An active set feasible method for largescale minimization problems with bound constraints
 M. De Santis,
 G. Di Pillo,
 S. Lucidi
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We are concerned with the solution of the bound constrained minimization problem {minf(x), l≤x≤u}. For the solution of this problem we propose an active set method that combines ideas from projected and nonmonotone Newtontype methods. It is based on an iteration of the form x ^{ k+1}=[x ^{ k }+α ^{ k } d ^{ k }]^{♯}, where α ^{ k } is the steplength, d ^{ k } 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 d ^{ k } corresponding to the free variables are determined by a truncated Newton method, and the components \(d_{A}^{k}\) of d ^{ k } corresponding to the active variables are computed by a BarzilaiBorwein gradient method. The steplength α ^{ k } 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 x ^{ k }, and is well suited for largescale problems, since it uses matrixvector 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.
Inside
Within this Article
 Introduction
 Estimation of active sets
 Optimality results
 Search direction
 The Non Monotone Bound Constrained (NMBC) algorithm
 Implementation of the search direction
 Computational results
 Conclusions
 References
 References
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 Title
 An active set feasible method for largescale minimization problems with bound constraints
 Journal

Computational Optimization and Applications
Volume 53, Issue 2 , pp 395423
 Cover Date
 20121001
 DOI
 10.1007/s1058901295067
 Print ISSN
 09266003
 Online ISSN
 15732894
 Publisher
 Springer US
 Additional Links
 Topics
 Keywords

 Bound constrained minimization problems
 Largescale minimization problems
 Active set methods
 Projected Newtontype methods
 Nonmonotone Newtontype methods
 BarzilaiBorwein gradient methods
 Industry Sectors
 Authors

 M. De Santis ^{(1)}
 G. Di Pillo ^{(1)}
 S. Lucidi ^{(1)}
 Author Affiliations

 1. Dipartimento di Informatica e Sistemistica, Sapienza Università di Roma, Via Ariosto, 25, 00185, Roma, Italy