What Is Optimization Toolbox?

Unconstrained OptimizationFigure 3-1: Steepest Descent Method on Rosenbrock’s Function (Eq. 3-2)This function, also known as the banana function, is notorious inunconstrained examples because of the way the curvature bends around theorigin. Rosenbrock’s function is used throughout this section to illustrate theuse of a variety of optimization techniques. The contours have been plottedin exponential increments because of the steepness of the slope surroundingthe U-shaped valley.Quasi-Newton MethodsOf the methods that use gradient information, the most favored are thequasi-Newton methods. These methods build up curvature information ateach iteration to formulate a quadratic model problem of the formwhere the Hessian matrix, H, is a positive definite symmetric matrix, c is aconstant vector, and b is a constant. The optimal solution for this problemoccurs when the partial derivatives of x go to zero, i.e.,(3-3)3-7