Optimality Conditions for Convex Semi-infinite Programming Problems with Finitely Representable Compact Index Sets

Existence of Generalized Augmented Lagrange Multipliers for Constrained Optimization Problems

“…, which further implies that x k ∈ Λ by (42). We can assume without loss of generality that x k converges tox.…”

“…Moreover, CQ-free duality was proposed in the classical monograph [39] by Bonnans and Shapiro. The stronger results on CQ-free strong duality for semidefinite and general convex programming can be found in [40,41], and in more recent publications for semi-infinite, semidefinite, and copositive programming by Kostyukova and others [42,43]. Recently, Dolgopolik [44] studied the existence of augmented Lagrange multipliers for geometric constraint optimization by using the localization principle.…”

CQ-free optimality conditions and strong dual formulations for a special conic optimization problem

“…We will use here the approach developed in our previous papers (see e.g. [16,18,20]) for different classes of convex problems.…”

“…In our papers [15,16,19], and others, we introduced for convex SIP problems the concept of immobile indices (the indices of constraints which are active for all feasible solutions) and showed that these indices play a special role in formulating the optimality conditions which do not need the fulfillment of the Slater condition. Here, we will show how our approach can be applied to the conic problem under consideration.…”

“…As a rule, these conditions are either of the Slater type (see [7,26,28,29] and the references therein) or include some weaker assumptions (see, e.g. [15,16,19]).…”