What Is Optimization Toolbox?

3 Standard AlgorithmsThe functions use one of two line search strategies, depending on whethergradient information is readily available or whether it must be calculatedusing a finite difference method:• When gradient information is available, the default is to use a cubicpolynomial method.• When gradient information is not available, the default is to use a mixedcubic and quadratic polynomial method.Cubic Polynomial MethodIn the proposed cubic polynomial method, a gradient and a function evaluationaremadeateveryiterationk. At each iteration an update is performed whena new point is found, , that satisfies the condition(3-12)At each iteration a step,, is attempted to form a new iterate of the form(3-13)If this step does not satisfy the condition (Equation 3-12), then is reducedto form a new step, . The usual method for this reduction is to usebisection, i.e., to continually halve the step length until a reduction is achievedin f(x). However, this procedure is slow when compared to an approachthat involves using gradient and function evaluations together with cubicinterpolation/extrapolation methods to identify estimates of step length.When a point is found that satisfies the condition (Equation 3-12), an updateis performed if is positive. If it is not, then further cubic interpolationsare performed until the univariate gradient termis sufficientlysmall so that is positive.It is usual practice to reset to unity after every iteration. However, notethat the quadratic model (Equation 3-3) is generally only a good one nearto the solution point. Therefore, is modified at each major iterationto compensate for the case when the approximation to the Hessian ismonotonically increasing or decreasing. To ensure that, as approachesthe solution point, the procedure reverts to a value of close to unity, the3-12