## Description

Problem 1 (25pts). Suppose that f : R → R is convex, and a, b ∈ dom f with a < b, where dom

denotes the domain of the function. More specifically, f : Rp → Rq means that f is an Rp

-valued

function on some subset of Rp

, and this subset of Rp

is the domain of the function f. Show that

(a)

f(x) ≤

b − x

b − a

f(a) + x − a

b − a

f(b), for all x ∈ (a, b)

Hint: (Jensen’s Inequality) If p1, …, pn are positive numbers which sum to 1 and f is a real

continuous function that is convex, then

f

Xn

i=1

pixi

!

≤

Xn

i=1

pif (xi)

(b)

f(x) − f(a)

x − a

≤

f(b) − f(a)

b − a

≤

f(b) − f(x)

b − x

for all x ∈ (a, b). Draw a sketch that illustrates this inequality.

(c) Suppose f is differentiable. Use the result in (b) to show that:

f

′

(a) ≤

f(b) − f(a)

b − a

≤ f

′

(b)

Note that these inequalities also follow form:

f(b) ≥ f(a) + f

′

(a)(b − a), f(a) ≥ f(b) + f

′

(b)(a − b)

(d) Suppose f is twice differentiable. Use the result in (c) to show that f

′′(a) ≥ 0 and f

′′(b) ≥ 0.

Problem 2 (30pts) Show that the following functions are convex:

(a) f(x) = − log

− log Pm

i=1 e

a

T

i

x+bi

on dom f =

n

x |

Pm

i=1 e

a

T

i

x+bi < 1

o

.

You can use the

fact that log (Pn

i=1 e

yi ) is convex.

(b)

f(x, u, v) = − log

uv − x

T x

on dom f =

(x, u, v) | uv > xT x, u, v > 0

(c) Let T(x, ω) denote the trigonometric polynomial

T(x, ω) = x1 + x2 cos ω + x3 cos 2ω + · · · + xn cos(n − 1)ω

Show that the function

f(x) = −

Z 2π

0

log T(x, ω)dω

is convex on {x ∈ Rn

| T(x, ω) > 0, 0 ≤ ω ≤ 2π}.

Hint: Nonnegative weighted sum of convex functions is still convex. Let this property extend

to infinite sums and integrals. Assume that f(x, y) is convex in x for each y ∈ A and w(y) ≥ 0

for each y ∈ A and integral exists. Then the function g defined as

g(x) = Z

A

w(y)f(x, y)dy

is convex in x.

Problem 3 (20pts). Consider the following function:

minimize −x1 − x2 + max {x3, x4}

s.t. (x1 − x2)

2 + (x3 + 2×4)

4 ≤ 5

x1 + 2×2 + x3 + 2×4 ≤ 6

x1, x2, x3, x4 ≥ 0

(a) Verify this is a convex optimization problem.

(b) Use CVX to solve the problem.

Problem 4 (25pts). To model the influence of price on customer purchase probability, the

following logit model is often used:

λ(p) = e

−p

1 + e−p

where p is the price, λ(p) is the purchase probability.

Assume the variable cost of the product is 0 (e.g., iPhone Apps). As the seller, you want to

maximize the expected revenue by choosing the optimal price. That is, you want to solve:

maximizep pλ(p)

(a) Draw a picture of r(p) = pλ(p) (for p from 0 to 10) and use the picture to show that r(p) is

not concave (thus maximize r(p) is not a convex optimization problem)

(b) Write down p as a function of λ (the inverse function of λ(p) ). Show that you can write the

objective function as a function of λ : ˜r(λ), where ˜r(λ) is concave in λ.

(c) From part 2, write the KKT condition for the optimal λ. Then transform it back to an optimal

condition in p.