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Global Complexity Bound of a Proximal ADMM for Linearly-Constrained Nonseperable Nonconvex Composite Programming

Weiwei Kong (wwkong92***at***gmail.com)
Renato D.C. Monteiro (renato.monteiro***at***isye.gatech.edu)

Abstract: This paper proposes and analyzes a dampened proximal alternating direction method of multipliers (DP.ADMM) for solving linearly-constrained nonconvex optimization problems where the smooth part of the objective function is nonseparable. Each iteration of DP.ADMM consists of: (ii) a sequence of partial proximal augmented Lagrangian (AL) updates, (ii) an under-relaxed Lagrange multiplier update, and (iii) a novel test to check whether the penalty parameter of the AL function should be updated. Under a basic Slater point condition and some requirements on the dampening factor and under-relaxation parameter, it is shown that DP.ADMM obtains a first-order stationary point of the constrained problem in O(1/pł) iterations for a given numerical tolerance p > 0. One of the main novelties of the paper is that convergence of the method is obtained without requiring any rank assumptions on the constraint matrices.


Category 1: Nonlinear Optimization

Category 2: Convex and Nonsmooth Optimization (Nonsmooth Optimization )


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Entry Submitted: 10/24/2021
Entry Accepted: 10/24/2021
Entry Last Modified: 12/08/2021

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