Computational Optimization and Applications

, Volume 53, Issue 2, pp 395–423 | Cite as

An active set feasible method for large-scale minimization problems with bound constraints

Article

Abstract

We are concerned with the solution of the bound constrained minimization problem {minf(x), lxu}. For the solution of this problem we propose an active set method that combines ideas from projected and nonmonotone Newton-type methods. It is based on an iteration of the form xk+1=[xk+αkdk], where αk is the steplength, dk is the search direction and [⋅] is the projection operator on the set [l,u]. At each iteration a new formula to estimate the active set is first employed. Then the components \(d_{N}^{k}\) of dk corresponding to the free variables are determined by a truncated Newton method, and the components \(d_{A}^{k}\) of dk corresponding to the active variables are computed by a Barzilai-Borwein gradient method. The steplength αk is computed by an adaptation of the nonmonotone stabilization technique proposed in Grippo et al. (Numer. Math. 59:779–805, 1991). The method is a feasible one, since it maintains feasibility of the iterates xk, and is well suited for large-scale problems, since it uses matrix-vector products only in the truncated Newton method for computing \(d_{N}^{k}\). We prove the convergence of the method, with superlinear rate under usual additional assumptions. An extensive numerical experimentation performed on an algorithmic implementation shows that the algorithm compares favorably with other widely used codes for bound constrained problems.

Keywords

Bound constrained minimization problems Large-scale minimization problems Active set methods Projected Newton-type methods Nonmonotone Newton-type methods Barzilai-Borwein gradient methods 

Supplementary material

10589_2012_9506_MOESM1_ESM.pdf (90 kb)
Computational Results for NMBC_1, NMBC_2, GENCAN, LANCELOT B, ASA_CG (PDF 90 kB)

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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Dipartimento di Informatica e SistemisticaSapienza Università di RomaRomaItaly

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