# How to prove convexity of an optimization problem?

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?

• Maximizing a concave function over a standard simplex, thus convex. – Rodrigo de Azevedo Apr 14 at 8:45
• @RodrigodeAzevedo Does the form of the constraints (in this case linear in $p_i$), affect whether the problem is convex or not? – Anush Apr 14 at 17:47
• If the feasible region is not convex, then the problem is non-convex. – Rodrigo de Azevedo Apr 14 at 17:48
• @RodrigodeAzevedo Given that the objective function is linear in $\log(p_i)$ and the constraints are linear in $p_i$, does this change how you consider the convexity of the feasible region? – Anush Apr 14 at 17:49
• I don't understand your question. A convex program is either minimizing a convex function or maximizing a concave function over a convex feasible region. Tucker's answers deals with the concavity of the objective function to be maximized, but does not touch the constraints. – Rodrigo de Azevedo Apr 14 at 18:00

## 1 Answer

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

• I am guessing the constraints also need to be of a particular form for the problem to be convex? – Anush Apr 14 at 11:26
• Yes, to be a convex problem, the set of solutions that satisfies the constraints should be convex. In this case the constraints are linear so they form a convex set. (Note - in Rodrigo's first comment yesterday, when he says "over a standard simplex," this is referring to the constraints.) If you're interested in learning more, the free Boyd and Vandenberghe book may be helpful (e.g., section 2.1.4) web.stanford.edu/~boyd/cvxbook – E. Tucker Apr 15 at 13:21