Nonlinear Programming (ME 391Q)
Problem Set 1
1)A passenger wishes to catch a train at the last possible moment. The train is scheduled to depart at time T and the passenger must choose a time x when he will arrive at the station. His objective function is:
f(T–x) =
reflecting the assumption that he will miss the train if he arrives at precisely the same time that the train departs.
a.Does the passenger's problem as formulated here have a solution? Comment.
b.Suppose that the departure time T is a random variable having a known continuous density function, g(T), with a finite mean. Formulate the passenger's problem if he wants to minimize the expected value of his objective function f, as defined above.
c.Does the problem in part (b) have a solution? Comment. You may assume that the traveler must choose his time to lie between two fixed times, T1 and T2, and that g(T) only has nonzero values in this range.
2)Prove that the intersection of a (possibly infinite) family of closed sets is itself a closed set. An example of the intersection of an infinite number of closed sets might be
S = LimnS1S2• • •Sn, where Sk= for all k
One way to prove this result is to make use of the following definition of a closed set:
The set Sn is closed if, for any sequence of points {xk} S convergent to a limit point, the limit point belongs to S.
3)Show by example that the union of a family of closed sets need not be closed.
4)Let f:n1 be a continuous function of x = (x1,...,xn). Show that the set S = {x : f(x) ≤ 0} is closed.
5)Give an example of a discontinuous function f for which the set S = {x : f(x) ≤ 0} is not closed. Explain.
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