What Is Optimization Toolbox?

3 Standard Algorithms(3-4)The optimal solution point,,canbewrittenasNewton-type methods (as opposed to quasi-Newton methods) calculate Hdirectly and proceed in a direction of descent to locate the minimum after anumber of iterations. Calculating H numerically involves a large amountof computation. Quasi-Newton methods avoid this by using the observedbehavior of f(x) and to build up curvature information to make anapproximation to H using an appropriate updating technique.(3-5)A large number of Hessian updating methods have been developed. However,the formula of Broyden [3], Fletcher [14], Goldfarb [22], and Shanno [39](BFGS) is thought to be the most effective for use in a General Purposemethod.The formula given by BFGS iswhere(3-6)As a starting point, can be set to any symmetric positive definite matrix,for example, the identity matrix I. To avoid the inversion of the Hessian H,you can derive an updating method that avoids the direct inversion of H byusing a formula that makes an approximation of the inverse Hessian ateach update. A well-known procedure is the DFP formula of Davidon [9],Fletcher,andPowell[16].ThisusesthesameformulaastheBFGSmethod(Equation 3-6) except that is substituted for .The gradient information is either supplied through analytically calculatedgradients, or derived by partial derivatives using a numerical differentiationmethod via finite differences. This involves perturbing each of the design3-8