What Is Optimization Toolbox?

4 Large-Scale AlgorithmsEquation 4-2 (see [8]); such algorithms typically involve the computation of afull eigensystem and a Newton process applied to the secular equationSuch algorithms provide an accurate solution to Equation 4-2. However,they require time proportional to several factorizations of H. Therefore, forlarge-scale problems a different approach is needed. Several approximationand heuristic strategies, based on Equation 4-2, have been proposed in theliterature ([2] and [10]). The approximation approach followed in OptimizationToolbox is to restrict the trust-region subproblem to a two-dimensionalsubspace ([1] and [2]). Once the subspace has been computed, the work tosolve Equation 4-2 is trivial even if full eigenvalue/eigenvector informationis needed (since in the subspace, the problem is only two-dimensional). Thedominant work has now shifted to the determination of the subspace.The two-dimensional subspace is determined with the aid of apreconditioned conjugate gradient process described below. The toolboxassigns ,where is in the direction of the gradient g, and iseither an approximate Newton direction, i.e., a solution toor a direction of negative curvature,The philosophy behind this choice of is to force global convergence (via thesteepest descent direction or negative curvature direction) and achieve fastlocal convergence (via the Newton step, when it exists).A framework for Optimization Toolbox approach to unconstrainedminimization using trust-region ideas is now easy to describe:1 Formulate the two-dimensional trust-region subproblem.2 Solve Equation 4-2 to determine the trial step .3 If ,then .(4-3)(4-4)4-4