Partial Proximal Minimization Algorithms for Convex Pprogramming

An extension of the proximal minimization algorithm is considered where only some of the minimization variables appear in the quadratic proximal term. The resulting iterates are interpreted in terms of the iterates of the standard algorithm, and a uniform descent property is shown that holds independently of the proximal terms used. This property is used to give simple convergence proofs of parallel algorithms where multiple processors simultaneously execute proximal iterations using different partial proximal terms. It is also shown that partial proximal minimization algorithms are dual to multiplier methods with partial elimination of constraints, and a relation is established between parallel proximal minimization algorithms and parallel constraint distribution algorithms.

[1]  R. Tyrrell Rockafellar,et al.  Convex Analysis , 1970, Princeton Landmarks in Mathematics and Physics.

[2]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

[3]  D. Bertsekas,et al.  A new penalty function method for constrained minimization , 1972, CDC 1972.

[4]  R. Tyrrell Rockafellar,et al.  A dual approach to solving nonlinear programming problems by unconstrained optimization , 1973, Math. Program..

[5]  Dimitri P. Bertsekas,et al.  Necessary and sufficient conditions for a penalty method to be exact , 1975, Math. Program..

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

[7]  D. Bertsekas,et al.  Combined Primal–Dual and Penalty Methods for Convex Programming , 1976 .

[8]  E. G. Gol'shtein,et al.  Modified Lagrangians in Convex Programming and their Generalizations , 1979 .

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

[10]  D. Bertsekas,et al.  TWO-METRIC PROJECTION METHODS FOR CONSTRAINED OPTIMIZATION* , 1984 .

[11]  C. Ha A generalization of the proximal point algorithm , 1987, 26th IEEE Conference on Decision and Control.

[12]  Jong-Shi Pang,et al.  A Posteriori Error Bounds for the Linearly-Constrained Variational Inequality Problem , 1987, Math. Oper. Res..

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

[14]  J. Dunn Formal augmented Newtonian projection methods for continuous-time optimal control problems , 1989, Proceedings of the 28th IEEE Conference on Decision and Control,.

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

[16]  M C Ferris,et al.  Parallel Constraint Distribution , 1991, SIAM J. Optim..

[17]  Michael C. Ferris,et al.  Finite termination of the proximal point algorithm , 1991, Math. Program..

[18]  P. Tseng,et al.  On the linear convergence of descent methods for convex essentially smooth minimization , 1992 .

[19]  Y. Censor,et al.  On the proximal minimization algorithm with D-Functions , 1992 .

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

[21]  Jonathan Eckstein,et al.  Nonlinear Proximal Point Algorithms Using Bregman Functions, with Applications to Convex Programming , 1993, Math. Oper. Res..

[22]  Paul Tseng,et al.  On the convergence of the exponential multiplier method for convex programming , 1993, Math. Program..

[23]  Paul Tseng,et al.  On the Convergence Rate of Dual Ascent Methods for Linearly Constrained Convex Minimization , 1993, Math. Oper. Res..

[24]  Marc Teboulle,et al.  Convergence Analysis of a Proximal-Like Minimization Algorithm Using Bregman Functions , 1993, SIAM J. Optim..

[25]  Michael C. Ferris Parallel Constraint Distribution in Convex Quadratic Programming , 1994, Math. Oper. Res..