. Motivated by block partitioned problems arising from group sparsity representation
and generalized noncooperative potential games, this paper presents a basic decomposition method
for a broad class of multiblock nonsmooth optimization problems subject to coupled linear constraints on the variables that may additionally be individually constrained. The objective of such an
optimization problem is given by the sum of two nonseparable functions minus a sum of separable,
pointwise maxima of finitely many convex differentiable functions. One of the former two nonseparable functions is of the class LC1
, i.e., differentiable with a Lipschitz gradient, while the other
summand is multiconvex. The subtraction of the separable, pointwise maxima of convex functions
induces a partial difference-of-convex (DC) structure in the overall objective; yet with all three terms
together, the objective is nonsmooth and non-DC, but is blockwise directionally differentiable. By
taking advantage of the (negative) pointwise maximum structure in the objective, the developed
algorithm and its convergence result are aimed at the computation of a blockwise directional stationary solution, which arguably is the sharpest kind of stationary solutions for this class of nonsmooth
problems. This aim is accomplished by combining the alternating direction method of multipliers
(ADMM) with a semilinearized Gauss–Seidel scheme, resulting in a decomposition of the overall
problem into subproblems each involving the individual blocks. To arrive at a stationary solution of
the desired kind, our algorithm solves multiple convex subprograms at each iteration, one per convex
function in each pointwise maximum. In order to lessen the potential computational burden in each
iteration, a probabilistic version of the algorithm is presented and its almost sure convergence is
established.