A Semideenite Framework for Trust Region Subproblems with Applications to Large Scale Minimization

Primal-dual pairs of semideenite programs provide a general framework for the theory and algorithms for the trust region subproblem (TRS). This latter problem consists in minimizing a general quadratic function subject to a convex quadratic constraint and, therefore, it is a generalization of the minimum eigenvalue problem. The importance of (TRS) is due to the fact that it provides the step in trust region minimization algorithms. The semideenite framework is studied as an interesting instance of semideenite programming as well as a tool for viewing known algorithms and deriving new algorithms for (TRS). In particular, a dual simplex type method is studied that solves (TRS) as a parametric eigenvalue problem. This method uses the Lanczos algorithm for the smallest eigenvalue as a black box. Therefore, the essential cost of the algorithm is the matrix-vector multiplication and, thus, sparsity can be exploited. A primal simplex type method provides steps for the so-called hard case. Extensive numerical tests for This is an abbreviated revision of the University of Waterloo research report CORR 94-32. 1 large sparse problems are discussed. These tests show that the cost of the algorithm is 1 + (n) times the cost of nding a minimum eigen-value using the Lanczos algorithm, where 0 < (n) < 1 is a fraction which decreases as the dimension increases.