Nonlinear Lagrangian for Multiobjective Optimization and Applications to Duality and Exact Penalization

Abstract

Abstract. Duality and penalty methods are popular in optimization. The study on duality and penalty methods for nonconvex multiobjective optimization problems is very limited. In this paper, we introduce vector-valued nonlinear Lagrangian and penalty functions and formulate nonlinear Lagrangian dual problems and nonlinear penalty problems for multiobjective constrained optimization problems. We establish strong duality and exact penalization results. The strong duality is an inclusion between the set of infimum points of the original multiobjective constrained optimization problem and that of the nonlinear Lagrangian dual problem. Exact penalization is established via a generalized calmness-type condition.

Calmness and Exact Penalization in Vector Optimization with Cone Constraints

“…Existence of normal Lagrange multipliers was established under a local calmness assumption. When X = R n , Y = R l , C = R l + , Z = R m , K = −R m + , results similar to those of Theorems 2.1, 2.2, 2.4, 2.5, 2.6, 2.8 and 2.10 and Proposition 2.1 were obtained in [8]. Results concerning (local) properly efficient solutions (Corollary 2.1, Theorems 2.7, 2.9 and 3.2) have never been investigated even in the finite dimensional case.…”

“…(iii) When Y = R l , C = R l + , Z = R m and K = R m + , Definition 1.3 reduces to the uniform weak stability of rank α given in [8].…”

“…Clearly, uniform calmness of order α implies calmness of order α. However, the converse may not be true (see Example 3.1 in [8]). …”

“…Compared with penalization technique for scalar constrained optimization, the study of penalization technique for vector constrained optimization is very limited: penalty functions are studied in [8,12,16] and the exact penalty representation is obtained in [8]. Moreover, these studies are restricted to finite dimensional Euclidean spaces and Pareto orderings.…”

Characterizations of Nonemptiness and Compactness of the Set of Weakly Efficient Solutions for Convex Vector Optimization and Applications

“…We also assume throughout this section that the feasible set X 0 = x ∈ X g j x 0 j = 1 m is nonempty. Now we recall the nonlinear Lagrangian for a constrained vector optimization problem (see [5] for details). Let p R l + × R m → R l be a vectorvalued function such that each component function p i i = 1 l of p is l.s.c.…”