MIME 6740 Exam 1 Spring 2003

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1.  (40 points) The axes of an ellipse are x1 and x2. We want to find the ellipse with maximum area whose sum of the lengths of the axes is equal to one. This problem is formulated as follows:

Find x1 and x2

to maximize F = area = px1x2

so that x1 +x2=1

a)  Formulate the above problem as an one-dimensional optimization problem.

b)  Solve the problem in a) analytically.

c)  Solve the problem using the golden section method. Complete as many iterations as needed so that the tolerance interval is less or equal to 0.05. YOU CAN ONLY USE A PROGRAM TO CALCULATE THE VALUE OF THE OBJECTIVE FUNCTION HERE. THE REST OF THE CALCULATIONS MUST BE DONE BY HAND.

2. (40 points) Consider function .

a)  Find the minimum of this function analytically.

b)  Minimize the function using the conjugate gradient method starting from point x1=2, x2=2. You can use a program such as MS-Excel only for one-dimensional minimization. Show all steps for the rest of the solution.

c)  Check if the search directions S1 and S2 in the first two iterations are conjugate.

3) (20 points) Answer to the following true-false questions. You do not need to justify your answers but you may write something if you think a question is vague or ambiguous. Your answers can be marked on the problem statement or on a separate page but not on both.

a)  The Kuhn-Tucker optimality conditions for a constrained optimization problem include all the inequality constraints at the particular design point of interest regardless if they are active or inactive. (T-F)

b)  The Lagrange multipliers of the inequality constraints at the optimum can be negative. (T-F)

c)  The Lagrange multipliers of the equality constraints are non zero at the optimum. (T-F)

d)  Suppose you use the steepest descent method for unconstrained minimization. The search directions in two subsequent iterations are perpendicular. (T-F)

e)  Powell’s method finds the optimum of a function with n variables in n iterations (each iteration includes n one-dimensional searches for finding a conjugate direction plus an additional one-dimensional search in the conjugate direction). (T-F)

f)  Conjugate directions are perpendicular to each other. (T-F)