Extensions to the Proximal Distance Method of Constrained Optimization.

Abstract

The current paper studies the problem of minimizing a loss f(x) subject to constraints of the form Dx ∈ S, where S is a closed set, convex or not, and D is a matrix that fuses parameters. Fusion constraints can capture smoothness, sparsity, or more general constraint patterns. To tackle this generic class of problems, we combine the Beltrami-Courant penalty method of optimization with the proximal distance principle. The latter is driven by minimization of penalized objectives $f\left(x\right)+\frac{\rho }{2}\text{dist}{(Dx,S)}^2$ involving large tuning constants ρ and the squared Euclidean distance of Dx from S. The next iterate x_n+1 of the corresponding proximal distance algorithm is constructed from the current iterate x_n by minimizing the majorizing surrogate function $f\left(x\right)+\frac{\rho }{2}{\Vert Dx-{𝒫}_S\left(D{x}_n\right)\Vert }^2$. For fixed ρ and a subanalytic loss f(x) and a subanalytic constraint set S, we prove convergence to a stationary point. Under stronger assumptions, we provide convergence rates and demonstrate linear local convergence. We also construct a steepest descent (SD) variant to avoid costly linear system solves. To benchmark our algorithms, we compare their results to those delivered by the alternating direction method of multipliers (ADMM). Our extensive numerical tests include problems on metric projection, convex regression, convex clustering, total variation image denoising, and projection of a matrix to a good condition number. These experiments demonstrate the superior speed and acceptable accuracy of our steepest variant on high-dimensional problems. Julia code to replicate all of our experiments can be found at https://github.com/alanderos91/ProximalDistanceAlgorithms.jl.