A Two-Stage Active-Set Algorithm for Bound-Constrained Optimization

TL;DR

This paper introduces a novel two-stage active-set algorithm for bound-constrained optimization that combines active-set estimation with a truncated-Newton approach, achieving global convergence and superlinear rates, supported by promising experimental results.

Contribution

The paper presents a new two-stage method integrating active-set estimates with a truncated-Newton strategy for bound-constrained optimization, ensuring convergence and efficiency.

Findings

Proves global convergence of the algorithm.

Shows superlinear convergence under standard assumptions.

Demonstrates effectiveness through experimental results.

Abstract

In this paper, we describe a two-stage method for solving optimization problems with bound constraints. It combines the active-set estimate described in [Facchinei and Lucidi, 1995] with a modification of the non-monotone line search framework recently proposed in [De Santis et al., 2012]. In the first stage, the algorithm exploits a property of the active-set estimate that ensures a significant reduction in the objective function when setting to the bounds all those variables estimated active. In the second stage, a truncated-Newton strategy is used in the subspace of the variables estimated non-active. In order to properly combine the two phases, a proximity check is included in the scheme. This new tool, together with the other theoretical features of the two stages, enables us to prove global convergence. Furthermore, under additional standard assumptions, we can show that the algorithm converges at a superlinear rate. Promising experimental results demonstrate the effectiveness of the proposed method.