Douglas–Rachford splitting for nonconvex optimization with application to nonconvex feasibility problems

Abstract

We adapt the Douglas-Rachford (DR) splitting method to solve nonconvex feasibility problems by studying this method for a class of nonconvex optimization problem. While the convergence properties of the method for convex problems have been well studied, far less is known in the nonconvex setting. In this paper, for the direct adaptation of the method to minimize the sum of a proper closed function g and a smooth function f with a Lipschitz continuous gradient, we show that if the step-size parameter is smaller than a computable threshold and the sequence generated has a cluster point, then it gives a stationary point of the optimization problem. Convergence of the whole sequence and a local convergence rate are also established under the additional assumption that f and g are semi-algebraic. We also give simple sufficient conditions guaranteeing the boundedness of the sequence generated. We then apply our nonconvex DR splitting method to finding a point in the intersection of a closed convex set C and a general closed set D by minimizing the squared distance to C subject to D. We show that if either set is bounded and the step-size parameter is smaller than a computable threshold, then the sequence generated from the DR splitting method is actually bounded. Consequently, the sequence generated will have cluster points that are stationary for an optimization problem, and the whole sequence is convergent under an additional assumption that C and D are semi-algebraic. We achieve these results based on a new merit function constructed particularly for the DR splitting method. Our preliminary numerical results indicate that our DR splitting method usually outperforms the alternating projection method in finding a sparse solution of a linear system, in terms of both the solution quality and the number of iterations taken.

Computer Assisted Localization of a Heart Arrhythmia

“…Nonconvex feasibility problems can be solved using optimization techniques such as alternating optimization methods or Douglas-Rachford splitting [2].…”

“…• Any x s satisfying (2) gives a global minimizer of (4) with objective value 0, by w i = x s − x i . • Any solution with zero objective value gives x s feasible with respect to (2) or (3). Problem (4) may have a nonzero optimal value, in which case the 'relaxed' solution x s will not satisfy the original formulation.…”

“…The data collection process by a diagnostic catheter is inherently noisy, and some trial points give anomalous timing data. These anomalies then give incorrect information about the feasibility region (2). To make the method more robust, we detect and remove potential outliers in the course of solving each optimization problem (4).…”

“…Left: simple ventricle dataset with color-coded arrival times. Right: distance to source as a function of touches using simple feasible region (2). In this simple dataset, the proposed approach quickly finds the source.…”

Accelerating the DC algorithm for smooth functions

“…Algorithms for minimising composite functions have been extensively investigated and found applications to many problems such as: inverse covariance estimate, logistic regression, sparse least squares and feasibility problems, see e.g. [9,14,15,19] and the references quoted therein.…”