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starting from an arbitary initial condition <math> x^{(0)} \in \mathbb{R}^{n} </math>, where <math> x \in \mathbb{R}^{n} </math>, and <math>Q = Q^{T} > 0</math>. <br/> | starting from an arbitary initial condition <math> x^{(0)} \in \mathbb{R}^{n} </math>, where <math> x \in \mathbb{R}^{n} </math>, and <math>Q = Q^{T} > 0</math>. <br/> | ||
*(5 pts) Find the largest range of the step size, <math>\alpha</math>, for which the fixed step gradient descent algorithm is guaranteed to converge to the minimizer of the quadratic function<br/> | *(5 pts) Find the largest range of the step size, <math>\alpha</math>, for which the fixed step gradient descent algorithm is guaranteed to converge to the minimizer of the quadratic function<br/> | ||
| − | <center><math> f= 6x_{1}^{2}+2x_{2}^{2}-5 </math></center> | + | <center><math> f= 6x_{1}^{2}+2x_{2}^{2}-5 </math></center> <br/> |
starting from an arbitrary initial condition <math>x^{(0)} \in \mathbb{R}^{n}</math> | starting from an arbitrary initial condition <math>x^{(0)} \in \mathbb{R}^{n}</math> | ||
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---- | ---- | ||
3. (20 pts) Is the function <br/> | 3. (20 pts) Is the function <br/> | ||
| + | <center><math> f(x_{1}, x_{2})=\frac{1}{(x_{1}-2)^{2} + (x_{2}+1)^{2}+3} </math></center><br> | ||
| + | locally convex, concave, or neither in the neighborhood of the point <math> [2 -1]^{T} </math>? Justify your answer by giving all the details of your argument. | ||
| + | |||
| + | |||
| + | ---- | ||
| + | 4. (20 pts) Solve the following optimization problem: | ||
| + | <center>optimize <math> x_{1}x_{2} </math> </center><br> | ||
| + | <center>subject to <math> x_{1}+x_{2}+x_{3}=1 </math> </center><br> | ||
| + | <center><math> x_{1}+x_{2}-x_{3}=0 </math></center><br> | ||
| + | |||
| + | ---- | ||
| + | 5. (20 pts) Solve the following optimization problem:<br/> | ||
| + | <center>maximize <math> 14x_{1}-x_{1}^{2}+6x_{2}-x_{2}^{2}+7 </math></center><br> | ||
| + | <center>subject to <math>x_{1}+x_{2} \leq 2</math></center><br> | ||
| + | <center><math> x_{1}+2x_{2} \leq 3 </math></center> | ||
| + | |||
| + | ---- | ||
| + | ---- | ||
| + | |||
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Revision as of 22:52, 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} $, where $ x \in \mathbb{R}^{n} $, and $ Q = Q^{T} > 0 $.
- (5 pts) Find the largest range of the step size, $ \alpha $, for which the fixed step gradient descent algorithm is guaranteed to converge to the minimizer of the quadratic function
starting from an arbitrary initial condition $ x^{(0)} \in \mathbb{R}^{n} $
3. (20 pts) Is the function
locally convex, concave, or neither in the neighborhood of the point $ [2 -1]^{T} $? Justify your answer by giving all the details of your argument.
4. (20 pts) Solve the following optimization problem:
5. (20 pts) Solve the following optimization problem:
