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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]] | :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]] | ||
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| − | 1.(20 pts) Considern the following linear program, minimize <math>2x_{1} + x_{2}</math>, | + | 1.(20 pts) Considern the following linear program, <br/> |
| − | <math>x_{1} + 3x_{2} \geq 6 </math> <br/> | + | <center> minimize <math>2x_{1} + x_{2}</math>, </center> <br/> |
| − | <math>2x_{1} + x_{2} \geq 4</math> <br/> | + | <center> subject to <math>x_{1} + 3x_{2} \geq 6 </math> </center> <br/> |
| − | <math> x_{1} + x_{2} \leq 3 </math> <br/> | + | <center> <math>2x_{1} + x_{2} \geq 4</math> </center> <br/> |
| − | <math> x_{1} \geq 0 </math>, <math> x_{2} \geq 0 </math>. <br/> | + | <center> <math> x_{1} + x_{2} \leq 3 </math> </center> <br/> |
| − | Convert the above linear program into standard form and find an initial basix feasible solution for the program in | + | <center> <math> x_{1} \geq 0 </math>, <math> x_{2} \geq 0 </math>. </center> <br/> |
| + | Convert the above linear program into standard form and find an initial basix feasible solution for the program in standard form. <br/> | ||
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2.(20 pts) | 2.(20 pts) | ||
| − | + | *(15 pts) FInd the largest range of the step-size, <math> \alpha </math>, for which the fixed step gradient descent algorithm is guaranteed to convege to the minimizer of the quadratic function <br/> | |
| + | <center> <math> f = \frac{1}{2} x^{T}Qx - b^{T}x </math> </center> <br/> | ||
| + | starting from an arbitary initial condition <math> x^{(0)} \in \mathbb{R}^{n} </math> | ||
[[ECE_PhD_Qualifying_Exams|Back to ECE QE page]] | [[ECE_PhD_Qualifying_Exams|Back to ECE QE page]] | ||
Revision as of 23:16, 27 January 2019
Automatic Control (AC)
Question 3: Optimization
August 2017
1.(20 pts) Considern the following linear program,
Convert the above linear program into standard form and find an initial basix feasible solution for the program in standard form.
2.(20 pts)
- (15 pts) FInd the largest range of the step-size, $ \alpha $, for which the fixed step gradient descent algorithm is guaranteed to convege to the minimizer of the quadratic function
starting from an arbitary initial condition $ x^{(0)} \in \mathbb{R}^{n} $
