Optimization Methods
Sample Exam
Fall 2005
Instructor: Muhammad Al-Salamah

1. Consider the following function



for x[0, 6].

1. Use 2 iterations of the golden section search method to find the minimizer of f(x). At the end of procedure, write the interval containing the minimum.
2. Perform 3 iterations of the bisection method to locate the minimizer of f(x). At the end of procedure, write the interval containing the minimizer.

 

2. Estimate the minimizer of the function



starting with x⁽⁰⁾ = (2, 2)^T and by using
1. two iterations of the simplex method with scaling factor a = 2.
2. two iterations of the Hooke-Jeeves pattern search method with D = (1, 1)^T.

 

3. Solve the unconstrained optimization problem



using Powell’s conjugate direction method staring with x⁽⁰⁾ = (1, -1)^T.

 

4. Consider the unconstrained optimization problem



Starting with x⁽⁰⁾ = (0.3, 0.8)^T, find the minimizing direction using
1. the steepest descent method,
2. the Newton’s method.

 

5. Solve the optimization problem



using the method of Lagrange multipliers. Determine by how much the optimal value of the objective function will increase or decrease as a result of a unit increase in the right-hand-side of the constraint.