What is the logarithmic transformation of this optimization problem?

Question

Given $$n, C, r_i, p_i,a_i \quad∀ i={1,2,...,n} $$ I want to solve this optimization problem: $$\text{maximize} \quad f(x_1,x_2,...,x_n)=\prod_{i=1}^n {{(x_i-a_i)}^{p_i}} $$ $$\text{s.t.} \quad {(x_i/r_i)≤1}, \quad {(\sum_{i=1}^n x_i)}/C ≤1,\quad {(a_i/x_i)≤1}$$ where $$0≤p_i≤1 \quad\quad, \sum_{i=0}^n p_i =1\quad \quad\quad, ∀x_i,r_i.a_i,p_i∈R,\quad x_i,r_i,a_i>0,\quad C>0 $$ I want to know what is the logarithmic transformation of this problem and I want to convert it to an appropriate form to be solved by CVX. Thank you for your help

Answer 1

Minimizing a function is the same as minimizing the log, so we can transform the objective $$\prod_{i=1}^n (x_i - a_i)^{p_i} \mapsto \log\prod_{i=1}^n (x_i - a_i)^{p_i} = \sum_{i=1}^n p_i \log(x_i - a_i),$$ which is concave since each $p_i\geq 0$. Simple rearrangement of the denominator gives the following problem which is suitable for CVX. $$ \begin{align*} \text{maximize: }& \sum_{i=1}^n p_i \log(x_i - a_i)\\ \text{subj. to: }& a_i \leq x_i \leq r_i \text{ for all $i=1,2,\ldots,n$}\\ & \sum_{i=1}^n x_i \leq C. \end{align*} $$ Since the constraints are linear and the objective is a concave maximization, this problem is convex.