ECE Ph.D. Qualifying Exam

Automatic Control (AC)

Question 3: Optimization

August 2016




1.(20 pts) Considern the following linear program,

minimize $ 2x_{1} + x_{2} $,

subject to $ x_{1} + 3x_{2} \geq 6 $

$ 2x_{1} + x_{2} \geq 4 $

$ x_{1} + x_{2} \leq 3 $

$ x_{1} \geq 0 $, $ x_{2} \geq 0 $.

Convert the above linear program into standard form and find an initial basic feasible solution for the program in standard form.

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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 converge to the minimizer of the quadratic function
$ f = \frac{1}{2} x^{T}Qx - b^{T}x $

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
$ f= 6x_{1}^{2}+2x_{2}^{2}-5 $

starting from an arbitrary initial condition $ x^{(0)} \in \mathbb{R}^{n} $

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3. (20 pts) Is the function

$ f(x_{1}, x_{2})=\frac{1}{(x_{1}-2)^{2} + (x_{2}+1)^{2}+3} $

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.

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4. (20 pts) Solve the following optimization problem:

optimize $ x_{1}x_{2} $

subject to $ x_{1}+x_{2}+x_{3}=1 $

$ x_{1}+x_{2}-x_{3}=0 $

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5. (20 pts) Solve the following optimization problem:

maximize $ 14x_{1}-x_{1}^{2}+6x_{2}-x_{2}^{2}+7 $

subject to $ x_{1}+x_{2} \leq 2 $

$ x_{1}+2x_{2} \leq 3 $
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