In this paper, we are concerned with convex problems of infinite Optimization, namely problems of convex Semi-Infinite Programming (SIP), linear problems of Semidefinite Programming (SDP), and linear Copositive Programming (LCP) problems that are closely related.
Semi-Infinite Programming deals with extremal problems that consist in minimization of an objective function of finitely many variables in a set described by an infinite system of constraints. SIP models appear in different fields of modern science and engineering where it is necessary to simulate a behavior of complex processes whose models contain at least one inequality constraint for each value of some parameter (for example, time) varying in a given compact domain.
In SDP, an objective function is minimized under the condition that some matrix valued function is positive semidefinite. When the objective function is linear and the matrix valued function is an affine combination of some symmetric matrices, we get a convex problem. There are many applications of SDP models to combinatorial optimization, control theory, approximation theory, etc.
Copositive Programming is a relatively new field of Conic Optimization, which is most actively developing in recent years. A general problem of LCP consists in optimizing over the cone of matrices which are positively semidefined on the non-negative orthant (so-called copositive matrices).
Optimality conditions for Optimization problems are of special interest both from theoretical and practical points of view. A special attention is devoted to the results that do not need additional conditions on the constraints, so called constraint qualifications (CQ).
In the talk, we present our recent results on optimality for convex Semi-Infinite Programming and apply them to problems of linear SDP and LCP. Our approach is based on the notions of immobile indices and their immobility orders for problems of LSIP and LCP and of a subspace of immobile indices for problems of linear SDP. We show how these concepts can be used to obtain new CQ-free optimality conditions for the considered classes of Optimization problems.
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