Polynomial vector variational inequalities under polynomial constraints and applications

By using a scalarization method and some properties of semi-algebraic sets, we prove that both the proper Pareto solution set and the weak Pareto solution set of a vector variational inequality, where the convex constraint set is given by polynomial functions and all the components of the basic operators are polynomial functions, have finitely many connected components, provided that the Mangasarian-Fromovitz constraint qualification is satisfied at every point of the constraint set. In addition, if the proper Pareto solution set is dense in the Pareto solution set, then the latter also has finitely many connected components. Consequences of the results for vector optimization problems are discussed in detail. © 2016 Societ y for Industrial and Applied Mathematics.