Optimization 1: BFGS 1. Write a MATLAB function BFGS.m that implements the ba- sic BFGS algorithm on

Optimization 1: BFGS 1. Write a MATLAB function BFGS.m that implements the ba- sic BFGS algorithm on page 140 of your book. Use backtracking (use an initial step a = 1 in backtracking). The first line of the matlab file should be function [xstar , fval, iter]=bfgs (x0,Ho,func , gradfunc , maxit , tol) where Argument Definition vector giving the initial guess (n _1 matrix giving the initial guess to the inverse of the Hessian (nx n) name of a matlab function that returns the value of the objective function f(x) given an n _1 vector x HO func gradfunc name of a matlab function that returns

Optimization 1: BFGS 1. Write a MATLAB function BFGS.m that implements the ba- sic BFGS algorithm on page 140 of your book. Use backtracking (use an initial step a = 1 in backtracking). The first line of the matlab file should be function [xstar , fval, iter]=bfgs (x0,Ho,func , gradfunc , maxit , tol) where Argument Definition vector giving the initial guess (n _1 matrix giving the initial guess to the inverse of the Hessian (nx n) name of a matlab function that returns the value of the objective function f(x) given an n _1 vector x HO func gradfunc name of a matlab function that returns the gradient of the objective function ?f(x) as an n _1 vector given an n _1 vector x the maximum number of iterations permitted stopping tolerance (denoted e in Algorithm 6.1) maxit The items retued by bfgs.m are Argument | Definition xstar fval iter vector giving the final estimate of the minimizer (n _1) the value of the objective function f(x) at the final iterate the actual number of iterations taken 2. Test the algorithm using f(x) = 3×1+2x1x2+ :0 = [1:1], H0 = eye(2) (choose your own stopping criterion. You should be able to approximate the optimal point x* = (0,0) 3. Run your algorithm on the Rosenbrock function in 3.1 on page 63 of your book using the two initial guesses from the book and a third initial guess 20 = ( 1.9, 2) Plot a contour plot of the Rosenbrock function and the iterates (this will require a change to your BFGS function to plot as the algorithm progresses). Make your own choice of the initial Hessian. Take tol = 10 . How many steps are needed to approximate the exact solution with an 12 relative error of 0.01%?

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