Exercise 4: [4 points, Matlab grader] Unconstrained optimization In this exercise you are going to implement a Levenberg-Marquardt iteration algorithm for a Rosen- brock function. Levenberg-Marquardt. The LM algorithm is a modification of Newton's method and is given as Tk+1=1 - and with du = (v2f(Tk) + 2])-f(TR) where .*>is the stepsize at iteration k, which can be either fixed or dynamic, ke > is the blending factor (or damping parameter) at iteration k. Implement LM with and a factors updated in the following manner. 1. Compute dx, if de is not in a descent direction, then replace Hk = weld up and recompute dy else keep de and update 2+1 = a/dorni 2. Evaluate 1k+1, the function value f(Tk+1) and the difference in function values Afx = f(Tk+1) - f(x); 3. As long as Afk > this is not a descend step and f(+k+1) is recomputed with one = /down, else if Afe then the descend step is accepted and the step size is increased as ak =c4Cup. 4. Set k=k+1 and go to (1.). Here down, up, down, and Qup and the initial values of Mo and are choices for the algorithm. Rosenbrock functions. Rosenbrock functions are a set of non-convex functions that are used to test the performance of iteration based optimization algorithms. The functions have the beneficial property that the minima are known but are still hard to find by an iterative process. For 2D functions, the Rosenbrock functions are given by: f(x,y) = (a 1) +b(y-2?), where a and b are real valued constants. For this function, the unique global minimum is f(x",y') = f(a,c) = 0. Assignment Implement the LM-algorithm with a initial blending factor of Mo = 1 and step size ag for any given . initial To dup Choose the update variables down up, down, and Add stopping conditions f(Tk+1)-f(Tk) with constants 1,69 > 0. Exercise 4: [4 points, Matlab grader] Unconstrained optimization In this exercise you are going to implement a Levenberg-Marquardt iteration algorithm for a Rosen- brock function. Levenberg-Marquardt. The LM algorithm is a modification of Newton's method and is given as Tk+1=1 - and with du = (v2f(Tk) + 2])-f(TR) where .*>is the stepsize at iteration k, which can be either fixed or dynamic, ke > is the blending factor (or damping parameter) at iteration k. Implement LM with and a factors updated in the following manner. 1. Compute dx, if de is not in a descent direction, then replace Hk = weld up and recompute dy else keep de and update 2+1 = a/dorni 2. Evaluate 1k+1, the function value f(Tk+1) and the difference in function values Afx = f(Tk+1) - f(x); 3. As long as Afk > this is not a descend step and f(+k+1) is recomputed with one = /down, else if Afe then the descend step is accepted and the step size is increased as ak =c4Cup. 4. Set k=k+1 and go to (1.). Here down, up, down, and Qup and the initial values of Mo and are choices for the algorithm. Rosenbrock functions. Rosenbrock functions are a set of non-convex functions that are used to test the performance of iteration based optimization algorithms. The functions have the beneficial property that the minima are known but are still hard to find by an iterative process. For 2D functions, the Rosenbrock functions are given by: f(x,y) = (a 1) +b(y-2?), where a and b are real valued constants. For this function, the unique global minimum is f(x",y') = f(a,c) = 0. Assignment Implement the LM-algorithm with a initial blending factor of Mo = 1 and step size ag for any given . initial To dup Choose the update variables down up, down, and Add stopping conditions f(Tk+1)-f(Tk) with constants 1,69 > 0
