Quiz Chapter 09.03: Multidimensional Direct Search Method

1. Which of the following statement is FALSE?

Multidimensional direct search methods are similar to one-dimensional direct search methods.

Enumerating all possible solutions in a search space and selecting the optimal solutions is an effective method for problems with very high dimensional solution spaces.

Multidimensional direct search methods do not require a twice differentiable function as an optimization function.

Genetic Algorithms belong to the family of multidimensional direct search methods.

2. Which of the following statements is FALSE?

Multidimensional direct search methods require an upper and lower bound for their search region.

Coordinate cycling method relies on single dimensional search methods to determine an optimal solution along each coordinate direction iteratively.

If the optimization function is twice differentiable, multidimensional direct search methods cannot be used to find an optimal solution.

Multidimensional direct search methods are not guaranteed to find the global optimum.

3. The first cycle of Example 1 in Chapter 09.03 results in an optimal solution of $f(2.6459, \, 0.8668)=4.8823$ for the gutter design problem. The next iteration starts with a search along dimension $l$ (length) looking for the optimal solution of the function $f(l, \, 0.8668)$ as shown in Table $3$ and reproduced below where $\theta = 0.8668$ and $f(x_{i}) = \left( 6-2l + l \cos (0.8668) l \sin (0.8668) \right)$ . What is the optimal solution for the length of the gutter side at the end of iteration $10$ ?

Iteration	$x_{l}$	$x_{u}$	$x_{1}$	$x_{2}$	$f(x_{1})$	$f(x_{2})$	$\varepsilon$
$1$	$0.0000$	$3.0000$	$1.8541$	$1.1459$	$4.9354$	$3.8871$	$3.0000$
$2$	$1.1459$	$3.0000$	$2.2918$	$1.8541$	$5.0660$	$4.9354$	$1.8541$
$3$	$1.8541$	$3.0000$	$2.5623$	$2.2918$	$4.9491$	$5.0660$	$1.1459$
$4$	$1.8541$	$2.5623$	$2.1246$	$2.1246$	$5.0660$	$5.0627$	$0.7082$
$5$	$2.1246$	$2.5623$	$2.3951$	$2.2918$	$5.0391$	$5.0660$	$0.4377$
$6$	$2.1246$	$2.3951$	$2.2918$	$2.2279$	$5.0660$	$5.0715$	$0.2705$
$7$	$2.1246$	$2.2918$	$2.2279$	$2.1885$	$5.0715$	$5.0708$	$0.1672$
$8$	$2.1885$	$2.2918$	$2.2523$	$2.2279$	$5.0704$	$5.0715$	$0.1033$
$9$	$2.1885$	$2.2523$	$2.2279$	$2.2129$	$5.0715$	$5.0716$	$0.0639$
$10$	$2.1885$	$2.2279$	$2.2129$	$2.2035$	$5.0716$	$5.0714$	$0.0395$

2.1885

2.2279

5.0715

2.2082

4. What is the maximum size for the area of gutter at the optimal point determined in multiple-choice question $3$ ? (Hint: You do not need to do any calculations to answer this question)

5.0716

5.0714

5.0715

2.2082

5. To find the minimum of the function $f(x,y)=5x^{2} - 6xy + 5y^{2} - 2$ hold $y=0$ and use $2$ and $-2$ as your upper and lower bounds for your one-dimensional search along the $x$ -coordinate using golden search method. What would be the optimal solution for $x$ after the first iteration?

3.1146

0.4721

0

0.0015

6. Considering the scenario in Question $5$ , what would be the optimal solution for after the first iteration? (Can you explain the difference?)

0

0.7639

0.4721

7.5728