# HW4 - ELE 704 Optimization HW4 Due 20 March 2007 1 Prove...

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ELE 704 Optimization HW4 Due 20 March 2007 1. Prove that 0 m I H ( x ) M I where H ( x ) is the Hessian matrix of a strongly convex function f ( x ) and m is the smallest and M is the largest eigenvalues of H ( x ). 2. Write the MATLAB code which performs the following line search algo- rithms for the function you used for HW3 (a) Exact Line Search (b) Bisection Line Search (c) Backtracking Line Search. Try to be as much flexible as possible. Write your code as a MAT- LAB function which can be called from another (possibly main) code with some parameters. This means you may also need to write your quadratic function, its gradient and its Hessian as a function also. 3. (a) Choose an arbitrary initial point. By assuming that the descent direction is d = -∇ f ( x (0) ) using the MATLAB command ”surfc” first draw the cost surface then add the curve h ( α ) = f ( x (0) + α d ) on top of the cost function. (b) For h ( α ) write the algorithm for the line search algorithms indicated in Q . 2 and run the algorithms for h ( α ) by hand.

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