- On the Existence of Pareto Solutions for Semi-algebraic Vector Optimization Problems Do Sang Kim (dskimpknu.ac.kr) Pham Tien Son (sonptdlu.edu.vn) Nguyen Van Tuyen (nguyenvantuyen83hpu2.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 ) Citation: Download: [PDF]Entry Submitted: 11/21/2016Entry Accepted: 11/21/2016Entry Last Modified: 04/28/2017Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.