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

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 Newton-type methods. It is based on an iteration of the form x <Superscript> k+1</Superscript>=[x <Superscript> k </Superscript>+α <Superscript> k </Superscript> d <Superscript> k </Superscript>]<Superscript>♯</Superscript>, where α <Superscript> k </Superscript> is the steplength, d <Superscript> k </Superscript> is the search direction and [⋅]<Superscript>♯</Superscript> 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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$d_{N}^{k}$</EquationSource> </InlineEquation> of d <Superscript> k </Superscript> corresponding to the free variables are determined by a truncated Newton method, and the components <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$d_{A}^{k}$</EquationSource> </InlineEquation> of d <Superscript> k </Superscript> corresponding to the active variables are computed by a Barzilai-Borwein gradient method. The steplength α <Superscript> k </Superscript> is computed by an adaptation of the nonmonotone stabilization technique proposed in Grippo et al. (Numer. Math. 59:779–805, <CitationRef CitationID="CR13">1991</CitationRef>). The method is a feasible one, since it maintains feasibility of the iterates x <Superscript> k </Superscript>, and is well suited for large-scale problems, since it uses matrix-vector products only in the truncated Newton method for computing <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$d_{N}^{k}$</EquationSource> </InlineEquation>. 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. Copyright Springer Science+Business Media New York 2012