Interior proximal methods for equilibrium programming: part I

Abstract

In this article a method for solving equilibrium problems introduced by Flåm and Antipin [S.D. Flåm and A.S. Antipin, Equilibrium programming using proximal-like algorithms, Math. Program. 77 (1997), pp. 29–41] is discussed. We extend this method to unbounded feasible sets which also leads to the necessity of a new discussion of solvability of the subproblems and appropriate stopping criteria. We also provide results permitting the use of zone-coercive regularizing functionals (of Bregman type). For example, when the boundary of the feasible set has a certain curvature, the regularized subproblems can be treated as unconstrained ones.

Keywords:

• equilibrium problems
• Bregman distances
• proximal-like methods
• interior-point effect

AMS Subject Classifications::

• 91A06
• 91A10
• 90C25
• 90C30
• 90D10

Acknowledgements

The author is grateful to two anonymous referees for reviewing the manuscript and giving helpful remarks to improve the quality of the article.