Solutions to Affine Generalized Equations Using Proximal Mappings

The normal map has proven to be a powerful tool for solving generalized equations of the form: find ; £ C, with 0 e F{z) + NiAz). where C is a convex set and Nr(z) is the normal cone to C at ;. In this paper, we use the 7"-map, a generalization of the normal map, to solve equations of the more general form: find : E dom(r), with 0 ^ E(z) + T(z). where T is a maximal monotone multifunction. We present a path-following algorithm that determines zeros of coherently oriented piecewise-affine functions, and we use this algorithm, together with the T-map, to solve the generalized equation for affine, coherently oriented functions F, and polyhedral multi functions T. The path-following algorithm we develop here extends the piecewise-linear homotopy framework of Eaves to the case where a representation of a subdivided manifold is unknown.