Solving the quadratic trust-region subproblem in a low-memory BFGS framework

Authors:
M. S. Apostolopoulou;D. G. Sotiropoulos;P. Pintelas
Affiliations:
Department of Mathematics, University of Patras, Patras, Greece;Department of Informatics, Ionian University, Corfu, Greece;Department of Mathematics, University of Patras, Patras, Greece
Venue:
Optimization Methods & Software - THE JOINT EUROPT-OMS CONFERENCE ON OPTIMIZATION, 4-7 JULY, 2007, PRAGUE, CZECH REPUBLIC, PART I
Year:
2008

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Abstract

We present a new matrix-free method for the large-scale trust-region subproblem, assuming that the approximate Hessian is updated by the L-BFGS formula with m=1 or 2. We determine via simple formulas the eigenvalues of these matrices and, at each iteration, we construct a positive definite matrix whose inverse can be expressed analytically, without using factorization. Consequently, a direction of negative curvature can be computed immediately by applying the inverse power method. The computation of the trial step is obtained by performing a sequence of inner products and vector summations. Furthermore, it immediately follows that the strong convergence properties of trust region methods are preserved. Numerical results are also presented.