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Applications of splitting algorithm to decomposition in convex programming and variational inequalities

Recently Han and Lou proposed a highly parallelizable decomposition algorithm for minimizing a strongly convex cost over the intersection of closed convex sets. It is shown that their algorithm is in fact a special case of a splitting algorithm analyzed by Gabay for finding a zero of the sum of two maximal monotone operators. Gabay’s convergence analysis for the splitting algorithm is sharpened, and new applications of this algorithm to variational inequalities, convex programming, and the solution of linear complementarily problems are proposed. For convex programs with a certain separable structure, a multiplier method that is closely related to the alternating direction method of multipliers of Gabay–Mercier and of Glowinski–Marrocco, but which uses both ordinary and augmented Lagrangians, is obtained.