# Quiz Chapter 09.03: Multidimensional Direct Search Method

 MULTIPLE CHOICE TEST MULTIDIMENSIONAL DIRECT SEARCH METHOD OPTIMIZATION

1. Which of the following statement is FALSE?

2. Which of the following statements is FALSE?

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

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. 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?

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