Existence of Solutions and Relative Regularity Conditions for Polynomial Vector Optimization Problems

In this paper, we establish the existence of the efficient solutions for polynomial vector optimization problems on a nonempty closed constraint set without any convexity and compactness assumptions. We first introduce the relative regularity conditions for vector optimization problems whose objective functions are a vector polynomial and investigate their properties and characterizations. Moreover, we establish relationships between the relative regularity conditions, Palais-Smale condition, weak Palais-Smale condition, M-tameness and properness with respect to some index set. Under the relative regularity and non-regularity conditions, we establish nonemptiness of the efficient solution sets of the polynomial vector optimization problems respectively. As a by-product, we infer Frank-Wolfe type theorems for a non-convex polynomial vector optimization problem. Finally, we study the local properties and genericity characteristics of the relative regularity conditions.