Dual Programming Problems as Hemi-Games

Abstract

It is proposed here a new and unifying formal framework for duality in optimization problems, relating more closely programs to games, by means of the concepts of "hemi-games" and "quasi-duality". A new generalization of the idea of La-grange multipliers is also presented. Associated with each programming problem, we consider (through a generalization of the Lagrange's function) one particular game (from the many possible ones) such that our programming problem is one of the two hemi-games of such a game. Each hemi-game can be considered as a programming problem. Then, for each game we have a pair of programming problems: the two hemi-games of the game considered. If the game has a solution, then so does each of the two associated programming problems; and the solution of each one is an optimal strategy for the respective player. This couple of programming problems constitute a pair of dual problems. And each pair of dual problems can be thought out as such a couple of programming problems associated with the two players of a solvable game. Various and seemingly disparate dualities, already considered in the literature, are then exhibited in order to show how they can be obtained from the hemi-game notion proposed. This conceptual framework of duality is not used here for obtaining "new" results, but seems sufficiently interesting in itself.