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This paper is concerned with two-block separable convex minimization problemswith linear constraints, for which it is either impossible or too expensive toobtain the exact solutions of the subproblems involved in the proximal ADMM(alternating direction method of multipliers). Such structured convexminimization problems often arise from the two-block regroup settlement ofthree or four-block separable convex optimization problems with linearconstraints, or from the constrained total-variation superresolution imagereconstruction problems in image processing. For them, we propose an inexactindefinite proximal ADMM of step-size $\tau\in\!(0,\frac{\sqrt{5}+1}{2})$ withtwo easily implementable inexactness criteria to control the solution accuracyof subproblems, and establish the convergence under a mild assumption onindefinite proximal terms. We apply the proposed inexact indefinite proximalADMMs to the three or four-block separable convex minimization problems withlinear constraints, which are from the duality of the important class of doublynonnegative semidefinite programming (DNNSDP) problems with many linearequality and/or inequality constraints. Numerical results indicate that theinexact indefinite proximal ADMM with the absolute error criterion has acomparable performance with the directly extended multi-block ADMM of step-size$\tau=1.618$ without convergence guarantee, whether in terms of the number ofiterations or the computation time.

Inexact indefinite proximal ADMMs for 2-block separable convex programs and applications to 4-block DNNSDPs