Perspective of log-sum-exp as exponential cone

Question

According to the Mosek documentation, Geometric Programming constraints of form log-sum-exp can be formulated with exponential cones. If the constraint is of form $$t \geq \log(\exp(x_1)+\ldots + \exp(x_n)), $$

we can write the constraint as $$\begin{cases} \sum_{i=1}^n u_i \leq 1 \\ (u_i,1,x_i-t) \in \mathcal{K}_{\exp} \ \forall i \end{cases}$$ where $\mathcal{K}_{\exp}$ stands for the exponential cone, defined as: $$ \mathcal{K}_{\exp} = \{(a_1,a_2,a_3):a_1 \geq a_2 e^{a_3 / a_2}, a_2>0 \}\cup \{(a_1,0,a_3): a_1 \geq 0, a_3 \leq 0 \}$$ So, to be able to use it in the above re-formulation we can show that this cone consists of points which satisfy $a_3 \leq a_2\log (a_1/a_2), a_1,a_2 > 0.$

My case is slightly different. Let $f:\mathbb{R}^n \mapsto \mathbb{R}$ be defined as $f(\mathbf{x}) = \log(\exp(x_1)+\ldots + \exp(x_n))$. The perspective of $f$ can be shown as $x_0f(\frac{x}{x_0})$ where $x_0 > 0$. The constraint I have is: $$t \geq x_0 f\left(\frac{x}{x_0}\right).$$ So I have the same constraint with the perspective function. How can I formulate this constraint with exponential cones (and linear constraints in the variables)? Note that in my constraint $t$ is a variable, too.

Answer 1

I will write up the solution for the sake of completeness and because the question seems to have appeared elsewhere http://ask.cvxr.com/t/convert-the-perspective-function-of-log-sum-exp-to-cvx/5232

The inequality $$t\geq x_0 f(x/x_0)$$ is $$t\geq x_0\cdot\log(\exp(x_1/x_0)+\cdots+\exp(x_n/x_0)).$$ Since $x_0>0$, dividing by it and then following the same procedure as for log-sum-exp in https://docs.mosek.com/modeling-cookbook/expo.html#log-sum-exp we arrive at the inequality $$1\geq \sum \exp\left(\frac{x_i-t}{x_0}\right)$$ which we can rewrite as $$x_0\geq \sum x_0\exp\left(\frac{x_i-t}{x_0}\right)$$ and eventually $$x_0\geq \sum z_i,\quad z_i\geq x_0\exp\left(\frac{x_i-t}{x_0}\right)$$ with exponential cones on the right-hand side.