Minimizing Sum of Truncated Convex Functions and Its Applications

ABSTRACT

In this article, we study a class of problems where the sum of truncated convex functions is minimized. In statistical applications, they are commonly encountered when ℓ₀-penalized models are fitted and usually lead to NP-Hard non-convex optimization problems. In this article, 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.

KEYWORDS:

• ℓ₀ penalty
• Non-convex optimization
• NP-Hard
• Outlier detection
• signal and image restoration

Acknowledgments

We thank David Eberly for his suggestion on the algorithm for finding intersections of ellipses. We thank the two anonymous reviewers and the associate editor for their suggestions on the image restoration application and the extension to high-dimensional settings. Their comments and suggestions have helped us improve the quality of this article substantially.