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AN APPROXIMATE ALTERNATING LINEARIZATION DECOMPOSITION METHOD  

Li, Dan (Institute of Operations Research and Control, School of Mathematical Sciences, Dalian University of Technology (DUT))
Pang, Li-Ping (Institute of Operations Research and Control, School of Mathematical Sciences, Dalian University of Technology (DUT))
Xia, Zun-Quan (Institute of Operations Research and Control, School of Mathematical Sciences, Dalian University of Technology (DUT))
Publication Information
Journal of applied mathematics & informatics / v.28, no.5_6, 2010 , pp. 1249-1262 More about this Journal
Abstract
An approximate alternating linearization decomposition method, for minimizing the sum of two convex functions with some separable structures, is presented in this paper. It can be viewed as an extension of the method with exact solutions proposed by Kiwiel, Rosa and Ruszczynski(1999). In this paper we use inexact optimal solutions instead of the exact ones that are not easily computed to construct the linear models and get the inexact solutions of both subproblems, and also we prove that the inexact optimal solution tends to proximal point, i.e., the inexact optimal solution tends to optimal solution.
Keywords
convex programming; decomposition method; proximal point method; approximate subdifferential;
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1 P. Tseng, Applications of a splitting algorithm to decomposition in convex programming and variational inequalities, SIAM J. Control Optim. 29 (1991) 119-138.   DOI
2 J. E. Spingarn, Applications of the method of partial inverses to convex programming:Decomposition, Math. Programming 32 (1985) 199-223.   DOI   ScienceOn
3 P. Tseng, Further applications of a splitting algorithm to decomposition in variational inequalities and convex programming, Math. Programming 48 (1990) 249-263.   DOI
4 P. Mahey and P. D. Tao, Partial regularization of the sum of two maximal monotoneoperators, RAIRO Model. Math. Anal. Number 27 (1993) 375-392.
5 R. T. Rockafellar, Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Math. Oper. Res. 1 (1976) 97-116.   DOI   ScienceOn
6 J. Eckstein and D. P. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Math. Programming 55 (1992) 293-318.   DOI
7 J. Eckstein and M. Fukushima, Some reformulations and applications of the alternating direction method of multipliers, in Large Scale Optimization: State of the Art, W. W. Hager, D. W. Hearn, and P. M. Pardalos, eds., Kluwer, Dordrecht, the Netherlands, 1994, 115-134.
8 A. Ruszczynski, Some advances in decomposition methods for stochastic linear programming Annals of Operations Research 85(1999) 153-172.
9 R. T. Rockafellar, Convex Analysis, Princeton University Press, NJ, 1970.
10 R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM. Control Optim. 14 (1976), 877-898.   DOI   ScienceOn
11 K. C. Kiwiel, C.H. Rosa and A. Ruszczynski, Proximal decomposition via alternating linearization, SIAM Journal on Optimization 9 (1999) 153-172.
12 P. L. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM Journal on Numerical Analysis 16 (1979) 964-979.   DOI   ScienceOn
13 J. Eckstein, Approximate iterations in Bregman-function-based proximal algorithms, Mathematical. Programming 83 1998, 113-123.
14 J. F. Benders, Partitioning procedures for solving mixed-variable programming problems, Numerische Mathematik 4 (1962) 238-252.   DOI
15 D. P. Bertsekas and J. N. Tsitsiklis, Parallel and Distributed Computation: Numerical-Methods, Prentice-Hall, Englewood Cliffs, NJ, 1989.
16 M. Fukushima, Application of the alternating direction method of multipliers to separable convex programming problems, Comput. Optim. Appl. 1 (1992) 93-111.   DOI   ScienceOn
17 D. Gabay, Applications of the method of multipliers to variational inequalities, in Augmented Lagrangian Methods: Applications to the Solution of Boundary Value Problems, M. Fortin and R. Glowinski, eds., North-Holland, Amsterdam, 1983, 299-331.
18 Hiriart-Urruty, Jean-Baptiste and Lemarchal.Claude, Convex Analysis and Minimization Algorithms, Two volumes, Springer Verlag, Heidelberg, 1993.
19 G. B. Dantzig, P. Wolfe, The decomposition algorithm for linear programs, Econometrica, Vol. 29 No. 4 (1961) 767-778.   DOI   ScienceOn