Math 164: Homework #9, due on Wednesday, June 3rd or Friday, June 5
[1] Solve the problem: Minimize f(x1, x2) = 1
2x2
1+x2
2subject to
2x1+x2≥2
x1−x2≤1
x1≥0.
[2] Let Abe an m×nmatrix whose rows are linearly independent. Prove that there exists
a vector psuch that Ap =e1, where e1= (1,0,0, ..., 0)T.
[3] Consider the bound-constrained problem
minimize f(x)
subject to l≤x≤u,
where l, u are vectors of lower and upper bounds, such that l < u. Let x∗be a local
minimizer. Show that:
if x∗,i =li, then ∂f(x∗)
∂xi≥0,
if x∗,i =ui, then ∂f(x∗)
∂xi≤0,
if li< x∗,i < ui, then ∂f(x∗)
∂xi= 0.
[4] Use the optimality conditions for nonlinear equality constraints to find all local solutions
to the problem
minimize f(x) = −x2
1+x2
2
subject to 2x2
1+x2
2= 4.
[5] Use the optimality conditions for nonlinear inequality constraints to find all local solutions
to the problem
minimize f(x) = x1+x2
subject to (x1−1)2+x2
2≤2
(x1+ 1)2+x2
2≥2
1