Abstract

This paper shows that, for vector optimization problems, the set of Karush-Kuhn-Tucker (KKT) points and strong KKT points can be characterized as sets of stationary points and proper stationary points, respectively, provided that the Mangasarian-Fromovitz constraint qualification (MFCQ) holds everywhere in the convex constraint set. As a consequence, by exploiting properties of semi-algebraic sets, it has been shown, via a direct approach, that the sets of stationary points, proper stationary points, and weak Pareto solutions all have finitely many connected components. This result holds for vector optimization problems in which both the constraint set and all objective function components are defined by polynomial functions, assuming MFCQ is satisfied everywhere in the convex constraint set. Moreover, if the set of proper stationary points is dense in the Pareto solution set, then the Pareto solution set itself also has finitely many connected components. Under convexity and the stated constraint qualification, an explicit upper bound on the number of connected components is derived. These bounds are significantly sharper than those previously reported in the literature. Finally, the connectedness structure of solution sets for special cases, including polynomial vector optimization and linear fractional vector optimization problems under polyhedral convex constraint sets, is explored.

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