Calmness and Exact Penalization in Vector Optimization with Cone Constraints

Abstract

In this paper, a (local) calmness condition of order α is introduced for a general vector optimization problem with cone constraints in infinite dimensional spaces. It is shown that the (local) calmness is equivalent to the (local) exact penalization of a vector-valued penalty function for the constrained vector optimization problem. Several necessary and sufficient conditions for the local calmness of order α are established. Finally, it is shown that the local calmness of order 1 implies the existence of normal Lagrange multipliers.

Connections among constrained continuous and combinatorial vector optimization

“…Now we establish a general equivalence result (without the restriction that P o ¼ R p þ ) between problems (3.3) and (3.5) by using an exact penalty result from [7]. THEOREM 3.2 Let P be a pointed (P \ ÀP ¼ {0}), closed and convex cone with nonempty interior P o andê 2 P o .…”

“…þ1, it follows that (3.14) cannot hold. It is easy to check that all the other conditions of Theorem 2.6 in [7] are satisfied. Therefore, that there exists ar 4 0 such that V(0) Q(r) for all r !r.…”

“…For this purpose, we only need to check all the conditions of Theorem 2.6 in [7] hold for ¼ 1 and K ¼ {0}. We show that there exists a scalar M40 such that…”

“…Penalty methods for constrained VOPs have been investigated by a number of researchers (see, e.g. [2,7,8,13]). …”

Calmness and Exact Penalization in Constrained Scalar Set-Valued Optimization

“…Subsequently, lower penalization technique was applied to mathematical programs with complementarity constraints [9] and nonlinear semidefinite programs [10]. It is worth mentioning that the study of general penalization and lower penalization technique has been extended to constrained vector optimization problems (see, e.g., [11][12][13][14]). However, to the best of our knowledge, exact penalization for constrained set-valued optimization problems has not been investigated in the literature although set-valued optimization has attracted increasing attention in the optimization community (see, e.g., [15][16][17][18][19][20][21][22][23] and the references therein).…”