On Constraint Qualifications for Multiobjective Optimization Problems with Vanishing Constraints

Abstract

In this chapter, we consider a class of multiobjective optimization problems with inequality, equality and vanishing constraints. For the scalar case, this class of problems reduces to the class of mathematical programs with vanishing constraints recently appeared in literature. We show that under fairly mild assumptions some constraint qualifications like Cottle constraint qualification, Slater constraint qualification, Mangasarian-Fromovitz constraint qualification, linear independence constraint qualification, linear objective constraint qualification and linear constraint qualification do not hold at an efficient solution, whereas the standard generalized Guignard constraint qualification is sometimes satisfied. We introduce suitable modifications of above mentioned constraint qualifications, establish relationships among them and derive the Karush-Kuhn-Tucker type necessary optimality conditions for efficiency.

Sufficiency and Duality for Interval-Valued Optimization Problems with Vanishing Constraints Using Weak Constraint Qualifications

“…The modified Guignard constraint qualification was introduced by Mishra et al ( [12], Definition 6.14)…”

“…Mishra et al [12] proved the Karush-Kuhn-Tucker type necessary optimality conditions for a multiob- Theorem 2.1. Let x * ∈ F be a LU optimal solution of (IVVC) such that (IVVC-GCQ) holds at x * .…”

“…This problem was first studied by Achtziger and Kanzow in [2] and this serves as a model for many problems from topology and structural optimization (see [2,5]). For Mathematical programming problems with vanishing constraints, it is well known that the usual nonlinear programming constraint qualifications such as Slater constraint qualification, Mangasarian-Fromovitz constraint qualification, Cottle constraint qualification and linear independence constraint qualification do not hold (see [12]), while Mishra et al [12] proved that the standard generalized Guignard constraint qualification holds in many situations, and some sufficient conditions are presented in [12].…”

Karush-Kuhn-Tucker optimality conditions and duality for semi-infinite programming problems with vanishing constraints

“…The concepts of stationary points of mathematical programming problems with vanishing constraints were studied in [3] under a topological point of view on critical point theory. Strong KKT necessary optimality conditions for multiobjective mathematical programming problems with vanishing constraints were discussed in [16]. The KKT necessary optimality conditions for mathematical programming problems with non-differentiable vanishing constraints were established in [14] via Clarke subdifferentials.…”

On strong KKT type sufficient optimality conditions for multiobjective semi-infinite programming problems with vanishing constraints

“…Hoheisel and Kanzow [ 12 ] established optimality conditions for weak constraint qualification. Mishra et al [ 13 ] obtained various constraint qualifications and established Karush-Kuhn-Tucker (KKT) type necessary optimality conditions for multiobjective MPVCs. We refer to [ 14 – 16 ] and references therein for more details as regards MPVCs.…”