Proximal Decomposition Via Alternating Linearization

A new approximate proximal point method for minimizing the sum of two convex functions is introduced. It replaces the original problem by a sequence of regularized subproblems in which the functions are alternately represented by linear models. The method updates the linear models and the prox center, as well as the prox coefficient. It is monotone in terms of the objective values and converges to a solution of the problem, if any. A dual version of the method is derived and analyzed. Applications of the methods to multistage stochastic programming problems are discussed and preliminary numerical experience is presented.

[1]  John M. Mulvey,et al.  A New Scenario Decomposition Method for Large-Scale Stochastic Optimization , 1995, Oper. Res..

[2]  John N. Tsitsiklis,et al.  Parallel and distributed computation , 1989 .

[3]  Krzysztof C. Kiwiel A bundle Bregman proximal method for convex nondifferentiable minimization , 1999, Math. Program..

[4]  B. Martinet Brève communication. Régularisation d'inéquations variationnelles par approximations successives , 1970 .

[5]  Michael C. Ferris,et al.  Operator-Splitting Methods for Monotone Affine Variational Inequalities, with a Parallel Application to Optimal Control , 1998, INFORMS J. Comput..

[6]  Claude Lemaréchal,et al.  Convergence of some algorithms for convex minimization , 1993, Math. Program..

[7]  Andrzej Ruszczynski,et al.  On augmented Lagrangian decomposition methods for multistage stochastic programs , 1996, Ann. Oper. Res..

[8]  Dimitri P. Bertsekas,et al.  On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators , 1992, Math. Program..

[9]  Marc Teboulle,et al.  A proximal-based decomposition method for convex minimization problems , 1994, Math. Program..

[10]  P. D. Tao,et al.  Partial regularization of the sum of two maximal monotone operators , 1993 .

[11]  Jonathan E. Spingarn,et al.  Applications of the method of partial inverses to convex programming: Decomposition , 1985, Math. Program..

[12]  Jonathan Eckstein Some Saddle-function splitting methods for convex programming , 1994 .

[13]  P. Tseng Applications of splitting algorithm to decomposition in convex programming and variational inequalities , 1991 .

[14]  A. Auslender Numerical methods for nondifferentiable convex optimization , 1987 .

[15]  Paul Tseng,et al.  Alternating Projection-Proximal Methods for Convex Programming and Variational Inequalities , 1997, SIAM J. Optim..

[16]  C. H. Rosa Pathways of Economic Development in an Uncertain Environment: A Finite Scenario Approach to the U.S. Region Under Carbon Emission Restrictions , 1994 .

[17]  Stephen M. Robinson Extended scenario analysis , 1991, Ann. Oper. Res..

[18]  S. Kontogiorgis,et al.  Alternating direction splitting for block Angular parallel optimization , 1996 .

[19]  Osman Güer On the convergence of the proximal point algorithm for convex minimization , 1991 .

[20]  Dimitri P. Bertsekas,et al.  Constrained Optimization and Lagrange Multiplier Methods , 1982 .

[21]  R. Rockafellar Monotone Operators and the Proximal Point Algorithm , 1976 .

[22]  M. J. D. Powell,et al.  A method for nonlinear constraints in minimization problems , 1969 .

[23]  Shih-Ping Han A parallel algorithm for a class of convex programs , 1988 .

[24]  R. Tyrrell Rockafellar,et al.  Augmented Lagrangians and Applications of the Proximal Point Algorithm in Convex Programming , 1976, Math. Oper. Res..

[25]  J. Hiriart-Urruty,et al.  Convex analysis and minimization algorithms , 1993 .

[26]  Michael A. Saunders,et al.  A projected Lagrangian algorithm and its implementation for sparse nonlinear constraints , 1982 .

[27]  J. Strodiot,et al.  A perturbed parallel decomposition method for a class of nonsmooth convex minimization problems , 1991 .

[28]  M. Hestenes Multiplier and gradient methods , 1969 .

[29]  Stephen M. Robinson,et al.  Scenario analysis via bundle decomposition , 1995, Ann. Oper. Res..

[30]  David Kendrick,et al.  GAMS, a user's guide , 1988, SGNM.

[31]  Masao Fukushima,et al.  A parallel descent algorithm for convex programming , 1996, Comput. Optim. Appl..

[32]  Tao Pham Dinh,et al.  Proximal Decomposition on the Graph of a Maximal Monotone Operator , 1995, SIAM J. Optim..

[33]  K. Kiwiel Proximal Minimization Methods with Generalized Bregman Functions , 1997 .

[34]  Masao Fukushima,et al.  Some Reformulations and Applications of the Alternating Direction Method of Multipliers , 1994 .

[35]  R. Tyrrell Rockafellar,et al.  Scenarios and Policy Aggregation in Optimization Under Uncertainty , 1991, Math. Oper. Res..

[36]  Masao Fukushima,et al.  Application of the alternating direction method of multipliers to separable convex programming problems , 1992, Comput. Optim. Appl..

[37]  Marc Teboulle,et al.  Convergence of Proximal-Like Algorithms , 1997, SIAM J. Optim..