# An inexact strategy for the projected gradient algorithm in vector optimization problems on variable ordered spaces

## TL;DR

This paper introduces an inexact projected gradient method for smooth constrained vector optimization problems on variable ordered spaces, establishing convergence to weakly efficient points and demonstrating computational efficiency through experiments.

## Contribution

It presents a novel inexact gradient algorithm for variable ordered spaces, with convergence analysis and practical computational comparisons.

## Key findings

- Convergence of accumulation points to weakly efficient solutions.
- The inexact method reduces CPU time compared to exact methods.
- Applicability to unconstrained and domain-specific optimization models.

## Abstract

Variable order structures model situations in which the comparison between two points depends on a point-to-cone map. In this paper, an inexact projected gradient method for solving smooth constrained vector optimization problems on variable ordered spaces is presented. It is shown that every accumulation point of the generated sequence satisfies the first order necessary optimality condition. The convergence of all accumulation points to a weakly efficient point is established under suitable convexity assumptions for the objective function. The convergence results are also derived in the particular case in which the problem is unconstrained and if exact directions are taken as descent directions. Furthermore, we investigate the application of the proposed method to optimization models where the domain of the variable order map and the objective function are the same. In this case, similar concepts and convergence results are presented. Finally, some computational experiments designed to illustrate the behavior of the proposed inexact methods versus the exact ones (in terms of CPU time) are performed.