A 2-BFGS updating in a trust region framework

Marianna S. Apostolopoulou (msamath.upatras.gr)
Dimitris G. Sotiropoulos (dgsionio.gr)
Panagiotis Pintelas (pintelasmath.upatras.gr)

Abstract: We present a new matrix-free method for the trust region subproblem, assuming that the approximate Hessian is updated by the limited memory BFGS formula with m = 2. The resulting updating scheme, called 2-BFGS, give us the ability to determine via simple formulas the eigenvalues of the resulting approximation. Thus, at each iteration, we can 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. Thus, the computation of the trial step can be 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.

Keywords: large scale problems, unconstrained optimization, trust-region subproblem, limited memory BFGS, eigenvalues

Category 1: Nonlinear Optimization

Category 2: Nonlinear Optimization (Unconstrained Optimization )

Citation: Technical Report No. TR07-01, University of Patras, Department of Mathematics, June 2007.

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Entry Submitted: 07/15/2007
Entry Accepted: 07/15/2007
Entry Last Modified: 07/16/2007

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