On the existence of Pareto solutions for polynomial vector optimization problems

Abstract

We are interested in the existence of Pareto solutions to the vector optimization problem MinR+m{f(x)|x∈Rn},where f: Rn→ Rm is a polynomial map. By using the 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 the 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 having at least one Pareto solution.