Abstract
This work is about a splitting approach for solving separably smooth nonconvex linearly constrained optimization problems. Based on the ideas from two classical methods, namely the sequential quadratic programming (SQP) and the alternating direction method of multipliers (ADMM), we propose an ADMM-based SQP method. We focus on decomposing the quadratic programming (QP) subproblem of the primal problem into small-scale QP subproblems, which further embedded with Bregman distances can be solved effectively and followed by a dual ascent type update for the Lagrangian multipliers. Under suitable conditions as well as the crucial Kurdyka–Łojasiewicz property, we establish the global and strong convergence properties of the proposed method.
Highlights
Nonconvex optimization problems arise in a variety of applications ranging from the fields of signal and image processing, machine learning [1]
In this paper, motivated by the ideas of the splitting scheme applied to the quadratic programming (QP) subproblem in [22], and of the Bregman modification of alternating direction method of multipliers (ADMM) in [10], we focus on QP
2 Preliminaries and ADMM-based sequential quadratic programming (SQP) method we provide some preliminaries that are useful in the sequel, and describe the ADMM-based SQP method in detail
Summary
Nonconvex optimization problems arise in a variety of applications ranging from the fields of signal and image processing, machine learning [1]. We consider the following nonconvex optimization problems with linear constraints and a separable objective function: min f (x) + g(y), x,y s.t. Ax + By = b, where f : Rn1 → R and g : Rn2 → R are continuously differentiable, but not necessarily convex, matrices A ∈ Rm×n1 , B ∈ Rm×n2 and the vector b ∈ Rm are assumed to be given. Remark 1 At first glance, one might view the ADMM-based SQP method proposed in this paper as a special case of the well-known Bregman ADMM [10], whose iterative scheme is given as follows:
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