What Is Optimization Toolbox?

Linearly Constrained ProblemsLinearly Constrained ProblemsLinear constraints complicate the situation described for unconstrainedminimization. However, the underlying ideas described previously can becarried through in a clean and efficient way. The large-scale methods inOptimization Toolbox generate strictly feasible iterates.• “Linear Equality Constraints” on page 4-9• “Box Constraints” on page 4-10Linear Equality ConstraintsThe general linear equality constrained minimization problem can be writtenwhere is an m-by-n matrix( ). Optimization Toolbox preprocessesto remove strict linear dependencies using a technique based on theLU-factorization of [6]. Here isassumedtobeofrankm.(4-5)The method used to solve Equation 4-5 differs from the unconstrainedapproach in two significant ways. First, an initial feasible point iscomputed, using a sparse least-squares step, so that . Second,Algorithm PCG is replaced with Reduced Preconditioned Conjugate Gradients(RPCG), see [6], in order to compute an approximate reduced Newton step(or a direction of negative curvature in the null space of ). The key linearalgebra step involves solving systems of the form(4-6)where approximates (small nonzeros of are set to zero provided rank isnot lost) and is a sparse symmetric positive-definite approximation to H,i.e., .See[6] for more details.4-9