# Piecewise Linear Vector Optimization Problems on Locally Convex Hausdorff Topological Vector Spaces

## TL;DR

This paper studies piecewise linear vector optimization problems in locally convex Hausdorff topological vector spaces, showing their efficient solutions form unions of finitely many semi-closed generalized polyhedral convex sets, extending previous results from normed spaces.

## Contribution

It extends the characterization of solution sets of piecewise linear vector optimization problems to locally convex Hausdorff topological vector spaces, generalizing prior normed space results.

## Key findings

- Efficient solution sets are unions of finitely many semi-closed generalized polyhedral convex sets.
- Convex problems have solution sets that are unions of finitely many generalized polyhedral convex sets.
- Solution sets are connected by line segments in convex cases.

## Abstract

Piecewise linear vector optimization problems in a locally convex Hausdorff topological vector spaces setting are considered in this paper. The efficient solution set of these problems are shown to be the unions of finitely many semi-closed generalized polyhedral convex sets. If, in addition, the problem is convex, then the efficient solution set and the weakly efficient solution set are the unions of finitely many generalized polyhedral convex sets and they are connected by line segments. Our results develop the preceding ones of Zheng and Yang [Sci. China Ser. A. 51, 1243--1256 (2008)], and Yang and Yen [J. Optim. Theory Appl. 147, 113--124 (2010)], which were established in a normed spaces setting.