Approximate solution of the trust region problem by minimization over two-dimensional subspaces (Q1107449)

The following problem is considered: \(\min \{g^ Td+(1/2)d^ TBd\); \(\| d\| \leq \Delta \}\), where \(g\in R^ n\), \(B\in R^{n\times n}\) is symmetric, and \(\Delta >0\). Problems of this type arise in trust region algorithms for unconstrained optimization. In a previous paper [SIAM J. Numer. Anal. 22, 47-67 (1985; Zbl 0574.65061)] the authors introduced an approximate solution technique for this problem that involves the solution of a two-dimensional trust region problem. The present paper reports computational results and its main purpose is to show perhaps surprising computational evidence that minimization over a subspace spanned by two reasonably chosen directions gives in many cases the value obtained in exact minimization over \(R^ n\).