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Optimization Prof. CarlsonHomework 6
2.9 Suppose that f(X) and g(X) are convex functions defined on
aconvex set C in RN and that
h(X) = max{f(X), g(X)}, X C.
Show that h(X) is convex on C.Fix X, Y C and t with 0 t 1.
Assume that h((1 t)X + tY ) =
f((1 t)X + tY ). Then by convexity of f we have
h((1 t)X + tY ) = f((1 t)X + tY )
(1 t)f(X) + tf(Y ) (1 t)h(X) + th(Y ),proving the convexity of
h.
2.15 Use the A-G inequality
ni=1
xii n
i=1
ixi, xi > 0, i > 0,
i
i = 1.
a) Minimize x2 + y + z subject to xyz = 1 and x, y, z > 0.
Write thefunction as
5[x2/5 + (y/2)(2/5) + (z/2)(2/5)]
with 1 = 1/5, 2 = 2/5, 3 = 2/5 to get
x2 + y + z 5(x2)1/5(y/2)2/5(z/2)2/5 = 524/5
(xyz)2/5 =5
24/5.
Equality occurs if and only if
x2 = y/2 = z/2,
giving
5(x2)1/5(x2)2/5(x2)2/5 =5
24/5,
orx10/5 = x2 = y/2 = z/2 = 24/5, x = 22/5.
1
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b) Maximize xyz subject to 3x + 4y + 12z = 1 and x, y, z >
0.We have
1 = 3x + 4y + 12z = 9x/3 + 12y/3 + 36z/3
(9x)1/3(12y)1/3(36z)1/3
= 91/3121/3361/3(xyz)1/3,
so
xyz 19 12 36 ,
and equality holds if and only if
9x = 12y = 36z,
or1 = (9x)1/3(9x)1/3(9x)1/3 = 9x,
or x = 1/9, etc.2.16 Suppose that c1, c2, c3 are positive
constants and
f(x, y) = c1x + c2x2y3 + c3y
4.
Minimize f(x, y) over x, y > 0.Write f in the general
form
f(x, y) =c11
1x +c22
2x2y3 +
c33
3y4
(c1
1
)1( c22
)2(c33
)3,
ifx1x22y32y43 = 1.
That is, we want1 22 = 0,32 + 43 = 0,
along with1 + 2 + 3 = 1.
2
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This gives1 = 8/15, 2 = 4/15, 3 = 3/15.
2.17 Solve using A-G.a) Find the largest (volume) circular
cylinder inscribed in a sphere of
given radius.The volume of the cylinder is
Vc = pir2h.
If the sphere has radius R then (a picture shows that) an
inscribed cylindersdimensions will satisfy
R2 = (h/2)2 + r2.
We have
R2 =1
3
3
4h2 +
2
3
3
2r2,
so with 11/3, 2 = 2/3
R2 (34)1/3h2/3(
3
2)2/3r4/3 = (
27
16)1/3(r2h)2/3
Thus the maximum volume is
pir2h = pi4
3
3R3,
which is achieved when3
4h2 =
3
2r2.
b) Find the smallest radius R such that a circular cylinder of
volume8 cubic units can be inscribed in the sphere of radius R.
Using part a)
6
3
pi= R3.
2.21 Let f(x, y, z) = x2 + y 3z. Show that f is convex on R3. Is
itstrictly convex?
By theorem 2.3.7 the question boils down to the analysis of the
Hessianof f . The Hessian is
Hf =
2 0 00 0 00 0 0
,
which is positive semidefinite but not positive definite.
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