Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# Finding Vector Critical Points on Hadamard Manifolds: Nonsmooth Case

Version 1 : Received: 1 May 2020 / Approved: 2 May 2020 / Online: 2 May 2020 (12:23:46 CEST)

How to cite: Ruiz-Garzón, G.; Osuna-Gómez, R.; Rufián-Lizana, A. Finding Vector Critical Points on Hadamard Manifolds: Nonsmooth Case. Preprints 2020, 2020050008 (doi: 10.20944/preprints202005.0008.v1). Ruiz-Garzón, G.; Osuna-Gómez, R.; Rufián-Lizana, A. Finding Vector Critical Points on Hadamard Manifolds: Nonsmooth Case. Preprints 2020, 2020050008 (doi: 10.20944/preprints202005.0008.v1).

## 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.

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