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On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. (English) Zbl 0765.90073
The authors show that the Douglas-Rachford splitting method for finding a zero of the sum of two monotone operators is a special case of the proximal point algorithm by means of an operator called a splitting operator. Therefore, applications of Douglas-Rachford splitting, such as the alternating direction method of multipliers for convex programming decomposition, are also special cases of the proximal point algorithm. This observation allows the unification and generalization of a variety of convex programming algorithms. By introducing a modified version of the proximal point algorithm, the authors derive a new, generalized alternating direction method of multipliers for convex programming. Advances of this sort illustrate the power and generality gained by adopting monotone operator theory as a conceptual framework.

MSC:
90C25Convex programming
49M29Methods involving duality in calculus of variations
47H05Monotone operators (with respect to duality) and generalizations
90C48Programming in abstract spaces
49J40Variational methods including variational inequalities
90-08Computational methods (optimization)
47N10Applications of operator theory in optimization, convex analysis, programming, economics
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