Math 164: Homework #2, due on Wednesday, April 15
Please solve as many problems as you can from the textbook. Also, please
review the material in the Appendices, at the end of the textbook, and read
Sections 2.2. and 2.3 from the textbook.
[1] Consider the feasible region defined by the constraints
1−(x1)2−(x2)2≥0,1−x1−x2≥0,and x2≥0.
For each of the following points, determine whether the point is feasible or
infeasible, and (if it is feasible) whether it is interior to or on the boundary
of each of the constraints:
xa= (1
2,1
2)T, xb= (1,1)T, xc= (−1,0)T, xd= (−1
2,0)T, xe= ( 1
√2,1
√2)T.
[2] Consider the problem
minimize x1
subject to (x1)2+ (x2)2≤4,
(x1)2≥1.
Graph the feasible set. Use the graph to find all local minimizers for the
problem, and determine which of those are also global minimizers.
[3] Let S1={x= (x1, x2) : x1+x2≤1, x1≥0}, and S2={x= (x1, x2) :
x1−x2≥0, x1≤1}, and let S=S1∪S2. Prove that S1and S2are both
convex sets, but that Sis not a convex set.
(this shows that the union of convex sets is not necessarily convex, but
the intersection is, see exercise 1 page 24).
[4] Let f:Rn→Rbe a convex function, and let g:R→Rbe a convex
nondecreasing function. Prove that the composite function h:Rn→R
defined by h(x) = g(f(x)) is convex.
[5] Consider the one-variable function
f(x) = (x+ 1)x(x−2)(x−5) = x4−6x3+ 3x2+ 10x.
Graph this function and locate (approximately) the stationary points, local
minima, and global minima.
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