What Is Optimization Toolbox?

Least-Squares OptimizationIn problems of this kind, the residual is likely to be small at theoptimum since it is general practice to set realistically achievable targettrajectories. Although the function in LS can be minimized using a generalunconstrained minimization technique, as described in “UnconstrainedOptimization” on page 3-6, certain characteristics of the problem can often beexploited to improve the iterative efficiency of the solution procedure. Thegradient and Hessian matrix of LS have a special structure.Denoting the m-by-n Jacobian matrix of F(x) asJ(x), the gradient vector off(x) as , the Hessian matrix of f(x) as , and the Hessian matrix ofeach as ,youhavewhere(3-17)The matrix Q(x) has the property that when the residual tends to zeroas approaches the solution, then Q(x) alsotendstozero.Thuswhenissmallatthesolution,averyeffectivemethodistousetheGauss-Newtondirection as a basis for an optimization procedure.3-19