What Is Optimization Toolbox?

3 Standard AlgorithmsThe functions fmincon, fminimax, fgoalattain, andfseminf all use SQP.If Display is set to 'iter' in options, then various information is givensuch as function values and the maximum constraint violation. When theHessian has to be modified using the first phase of the preceding procedure tokeep it positive definite, then Hessian modified is displayed. If the Hessianhas to be modified again using the second phase of the approach describedabove, then Hessian modified twice is displayed. When the QP subproblemis infeasible, then infeasible is displayed. Such displays are usually nota cause for concern but indicate that the problem is highly nonlinear andthat convergence might take longer than usual. Sometimes the message noupdate is displayed, indicating that is nearly zero. This can be anindication that the problem setup is wrong or you are trying to minimize anoncontinuous function.Quadratic Programming SolutionAt each major iteration of the SQP method, a QP problem of the followingform is solved, where refers to the ith row of the m-by-n matrix .The method used in Optimization Toolbox is an active set strategy (alsoknown as a projection method) similar to that of Gill et. al., described in[20] and [19]. It has been modified for both Linear Programming (LP) andQuadratic Programming (QP) problems.(3-29)The solution procedure involves two phases. The first phase involves thecalculation of a feasible point (if one exists). The second phase involves thegeneration of an iterative sequence of feasible points that converge to thesolution. In this method an active set, , is maintained that is an estimateof the active constraints (i.e., those that are on the constraint boundaries) atthe solution point. Virtually all QP algorithms are active set methods. Thispoint is emphasized because there exist many different methods that are verysimilarinstructurebutthataredescribed in widely different terms.3-34