Partial Proximal Minimization Algorithms for Convex Pprogramming

We consider an extension of the proximal minimization algorithm where only some of the minimization variables appear in the quadratic proximal term. We interpret the resulting iterates in terms of the iterates of the standard algorithm and we show a uniform descent property, which 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. We also show that partial proximal minimization algorithms are dual to multiplier methods with partial elimination of constraints, and we establish a relation between parallel proximal minimization algorithms and parallel constraint distribution algorithms. 1 Supported by the National Science Foundation under Grant DDM-8903385 and Grant CCR-9103804, and the Army Research Office under Grant DAAL03-86-K-0171. 2 Department of Electrical Engineering and Computer Science, M. I. T., Cambridge, Mass., 02139. 3 Department of Mathematics, Univ. of Washington, Seattle, Wash., 98195.