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 (LCP). 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 LCP. 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.

KEYWORDS:

• Convex programming
• semi-infinite programming
• copositive programming
• constraint qualifications
• immobile index
• optimality conditions
• strong duality

AMS CLASSIFICATIONS:

• 90C25
• 90C30
• 90C34

Acknowledgements

The authors thank the anonymous referees for their very helpful comments and suggestions which aided us in considerably improving the presentation of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

T. V. Tchemisova http://orcid.org/0000-0002-2678-2552

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Additional information

Funding

This work was partially supported by the state research program “Convergence” (Republic Belarus), Task 1.3.01 and by Portuguese funds through CIDMA – Center for Research and Development in Mathematics and Applications, and FCT – Portuguese Foundation for Science and Technology, within the project UID/MAT/04106/2019 (CIDMA).