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| + | [[Category:automatic control]] | ||
| + | [[Category:optimization]] | ||
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| + | [[ECE_PhD_Qualifying_Exams|ECE Ph.D. Qualifying Exam]] | ||
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| + | Automatic Control (AC) | ||
| + | Question 3: Optimization | ||
| + | </font size> | ||
| − | + | August 2016 | |
| + | </center> | ||
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| + | :Student answers and discussions for [[QE2013_AC-3_ECE580-1|Part 1]],[[QE2013_AC-3_ECE580-2|2]],[[QE2013_AC-3_ECE580-3|3]],[[QE2013_AC-3_ECE580-4|4]],[[QE2013_AC-3_ECE580-5|5]] | ||
| + | ---- | ||
| + | '''1.(20 pts) In some of the optimization methods, when minimizing a given function f(x), we select an intial guess <math>x^{(0)}</math> and a real symmetric positive definite matrix <math>H_{0}</math>. Then we computed <math>d^{(k)} = -H_{k}g^{(k)}</math>, where <math>g^{(k)} = \nabla f( x^{(k)} )</math>, and <math>x^{(k+1)} = x^{(k)} + \alpha_{k}d^{(k)}</math>, where''' | ||
| + | <br> | ||
| + | <math> \alpha_{k} = arg\min_{\alpha \ge 0}f(x^{(k)} + \alpha d^{(k)}) .</math> | ||
| + | <br> | ||
| + | '''Suppose that the function we wish to minimize is a standard quadratic of the form,''' | ||
| + | <br> | ||
| + | <math> f(x) = \frac{1}{2} x^{T} Qx - x^{T}b+c, Q = Q^{T} > 0. </math> | ||
| + | <br><br> | ||
| + | '''(i)(10 pts) Find a closed form expression for <math>\alpha_k</math> in terms of <math>Q, H_k, g^{(k)}</math>, and <math>d^{(k)}; </math>''' | ||
| + | <br> | ||
| + | '''(ii)(10 pts) Give a sufficient condition on <math>H_k</math> for <math>\alpha_k</math> to be positive.''' | ||
| + | :'''Click [[QE2013_AC-3_ECE580-1|here]] to view [[QE2013_AC-3_ECE580-1|student answers and discussions]]''' | ||
| + | ---- | ||
| + | ---- | ||
| + | [[ECE_PhD_Qualifying_Exams|Back to ECE QE page]] | ||
[[ ECE PhD Qualifying Exams|Back to ECE PhD Qualifying Exams]] | [[ ECE PhD Qualifying Exams|Back to ECE PhD Qualifying Exams]] | ||
Revision as of 23:33, 27 January 2019
Automatic Control (AC)
Question 3: Optimization
August 2016
1.(20 pts) In some of the optimization methods, when minimizing a given function f(x), we select an intial guess $ x^{(0)} $ and a real symmetric positive definite matrix $ H_{0} $. Then we computed $ d^{(k)} = -H_{k}g^{(k)} $, where $ g^{(k)} = \nabla f( x^{(k)} ) $, and $ x^{(k+1)} = x^{(k)} + \alpha_{k}d^{(k)} $, where
$ \alpha_{k} = arg\min_{\alpha \ge 0}f(x^{(k)} + \alpha d^{(k)}) . $
Suppose that the function we wish to minimize is a standard quadratic of the form,
$ f(x) = \frac{1}{2} x^{T} Qx - x^{T}b+c, Q = Q^{T} > 0. $
(i)(10 pts) Find a closed form expression for $ \alpha_k $ in terms of $ Q, H_k, g^{(k)} $, and $ d^{(k)}; $
(ii)(10 pts) Give a sufficient condition on $ H_k $ for $ \alpha_k $ to be positive.
- Click here to view student answers and discussions
