Automatic Control (AC)

Question 3: Optimization

August 2011

## Question

Part 1. 20 pts

$\color{blue} \text{ Consider the optimization problem, }$

$\text{maximize} -x_{1}^{2}+x_{1}-x_{2}-x_{1}x_{2}$

$\text{subject to } x_{1}\geq0, x_{2}\geq0$

$\color{blue}\left( \text{i} \right) \text{ Characterize feasible directions at the point } x^{*}=\left[ \begin{array}{c} \frac{1}{2} \\ 0 \end{array} \right]$

$\color{blue}\left( \text{ii} \right) \text{Write down the second-order necessary condition for } x^{*} \text{. Does the point } x^{*} \text{ satisfy this condition?}$

Part 2.

$\color{blue} \text{ Use the simplex method to solve the problem, }$

maximize        x1 + x2

$\text{subject to } x_{1}-x_{2}\leq2$
$x_{1}+x_{2}\leq6$

$x_{1},-x_{2}\geq0.$

Part 3. (20 pts)

$\color{blue}\text{ Solve the following linear program, }$

maximize    − x1 − 3x2 + 4x3

subject to    x1 + 2x2x3 = 5

2x1 + 3x2x3 = 6

$x_{1} \text{ free, } x_{2}\geq0, x_{3}\leq0.$

Part 4. (20 pts)

$\color{blue} \text{ Consider the following model of a discrete-time system, }$

$x\left ( k+1 \right )=2x\left ( k \right )+u\left ( k \right ), x\left ( 0 \right )=0, 0\leq k\leq 2$

$\color{blue}\text{Use the Lagrange multiplier approach to calculate the optimal control sequence}$

$\left \{ u\left ( 0 \right ),u\left ( 1 \right ), u\left ( 2 \right ) \right \}$

$\color{blue}\text{that transfers the initial state } x\left( 0 \right) \text{ to } x\left( 3 \right)=7 \text{ while minimizing the performance index}$
$J=\frac{1}{2}\sum\limits_{k=0}^2 u\left ( k \right )^{2}$

Part 5. (20 pts)

$\color{blue} \text{ Consider the following optimization problem, }$

$\text{optimize} \left(x_{1}-2\right)^{2}+\left(x_{2}-1\right)^{2}$

$\text{subject to } x_{2}- x_{1}^{2}\geq0$

$2-x_{1}-x_{2}\geq0$

$x_{1}\geq0.$

$\color{blue} \text{The point } x^{*}=\begin{bmatrix} 0 & 0 \end{bmatrix}^{T} \text{ satisfies the KKT conditions.}$

$\color{blue}\left( \text{i} \right) \text{Does } x^{*} \text{ satisfy the FONC for minimum or maximum? Where are the KKT multipliers?}$

$\color{blue}\left( \text{ii} \right) \text{Does } x^{*} \text{ satisfy SOSC? Carefully justify your answer.}$ 