Incremental cutting-plane method and its application

TL;DR

This paper introduces an incremental regularized cutting-plane method for convex optimization, focusing on large-scale dual problems, demonstrating faster high-precision solutions compared to traditional methods.

Contribution

It proposes a novel incremental variant of regularized cutting-plane methods with limited-memory features and analyzes its convergence properties.

Findings

Incremental method achieves high-precision solutions faster.

Limited-memory approach effectively manages cut deletion.

Numerical experiments show improved efficiency on large-scale problems.

Abstract

We consider regularized cutting-plane methods to minimize a convex function that is the sum of a large number of component functions. One important example is the dual problem obtained from Lagrangian relaxation on a decomposable problem. In this paper, we focus on an incremental variant of the regularized cutting-plane methods, which only evaluates a subset of the component functions in each iteration. We first consider a limited-memory setup where the method deletes cuts after a finite number of iterations. The convergence properties of the limited-memory methods are studied under various conditions on regularization. We then provide numerical experiments where the incremental method is applied to the dual problems derived from large-scale unit commitment problems. In many settings, the incremental method is able to find a solution of high precision in a shorter time than the non-incremental method.