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The linearly constrained convex composite programming problems whoseobjective function contains two blocks with each block being the form ofnonsmooth+smooth arises frequently in multiple fields of applications. If bothof the smooth terms are quadratic, this problem can be solved efficiently byusing the symmetric Gaussian-Seidel (sGS) technique based proximal alternatingdirection method of multipliers (ADMM). However, in the non-quadratic case, thesGS technique can not be used any more, which leads to the separable structureof nonsmooth+smooth had to be ignored. In this paper, we present a generalizedADMM and particularly use a majorization technique to make the correspondingsubproblems more amenable to efficient computations. Under some appropriateconditions, we prove its global convergence for the relaxation factor in$(0,2)$. We apply the algorithm to solve a kind of simulated convex compositeoptimization problems and a type of sparse inverse covariance matrix estimationproblems which illustrates that the effectiveness of the algorithm are obvious.

A Majorized-Generalized Alternating Direction Method of Multipliers for Convex Composite Programming