What Is Optimization Toolbox?

Nonlinear Systems of EquationsNonlinear Systems of Equations• “Introduction” on page 3-25• “Gauss-Newton Method” on page 3-25• “Trust-Region Dogleg Method” on page 3-25• “Nonlinear Equations Implementation” on page 3-27IntroductionSolving a nonlinear system of equations involves finding a solutionsuch that every equation in the nonlinear system is 0. That is, there areequations and unknowns. The objective is to find x is an element of then-dimensional real numbers such that whereThe assumption is that a zero, or root, of the system exists. These equationsmay represent economic constraints, for example, that must all be satisfied.Gauss-Newton MethodOneapproachtosolvingthisproblemistouseaNonlinearLeast-Squaressolver, such those described in “Least-Squares Optimization” on page 3-18.Since the assumption is the system has a root, it would have a small residual;therefore, using the Gauss-Newton Method is effective. In this case, eachiteration solves a linear least-squares problem, as described in Equation 3-18,to find the search direction. (See “Gauss-Newton Method” on page 3-20 formore information.)Trust-Region Dogleg MethodAnother approach is to solve a linear system of equations to find the searchdirection, namely, Newton’s method says to solve for the search directionsuch that3-25