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Please upload a single pdf file on Canvas, containing the written answer for problem a and d, two figures and related codes of part b and c. Use verbatim package in latex to copy and paste the codes into a document. Consider “Rosenbrock” Function f(x1,x2)=100(x2 −x21)2 +(1−x1)2. This function is known as the banana function because of the shape of its level sets. a. Prove that [1,1]T is the unique global minimizer of f over R2. b. Write a method with signature function xsol = GradDescent (f, grad, x0) The input f and grad are function handles. The function f: RN → R is an arbitrary objective function, and grad: RN → RN is its gradient. The method should minimize f using gradient descent, and terminate when the gradient of f is small. I suggest stopping when ∥∇f(xk)∥ < ∥∇f(x0)∥ ∗ 10−4 Test your algorithm on Rosenbrock function, and plot ∥xk − x∗∥2 versus iteration numbers k for various fixed stepsize selection of α = 0.001, 0.05, 0.5, and explain your observation. c. Modify part b to create the new function xsol = GradDescentNesterov(f, grad, x0) This function should implement Nesterov’s method, that is xk =yk −α∇f(yk) √ δk+1 = 1+ 1+4(δk)2 2 yk+1 =xk +δk−1(xk −xk−1) δk+1 The method is initialized with x0 = y1 and δ1 = 1, and the the first iteration has index k = 1. Test your algorithm on Rosenbrock function, and plot ∥xk − x∗∥2 versus iteration numbers k for various fixed stepsize selection of α = 0.001, 0.05, 0.5, and explain your observation. d. With a starting point [0, 0]T, apply two iterations of Newton’s method to minimize Rosenbrock function. Hint: 􏰅 a b 􏰆−1 1 􏰅 d −b 􏰆 cd =ad−bc−ca 1

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