Minimizing Sum of Truncated Convex Functions and Its Applications

TL;DR

This paper introduces a new algorithm for globally minimizing sums of truncated convex functions, addressing NP-hard problems in low-dimensional cases and providing efficient approximate solutions in high-dimensional settings, with applications in statistics and image restoration.

Contribution

It presents a novel algorithm for global minimization of truncated convex functions, applicable to both low- and high-dimensional problems, with demonstrated effectiveness in various applications.

Findings

The proposed algorithm finds the global minimum in low-dimensional problems.

An efficient approximate solution is available for high-dimensional cases.

The method outperforms existing algorithms in simulations and real image restoration tasks.

Abstract

In this paper, we study a class of problems where the sum of truncated convex functions is minimized. In statistical applications, they are commonly encountered when $\ell_0$-penalized models are fitted and usually lead to NP-Hard non-convex optimization problems. In this paper, we propose a general algorithm for the global minimizer in low-dimensional settings. We also extend the algorithm to high-dimensional settings, where an approximate solution can be found efficiently. We introduce several applications where the sum of truncated convex functions is used, compare our proposed algorithm with other existing algorithms in simulation studies, and show its utility in edge-preserving image restoration on real data.