Automatic Control (AC)

Question 3: Optimization

August 2013

Student answers and discussions for Part 1,2,3,4,5

1.(20 pts) In some of the optimization methods, when minimizing a given function f(x), we select an intial guess $x^{(0)}$ and a real symmetric positive definite matrix $H_{0}$. Then we computed $d^{(k)} = -H_{k}g^{(k)}$, where $g^{(k)} = \nabla f( x^{(k)} )$, and $x^{(k+1)} = x^{(k)} + \alpha_{k}d^{(k)}$, where
$\alpha_{k} = arg\min_{\alpha \ge 0}f(x^{(k)} + \alpha d^{(k)}) .$
Suppose that the function we wish to minimize is a standard quadratic of the form,
$f(x) = \frac{1}{2} x^{T} Qx - x^{T}b+c, Q = Q^{T} > 0.$

(i)(10 pts) Find a closed form expression for $\alpha_k$ in terms of $Q, H_k, g^{(k)}$, and $d^{(k)};$
(ii)(10 pts) Give a sufficient condition on $H_k$ for $\alpha_k$ to be positive.

Problem 2. (20 pts) [(i) (10 pts)] Consider the one-point crossover of a chromosome in the schema
H = * 1 * 0 1 0 *
where the probability that a chromosome is chosen for crossover is $p_c = 0.5.$ Find an upper bound on the probability that a chromosome from H will be destroyed by the one-point crossover.

[(ii) (10 pts)] Consider a chromosome in the schema
H = * 1 * 0 * * *
Find the probability that the mutation operation destroys the schema, where the probability of random change of each symbol of the chromosome is $p_m = 0.1$ independently.

Problem 3. (20 pts) [(i) (10 pts)] Convert the following optimization problem into a linear programming problem and solve it;
maximize $-|x_1| -|x_2| -|x_3|$
subject to
$\begin{bmatrix} 1 & 1 &-1 \\ 0 & -1 & 0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 2 \\ 1 \end{bmatrix} .$

[(ii) (10 pts)] Construct the dual program of the linear program above and solve it.

Problem 4. (20pts) Consider the following model of a discrete-time system,
$x(k+1) = x(k) + 2u(k), x(0) = 3, 0 \le k \le 2$
Use the Lagrange multiplier approach to calculate the optimal control sequence
{u(0), u(1), u(2)}
that transfers the initial state x(0) to x(3) = 9 while minimizing the performance index
$J = \frac{1}{2} \sum_{k=0}^2 u(k)^2 = \frac{1}{2}u^Tu.$
Hint: Use the system model to obtain a constraint of the form, $Au = b, A \in R^{1 \times 3}.$

Problem 5. (20pts) Find minimizers and maximizers of the function,
$f(x) = (a^Tx)(b^Tx), x \in R^3,$
subject to
$x_1 + x_2 = 0$
$x_2 + x_3 = 0,$
where
$a = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$ and $b = \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}$