Linear copositive programming: strong dual formulations and their properties

In Copositive Programming, a cost function is optimized over a cone of matrices that are positive semidefinite in the non-negative ortant. Being a fairly new field of research, Copositive Programming has already gained popularity. Duality theory is a rich and powerful area of convex optimization, which is central to understanding sensitivity analysis and infeasibility issues as well as to development of numerical methods. In this paper, we continue our recent research on dual formulations for linear Copositive Programming. The dual problems obtained in the paper satisfy the strong duality relations and do not require any additional regularity assumptions such as constraint qualifications. Different dual formulations have their own special properties, the corresponding feasible sets are described in different ways, so they can have an independent application in practice.publishe

A study of one class of NLP problems arising in parametric Semi-Infinite Programming

The paper deals with a nonlinear programming (NLP) problem that depends on a finite number of integers (parameters). This problem has a special form, and arises as an auxiliary problem in study of solutions' properties of parametric semi-infinite programming (SIP) problems with finitely representable compact index sets. Therefore, it is important to provide a deep study of this NLP problem and its properties w.r.t. the values of the parameters. We are especially interested in the case when optimal solutions of the NLP problem satisfy certain properties due to some specific requirements arising in parametric SIP. We establish the values of the parameters for which optimal solutions of the corresponding NLP problem fulfil the needed properties, and suggest an algorithm that determines the right values of the parameters. An example is proposed to illustrate the application of the algorithm

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

We consider problems of linear copositive programming where feasible sets consist of vectors for which the quadratic forms induced by the corresponding linear matrix combinations are nonnegative over the nonnegative orthant. Given a linear copositive problem, we define immobile indices of its constraints and a normalized immobile index set. We prove that the normalized immobile index set is either empty or can be represented as a union of a finite number of convex closed bounded polyhedra. We show that the study of the structure of this set and the connected properties of the feasible set permits to obtain new optimality criteria for copositive problems. These criteria do not require the fulfillment of any additional conditions (constraint qualifications or other). An illustrative example shows that the optimality conditions formulated in the paper permit to detect the optimality of feasible solutions for which the known sufficient optimality conditions are not able to do this. We apply the approach based on the notion of immobile indices to obtain new formulations of regularized primal and dual problems which are explicit and guarantee strong duality. © 2020, Springer Nature B.V