Second order optimality conditions for generalized semi-infinite programming problems

Convex SIP Problems with Finitely Representable Compact Index Sets: Immobile Indices and the Properties of the Auxiliary NLP Problem

“…Hence the optimality of x 0 ∈ X in problem (25) implies its optimality in problem (23). Therefore if x 0 ∈ X satisfies the sufficient optimality conditions from [11] then it satisfies the optimality conditions from Theorem 2. The example presented below shows that the converse is false: the optimality of feasible x 0 ∈ X in problem (25) does not imply the fulfillment of the sufficient optimality conditions from [11].…”

“…Therefore if x 0 ∈ X satisfies the sufficient optimality conditions from [11] then it satisfies the optimality conditions from Theorem 2. The example presented below shows that the converse is false: the optimality of feasible x 0 ∈ X in problem (25) does not imply the fulfillment of the sufficient optimality conditions from [11]. Hence, in the case under consideration, the optimality conditions from Theorem 2 are more efficient than ones from [11].…”

“…• We prove (see Appendix) that in the case of the convex SIP problem and |T a (x 0 )| < ∞ if the sufficient optimality conditions from [11] are fulfilled for x 0 ∈ X, then there exist finite subsetsT ⊂ T * andĪ(t)…”

“…Therefore one can formulate the optimality conditions for SIP problem in terms of such conditions for the auxiliary NLP problem. The discovered properties of the NLP problem permit one to use the most efficient optimality conditions that should give new optimality conditions for SIP that differ from the known ones (see, e.g., [1,8,11,12,21,23,25]). Notice here that the assumptions we make for the convex SIP problem, are less restrictive than those that are usually made in the literature.…”

“…When, additionally, the set T depends on the decision variable x, one gets a problem of the generalized SIP, GSIP (see [11]). Sometimes (see e.g.…”