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Unformatted text preview: IEOR160: Operations Research I Midterm Exam
October 29, 2008 Professor Ilan Adler Name: (please print)
SID: 0 Clearly state all the mathematical expressions that are needed to solve the problems. No
credit will be given to numerical answers without the proper setup. 0 Answer each of the following questions in the space provided If you need more space to show major computations you performed to obtain your answer for a particular problem,
use the back of the preceding page. a You can quote and use any result stated in class or in the main body of the textbook as
well as wellknown genera] mathematical results but no references to other sources
(including homework and textbook exercises) are allowed. 0 Present your work in an organized and neat fashion. 0 Assume that all the functions are twice continuously differentiable.  Good luck! A score of 100 would be considered as perfect. Problem 1 (35 points) A farmer wishes to fence in a rectangular pen for her animals. She only has 200 feet of fencing to use, but the back wall of the barn is 40 feet long and she can use it for part or all of one side of
the pen. a) Formulate a nonlinear programnﬁng problem so that the area of the pen will be as large
as possible. Use only one variable in the formulation.
b) Write the KKT conditions for the problem 0) Solve the system you got in (b).
d) Is the solution you got from (c) the optimal solution for the problem? Explain. Problem 2 (35 points) Consider the following problem:
(P) z= max x2  y2
s.t. (x—l)2y2=b For questions (a)(c) assume that b=0.
3) Write the Lagrangian function for (P).
b) Determine all the points at which the gradient of the Lagrangian function is zero.
e) Given the fact that (P) has a global solution, ﬁnd it.
d) Find ayes. Problem 3 (35 points) . Read each of the following statements carefully to see whether it is true orfalse. Justify your
answers (no credit for answers without justiﬁcation!) For the following question suppose it is an ndimensionai column vector and let Vf(x) denote the
gradient of a function f at it (presented as a row vector). a) Suppose Vf{x*)d=0 for all ndimensional column vectors d. Then x* is a local maximum
point of f. b) If Vf(x*)=0 and all leading principal minors of the Hessian of f are negative for all the
points in R“, then x* is a global maximum point for 1'.
Note — there are no constraints in this question. 6) Consider:
(Pl) Max {(11)
s.t. gi(x)2bi, i=1,. ..m
Assume f and g; (i=1,. ..,m) are all concave functions. If 1* satisﬁes the KKT conditions
and the constraint qualiﬁcations, then 11* is a global maximum point. (:1) Consider:
I (1’2) Max f(x)
s.t. :1be (where a is an n—dimensional row vector) Assume f is a concave function. If x* satisﬁes ax*<b and Vf(x*)=0, then 1* is a global
maximum point. e) Consider:
(1‘3) Max f(x)
s.t. gi(x)Sbi, i=l,...m
Assume f is concave and g: (i=1,. . .,m) are convex. Let x*, y* be two (global) optimal
solutions for (PB). Then, 0.5x*+0.5y* is also a (global) optimal solution for (P3). ...
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 Spring '08
 Lim

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