Progressive Decoupling of Linkages in Optimization and Variational Inequalities with Elicitable Convexity or Monotonicity

Abstract

Algorithms for problem decomposition and splitting in optimization and the solving of variational inequalities have largely depended on assumptions of convexity or monotonicity. Here, a way of “eliciting” convexity or monotonicity is developed which can get around that limitation. It supports a procedure called the progressive decoupling algorithm, which is derived from the proximal point algorithm through passing to a partial inverse, localizing and rescaling. In the optimization setting, elicitability of convexity corresponds to a new and very general kind of second-order sufficient condition for a local minimum. Applications are thereby opened up to problem decomposition and splitting even in nonconvex optimization, moreover with augmented Lagrangians for subproblems assisting in the implementation.