What Is Optimization Toolbox?

Constrained OptimizationUpdating the Hessian MatrixAt each major iteration a positive definite quasi-Newton approximation of theHessian of the Lagrangian function, H, is calculated using the BFGS method,whereis an estimate of the Lagrange multipliers.where(3-27)Powell [35] recommends keeping the Hessian positive definite even thoughit might be positive indefinite at the solution point. A positive definiteHessian is maintained providing is positive at each update and that H isinitialized with a positive definite matrix. When is not positive, ismodified on an element-by-element basis so that . The general aimof this modification is to distort the elements of , which contribute to apositive definiteupdate,aslittleaspossible. Therefore, in the initial phase ofthe modification, the most negative element of is repeatedly halved.This procedure is continued until is greater than or equal to a smallnegative tolerance. If, after this procedure, is still not positive, modifyby adding a vector v multiplied by a constant scalar w, thatis,where(3-28)ifandotherwiseand increase w systematically untilbecomes positive.3-33