QE2016 AC-3 ECE580 - Rhea

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 convege 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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