Maximal monotonicity of piecewise polyhedral mappings

Abstract

Maximally monotone mappings that are piecewise polyhedral arise from subdifferentials of convex functions and saddle functions that are piecewise linear-quadratic and enter into algorithmic constructions of importance in linear-quadratic optimization and associated splitting methods. The question of whether those constructions preserve maximal monotonicity is then crucial. The usual answers to that invoke constraint qualifications involving the nonemptiness of intersections of certain relative interiors, but it is shown here that the piecewise polyhedral structure allows the relative interiors to be bypassed.