Dec 6, 2012
Student Name:
MATH 4/5420 - Optimization
Final Exam - Fall 2012
Take Home - Due Thursday, Dec 13, before 10am (no Exceptions!)
Problem 1
Solve the following linear programming problem using the simplex tableau method
max
3x1 + 2x2
subject to x1 ≤ 4,
x2 ≤ 1
2x1 + x2 ≤ 5
x1 , x2 ≥ 0
Problem 2
For the linear programming problem
min
x1 + x2 + 3x3
subject to x1 − x2 = 1
x1 + x3 ≥ 2
x1 ≥ 0, x2 ≥ 0, x3 ≥ 0
(a) Find the dual problem and solve it using the graphical method.
(b) Write down the KKT optimality conditions for the primal problem and solve it.
Problem 3
Consider the variational problem of minimizing
Z
J[x] =
1p
1 + [x0 (t)]2 dt
0
over all functions x = x(t), 0 ≤ t ≤ 1, with x(0) = 1 and x(1) = 0.
(a) Use the Euler-Lagrange equation to find the optimal solution. Can you verify that it is indeed a
minimizer?
(b) Using a discretization of the objective in (a) using N + 1 equal subintervals of [0, 1], solve the resulting
nonlinear optimization problem for X = (xj )1≤j≤N (in N-dimensions) to find an approximate solution to
(a). Does this approximation converge to the exact solution as N → ∞?
(c) [MATH 5420 Only] Redo (a) and (b) if, in addition, we imposes the additional constraint on x
Z
1
x(t) dt = 1
0
Problem 5 The fish population in a lake x = x(t) needs to be reduced in the next five years from the
current value x(0) = 1000 to half of that, x(5) = 500. The population of fish tends to grow by about 10%
a year, but it can be changed by controlling the number u(t) of fishermen. Then the population of fish will
evolve according to
dx
= 0.1x − u
dt
Since human intervention has its drawbacks for the environment, one wants to minimize the
Z 5
u2
min J[u] =
0
(a) Find the optimal function u∗ and the resulting fish population x∗ .
(b) Solve the same problem by converting it first to a calculus of variation problem [eliminate u] and using
the Euler-Lagrange equation. Can you verify that this is indeed a minimizer?