A comparison of alternative c-conjugate dual problems in infinite convex optimization

Abstract

In this work, we obtain a Fenchel–Lagrange dual problem for an infinite dimensional optimization primal one, via perturbational approach and using a conjugation scheme called c-conjugation instead of classical Fenchel conjugation. This scheme is based on the generalized convex conjugation theory. We analyse some inequalities between the optimal values of Fenchel, Lagrange and Fenchel–Lagrange dual problems and we establish sufficient conditions under which they are equal. Examples where such inequalities are strictly fulfilled are provided. Finally, we study the relations between the optimal solutions and the solvability of the three mentioned dual problems.

Keywords:

• Evenly convex function
• generalized convex conjugation
• Fenchel
• Lagrange and Fenchel–Lagrange dual problems

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was partially supported by MINECO of Spain and ERDF of EU [grant number MTM2014-59179-C2-1-P]; Consellería de la Educación de la Generalitat Valenciana, Spain, Pre-doc Program Vali+d, DOCV 6791/07.06.2012 [grant number ACIF-2013-156].