Douglas-Rachford splitting for nonconvex optimization with application to nonconvex feasibility problems
This work addresses convergence guarantees for a widely used optimization method in nonconvex settings, which is incremental but important for practitioners in fields like signal processing and machine learning dealing with nonconvex constraints.
The paper tackles the adaptation of the Douglas-Rachford splitting method to nonconvex optimization, proving convergence to stationary points under specific conditions and applying it to nonconvex feasibility problems. Preliminary numerical results show it outperforms alternating projection in finding sparse solutions of linear systems, with better solution quality and fewer iterations.
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.