How to prove convexity of an optimization problem?

Question

Consider the following optimization problem.

Let $d_3, d_2, d_1 > 0$.

Maximize $\log(p_1)+\log(p_2)+\log(p_3)$

Subject to:

$p_1d_1 + p_2d_2 + p_3d_3= 1$

$p_1 \geq p_2\geq p_3\geq 0$.

I believe this is a convex optimization problem, but how would one prove it?

Answer 1

A few hints:

• the log function is concave

• the sum of concave functions is concave

• a maximization problem is equivalent to a minimization problem with the objective function multiplied by -1