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Consider the following unconstrained optimization problem (UP) minimize 2xtQx-cx where x E Rn and Q is an n X n symmetric positive definite matrix. Let x* be the minimizer of problem (UP). Let V € Rn be an eigenvector of Q corresponding to the eigenvalue l. Suppose that the initial point for the steepest descent algorithm is (a) (10pts) Show that the gradient of = - x° is Vf(r))=lv. = (b) (10pts) Prove that if the steepest descent direction is taken, then the step-size which minimizes f in this direction is 00 = 1/X. (c) (20pts) Confirm the above assertions for the function f(x1,x2)=3x}-2x1x2 +3x2+2x1 -6x2. = - Compute the point x* obtained by one iteration of the steepest descent algo- rithm, starting at point x° = (1,2). Show that x* is the unique minimum of f. Verify that x° - x* is an eigenvector of the Hessian of f. Prove the following two parts independently. (a) (15pts) Consider the problem 1 (P) minimize T subject to where Q is a positive-definite matrix. Prove that Newton's method will determine the minimizer of f (x) = 1xtQx - cT. x in one iteration, regardless of the starting point. (b) Let f : Rn R be twice continuously differentiable function. Consider the following unconstrained optimization problem minimize f (x) subject to x E R and a linear transformation of variables x = Ay + b, where A is an n X n nonsingular matrix and y E Rn. i. (5pts) Find Newton's direction dk in the space of the variables y. ii. (5pts) Find the sequence {yk} generated by the Newton's method. iii. (5pts) Show that the Newton's method is not affected by linear scaling of the variables. Hint: Transform the sequence obtained in Part (b) into the sequence {xk} in the space of the variables x by replacing Syk by xk

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Unconstrained Optimization Problems
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