A New Active Set Algorithm for Box Constrained Optimization

• Published in: SIAM journal on optimization Vol. 17; no. 2; pp. 526 - 557
• Main Authors: Hager, William W., Zhang, Hongchao
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2006
• Subjects: Journal bearings; Methods; Optimization algorithms
• ISSN: 1052-6234; 1095-7189
• DOI: 10.1137/050635225

An active set algorithm (ASA) for box constrained optimization is developed. The algorithm consists of a nonmonotone gradient projection step, an unconstrained optimization step, and a set of rules for branching between the two steps. Global convergence to a stationary point is established. For a nondegenerate stationary point, the algorithm eventually reduces to unconstrained optimization without restarts. Similarly, for a degenerate stationary point, where the strong second-order sufficient optimality condition holds, the algorithm eventually reduces to unconstrained optimization without restarts. A specific implementation of the ASA is given which exploits the recently developed cyclic Barzilai-Borwein (CBB) algorithm for the gradient projection step and the recently developed conjugate gradient algorithm CG_DESCENT for unconstrained optimization. Numerical experiments are presented using box constrained problems in the CUTEr and MINPACK-2 test problem libraries.