Re-expressing this as a convex optimization problem.

Question

For some $n$, $\vec{\tau}^{\min},\vec{\tau}^{\max}, \vec{d} \succcurlyeq 0$, and convex,non-decreasing and positive $\Phi(s) : \mathbb{R} \rightarrow \mathbb{R}$, I have the following optimization program: \begin{align*} &\min \sum_{i=1}^n \tau_i \Phi(s_i) \\ &\text{Subject to: } \forall i,\\ & s_{\min} \le s_i \le s_{\max} \\ & \tau^{\min}_i \le \tau_i \le \tau^{\max}_i \\ & \tau_i s_i = d_i \end{align*} My goal is to convert this into a convex optimization problem. Would appreciate any help on this problem.

Answer 1

Let's see. All variables are nonnegative. First eliminate $\tau_i$: \begin{align*} \min \quad& \sum_{i=1}^n \frac{d_i}{s_i} \Phi(s_i) \\ \text{s.t.} \quad & \forall i,\\ &s_{\min} \le s_i \le s_{\max} \\ &\tau^{\min}_i s_i \le d_i \le \tau^{\max}_i s_i \\ & s_i \geq 0 \end{align*} then substitute $x_i = 1/s_i$: \begin{align*} \min \quad& \sum_{i=1}^n d_i x_i \Phi(1/x_i) \\ \text{s.t.} \quad & \forall i,\\ &s_{\min} x_i \le 1 \le s_{\max} x_i \\ &\tau^{\min}_i \le d_i x_i \le \tau^{\max}_i \\ & x_i \geq 0 \end{align*} Et voila, it is convex.