Strong KKT conditions and duality for Borwein and Geoffrion proper efficient solutions in $\mathbf{C^1}$ optimization

Abstract

This paper investigates higher-order optimality conditions for properly efficient solutions of $C^{1}$ multiobjective optimization problems with inequality constraints. By employing generalized radial directional derivatives, we introduce a higher-order Robinson-type constraint qualification and discuss its relationships with the Mangasarian-Fromovitz and Kurcyusz-Robinson-Zowe constraint qualifications. These conditions are then employed to derive strong Karush-Kuhn-Tucker optimality conditions for Borwein properly efficient solutions, featuring \textit{strictly positive multipliers} for the objective functions. Under relaxed higher-order generalized convexity assumptions, we establish sufficient conditions for Geoffrion properly efficient solutions. Furthermore, we develop weak and strong duality results within the Mond-Weir and Wolfe duality frameworks, thereby extending classical duality theory to a higher-order setting. Several examples are provided to illustrate the advantages of our results over existing ones.