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Unformatted text preview: ELE 704 Optimization
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.
- Spring '11