What Is Optimization Toolbox?

Constrained OptimizationGiven the problem description in GP (Equation 3-1) the principal idea is theformulation of a QP subproblem based on a quadratic approximation of theLagrangian function.(3-24)Here you simplify Equation 3-1 by assuming that bound constraints havebeen expressed as inequality constraints. You obtain the QP subproblem bylinearizing the nonlinear constraints.Quadratic Programming (QP) Subproblem(3-25)This subproblem can be solved using any QP algorithm (see, for instance,“Quadratic Programming Solution” on page 3-34). The solution is used toform a new iterateThe step length parameter is determined by an appropriate line searchprocedure so that a sufficient decrease in a merit function is obtained (see“Updating the Hessian Matrix” on page 3-33). The matrix is a positivedefinite approximation of the Hessian matrix of the Lagrangian function(Equation 3-24). can be updated by any of the quasi-Newton methods,although the BFGS method (see “Updating the Hessian Matrix” on page 3-33)appears tobethemostpopular.Anonlinearly constrained problem can often be solved in fewer iterationsthan an unconstrained problem using SQP. One of the reasons for this isthat, becauseoflimitsonthefeasiblearea,theoptimizercanmakeinformeddecisions regarding directions of search and step length.3-31