Dualization of generalized equations of maximal monotone type

This paper develops a simple duality framework for generalized equations defined by set-valued mappings from a linear space to another. The original problem is related to two auxiliary problems of the similar form, corresponding to Lagrangian and dual problems in the theory of convex programming. As in convex programming, the alternative formulations can be used to obtain information about a given problem and then used to solve it numerically. In particular, dualization can be used in deriving existence criteria for a given problem indirectly by considering one of the alternative formulations. Also, a given problem can often be solved more easily by way of a dual method. The strongest results of this paper concern monotone mappings. In this context, the duality framework yields several new criteria for maximal monotonicity of composite mappings. These results are useful theoretically as well as in numerical solution of generalized equations. The duality framework can also be used in problem decomposition since dualization can lead to reformulations to which operator-splitting methods and other special methods can be applied.