Existence of Pareto Solutions for Vector Polynomial Optimization Problems with Constraints

This paper deals with a vector polynomial optimization problem over basic closed semi-algebraic set. By invoking some powerful tools from real geometry, we first introduce the concept called tangency varieties; obtain relationships of Palais–Smale condition, Cerami M-tameness, and properness related to considered problem, in which condition Mangasarian–Fromovitz constraint qualification at infinity plays an essential role deriving these relationships. At last, according obtained connections, establish existence Pareto solutions consideration give examples illustrate our main findings.