Optimality criteria without constraint qualifications for linear semidefinite problems

Abstract

We consider two closely related optimization problems: a problem of convex SemiInfinite Programming with multidimensional index set and a linear problem of Semidefinite Programming. In study of these problems we apply the approach suggested in our recent paper [14] and based on the notions of immobile indices and their immobility orders. For the linear semidefinite problem, we define the subspace of immobile indices and formulate the first order optimality conditions in terms of a basic matrix of this subspace. These conditions are explicit, do not use constraint qualifications, and have the form of criterion. An algorithm determining a basis of the subspace of immobile indices in a finite number of steps is suggested. The optimality conditions obtained are compared with other known optimality conditions.

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.…”

“…Consider a subspace of R p in the form H: = {t ∈ R p : At = 0, Bt = 0} (18) and denote by b s ∈ R p , s ∈ S, |S| ≤ p, the vectors of an orthogonal basis of this subspace.…”

“…[25], p.65). Notice that for the polyhedral cone K in the form (17), it holds linK = H and K ∩ (linK) ⊥ = H * , where H and H * are defined in (18), (19).…”

“…where {b s ∈ R p , s ∈ S, |S| ≤ p} is the orthogonal basis of the subspace H defined in (18) and {a s , s ∈ S * , |S * | < ∞} is the set of extreme directions in the pointed cone H * defined in (19).…”

Existence of Generalized Augmented Lagrange Multipliers for Constrained Optimization Problems

“…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.…”