Algorithmic approaches to avoiding bad local minima in nonconvex inconsistent feasibility

TL;DR

This paper investigates how projection-based algorithms can be tuned to avoid bad local minima in nonconvex inconsistent feasibility problems, with applications in orbital tomographic imaging, demonstrating that some methods better escape local minima and approach global optima.

Contribution

It compares the effectiveness of cyclic projections, relaxed Douglas-Rachford, and splitting algorithms in avoiding bad local minima in nonconvex problems, with practical validation in imaging.

Findings

Relaxed Douglas-Rachford splitting can escape bad local minima more effectively.

Cyclic algorithms converge faster but may get trapped in local minima.

Theoretical results are validated with experimental ARPES imaging data.

Abstract

The main challenge of nonconvex optimization is to find a global optimum, or at least to avoid ``bad'' local minima and meaningless stationary points. We study here the extent to which algorithms, as opposed to optimization models and regularization, can be tuned to accomplish this goal. The model we consider is a nonconvex, inconsistent feasibility problem with many local minima, where these are points at which the gaps between the sets are smallest on neighborhoods of these points. The algorithms that we compare are all projection-based algorithms, specifically cyclic projections, the cyclic relaxed Douglas-Rachford algorithm, and relaxed Douglas-Rachford splitting on the product space. The local convergence and fixed points of these algorithms have already been characterized in pervious theoretical studies. We demonstrate the theory for these algorithms in the context of orbital tomographic imaging from angle-resolved photon emission spectroscopy (ARPES) measurements, both synthetically generated and experimental. Our results show that, while the cyclic projections and cyclic relaxed Douglas-Rachford algorithms generally converge the fastest, the method of relaxed Douglas-Rachford splitting on the product space does move away from bad local minima of the other two algorithms, settling eventually on clusters of local minima corresponding to globally optimal critical points.