanatory examples, and have presented an economics application showing that Nash’s critical and equilibrium points coincide in the case of invex payoff functions
Summary
Our area of interest is the Hadamard manifolds. This paper is concerned with the pursuit of solutions of optimization problems defined on Hadamard manifolds through critical points, where the objective function may be nonsmooth. The aim of our work is to characterize the types of nonsmooth functions for which the critical points are solutions to constrained and unconstrained optimization problems on Hadamard manifolds and to extend the results obtained by Gutiérrez et al [25] and Ruiz-Garzón et al [26] on linear spaces.
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