Optimality conditions for nonsmooth mathematical programs with equilibrium constraints, using convexificators

Optimality conditions and duality for interval-valued optimization problems using convexifactors

“…Clearly, (y, ξ L , ξ U , μ) = (0, 1, 1, 1) is a feasible solution to (IMWD) and objective value is [1,2]. We also note that x = 2 is a feasible solution to (IVP2) and corresponding objective value is [3,22].…”

“…By using a Guignard constraint qualification, some stronger Kuhn-Tucker type necessary optimality conditions for efficiency in terms of convexifactors were established. Further, Ardali et al [2] used the notion of convexifactor to study constraint qualifications, stationary concepts and optimality conditions for a nonsmooth mathematical programs with equilibrium constraints. Golestani and Nobakhtian [13] presented optimality conditions for nonsmooth semidefinite programming via convexificators.…”

Some Convexificators-Based Optimality Conditions for Nonsmooth Mathematical Program with Vanishing Constraints

“…Utilizing the above notations, motivated by [37], we are ready to introduce the Abadie type constraint qualification in the form of convexificator which is very important to establish the optimality conditions. Definition 3.1.…”

“…If the GS-ACQ holds at x , then x is a GS stationary point. Now, we provide the following example to illustrate Theorem 3.1, this example is a modified version of Example 4.7 in [37].…”

“…For nonsmooth optimization problems, various convexificators-based results with respect to the Fritz-John type and the Karush-Kuhn-Tucker type necessary optimality conditions have been developed in [32] [33] [34] [35] [36]. Very recently, Ansari, Movahedian and Nobakhtian [37] deal with constraint qualifications, stationary concepts and optimality conditions for a nonsmooth mathematical program with equilibrium constraints by using the notion of convexificators. However, the corresponding results about the nonsmooth mathematical program with vanishing constraints can be very few.…”