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On the Existence of Pareto Solutions for Semi-algebraic Vector Optimization Problems

Do Sang Kim (dskim***at***pknu.ac.kr)
Pham Tien Son (sonpt***at***dlu.edu.vn)
Nguyen Van Tuyen (nguyenvantuyen83***at***hpu2.edu.vn)

Abstract: We are interested in the existence of Pareto solutions to the vector optimization problem $$\text{\rm Min}_{\,\mathbb{R}^m_+} \{f(x) \,|\, x\in \mathbb{R}^n\},$$ where $f\colon\mathbb{R}^n\to \mathbb{R}^m$ be a differentiable semi-algebraic map. By using the so-called {\em tangency variety} of $f$ we first construct a semi-algebraic set of dimension at most $m - 1$ containing the set of Pareto values of the problem. Then we establish connections between Palais--Smale conditions, $M$-tameness, and properness for the map $f$. Based on these results, we provide some sufficient conditions for the existence of Pareto solutions of the problem. We also introduce a generic class of polynomial vector optimization problems, which have at least one Pareto solution.

Keywords: Existence theorems, Pareto optimal solutions, $M$-tameness, Palais--Smale conditions, Properness, Semi-algebraic

Category 1: Other Topics (Multi-Criteria Optimization )

Category 2: Nonlinear Optimization (Unconstrained Optimization )


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Entry Submitted: 11/21/2016
Entry Accepted: 11/21/2016
Entry Last Modified: 04/28/2017

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