HW9 - ELE 704 Optimization HW9 Due 15 May, 2007 1....

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Unformatted text preview: ELE 704 Optimization HW9 Due 15 May, 2007 1. Newton’s method with fixed step size α = 1 can diverge if the initial point is not close to x∗ . Consider the following examples: (a) f (x) = log(ex + e−x ) has a unique minimizer at x∗ = 0. i. Plot f (x) for x ∈ [−2, 2] (You may want to draw in the logarithmic scale.) ii. Run Newton’s method by hand with fixed step size α = 1, starting at x(0) = 1 and at x(0) = 1.1 iii. Plot the trajectory of x(k) on the figure of part (a.i) for the first 5 iterations and also draw f (x(k) ) − p∗ vs. number of iterations. (b) Repeat part (a) for f (x) = − log(x) + x which has the unique minimizer at x∗ = 1. Start at x(0) = 3. (c) Comment on your results. 2. Write the MATLAB code for pure Newton Method and run for the functions of Q.1 starting at the given initial points. Verify your answer with those of Q.1. 3. Consider the following problem min f (x) = log(ex + e−x ) s.t. x2 − 1.1x + 0.1 ≤ 0 (a) Plot f (x) and indicate the feasible region for x ∈ [−2, 2] (b) Write the MATLAB code for the penalty method with the quadratic penalty function, P (x). Starting at x(0) = −0.5 and at x(0) = 0.1 i. run the penalty method using the Newton’s method for the minimization procedure in the inner loop. ii. Plot the trajectory of the progress of the algorithm on the figure of part (a) until sufficiently large c is approached in the outer loop. Also draw f (x(k) ) − p∗ vs. number of total iteration (inner and outer). (You may want to draw in the logarithmic scale.) 1 ...
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This note was uploaded on 05/25/2011 for the course ELECTRONIC 704 taught by Professor Cenktoker during the Spring '11 term at Hacettepe Üniversitesi.

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