Smooth methods of multipliers for complementarity problems

Abstract

Abstract. This paper describes several methods for solving nonlinear complementarity problems. A general duality framework for pairs of monotone operators is developed and then applied to the monotone complementarity problem, obtaining primal, dual, and primal-dual formulations. We derive Bregman-function-based generalized proximal algorithms for each of these formulations, generating three classes of complementarity algorithms. The primal class is well-known. The dual class is new and constitutes a general collection of methods of multipliers, or augmented Lagrangian methods, for complementarity problems. In a special case, it corresponds to a class of variational inequality algorithms proposed by Gabay. By appropriate choice of Bregman function, the augmented Lagrangian subproblem in these methods can be made continuously differentiable. The primal-dual class of methods is entirely new and combines the best theoretical features of the primal and dual methods. Some preliminary computation shows that this class of algorithms is effective at solving many of the standard complementarity test problems.

Proximal Methods for Cohypomonotone Operators

“…The motivation for this line of work stems from the fact that proximal iterations have been observed to converge to zeros of nonmonotone operators in certain numerical experiments, e.g., [12]. Attempts to explain this behavior in the case of general variational inclusions can be traced back to [26], where a convergence proof is given which does not assume monotonicity.…”

A Class of Inexact Variable Metric Proximal Point Algorithms

“…Problems of this type appear, for example, in smooth multiplier methods for monotone complementarity problems [14]. We start with describing the method and giving its theoretical justification and then report on our numerical experiments.…”

Penalidades exatas para desigualdades variacionais

“…Os testes foram feitos com os problemas de complementaridade da MCPLIB, mesmo sem que eles satisfaçam todas as hipóteses do teorema de convergência, uma prática comum na literatura [11,19].…”

“…Recentemente, Eckstein e Ferris aproveitaram tais desenvolvimentos teóricos para estender as idéias de lagrangianos aumentados e o método de multiplicadores para problemas de complementaridade mistos, que são casos particulares de desigualdades variacionais [18]. Em seguida, Auslender e Teboulle estendem ainda mais tais resultados para lidar com restrições descritas por desigualdades convexas gerais [2].…”

“…Usando regularizações duplas, que combinam esses tipos de distâncias especiais e a distância quadrática clássica, Eckstein e Silva conseguem resultados ainda mais gerais [45]. Todas a provas exigem de uma forma ou outra a monotonicidade da função F , embora bons resultados práticos tenham sido obtidos com a aplicação dos algoritmos mesmo a problemas não monótonos [18,45].…”