Optimality conditions for linear copositive programming problems with isolated immobile indices

Abstract

In the present paper, we apply our recent results on optimality for convex Semi-Infinite Programming to a problem of Linear Copositive Programming. We prove explicit optimality conditions that use concepts of immobile indices and their immobility orders, and do not require the Slater constraint qualification to be satisfied. The only assumption that we impose here is that the set of immobile indices consists of isolated points, and hence is finite. This assumption is weaker than the Slater condition; therefore, the optimality conditions obtained in the paper are more general when compared with those usually used in Linear Copositive Programming. We present an example of a problem in which the new optimality conditions allow one to test the optimality of a given feasible solution while the known optimality conditions fail to do this. Further, we use the immobile indices to construct a pair of regularized dual copositive problems and show that regardless of whether the Slater condition is satisfied or not, the duality gap between the optimal values of these problems is zero. An example of a problem is presented for which the standard strict duality fails, but the duality gap obtained by using the regularized dual problem vanishes.

Immobile Indices and CQ-Free Optimality Criteria for Linear Copositive Programming Problems

“…In linear copositive programming, the usual approach to optimality conditions is that the given copositive problem is reformulated as an equivalent problem of semi-infinite programming (SIP), and then optimality conditions for the copositive problems are deduced from such conditions for their semi-infinite equivalents (see e.g. [1,23]). This allows one to use a rich arsenal of optimal conditions known in SIP.…”

“…In our previous papers [19][20][21][22][23][24] and others, we introduced notions of immobile indices and their immobility orders for different classes of convex optimization problems and formulated for these problems new CQ-free optimality conditions. These optimality conditions permit to test optimality of a given feasible solution even in the cases when other known conditions fail.…”

“…In [23], we reformulated the notion of immobile indices for linear copositive problems and used the set of immobile indices to obtain new optimality conditions for these problems. These conditions do not use any CQs, but are formulated under a less restrictive assumption about a finite number of immobile indices.…”

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

“…[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).…”

“…In this section, we will suppose that the cost function of the special conic problem (3) is linear and discuss some new dual formulations for this problem. In our papers [17] and [20], we derived regularized primal and dual formulations for linear copositive problems. These formulations are explicit and guarantee the strong duality.…”