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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?

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    $\begingroup$ Maximizing a concave function over a standard simplex, thus convex. $\endgroup$ – Rodrigo de Azevedo Apr 14 at 8:45
  • $\begingroup$ @RodrigodeAzevedo Does the form of the constraints (in this case linear in $p_i$), affect whether the problem is convex or not? $\endgroup$ – Anush Apr 14 at 17:47
  • $\begingroup$ If the feasible region is not convex, then the problem is non-convex. $\endgroup$ – Rodrigo de Azevedo Apr 14 at 17:48
  • $\begingroup$ @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? $\endgroup$ – Anush Apr 14 at 17:49
  • $\begingroup$ 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. $\endgroup$ – Rodrigo de Azevedo Apr 14 at 18:00
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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

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  • $\begingroup$ I am guessing the constraints also need to be of a particular form for the problem to be convex? $\endgroup$ – Anush Apr 14 at 11:26
  • $\begingroup$ 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 $\endgroup$ – E. Tucker Apr 15 at 13:21

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