How to Implement (Convex) Chance Constraints in Gurobi

Question

I would like to implement a chance constraint of the form $$ \mathbb{P}[a^\text{T}x\leqslant b]\geqslant\alpha$$ where $b\in\mathbb{R}$ is fixed, $a\in\mathbb{R}^n$ is a vector whose entries are iid and normally distributed with mean $\overline{a}$ and variance $\Sigma$ (that is, $a\sim\mathcal{N}(\overline{a},\Sigma)$), and $\alpha>1/2$ (so that the resulting problem is convex). It is well known that this constraint is equivalent to $$ F^{-1}(\alpha)\|\Sigma^{1/2}x\|_2\leqslant-\overline{a}^\text{T}x+b $$ which is a second-order cone constraint (SOC constraint). Moreover, Gurobi advertises that is has an SOCP solver. However, their documentation only advertises being able to solve the following three types of constraints: $$ x^\text{T}Qx+q^\text{T}x+b\leqslant0 \text{ where }Q\text{ is s.p.d.} $$ $$ x^\text{T}x\leqslant y^2, y\geqslant0 $$ $$ x^\text{T}x\leqslant yz, y,z\geqslant0 $$ I cannot wrangle the chance constraint into any of these three forms (I will be happy to show my attempts if desired). This seems like it should certainly be possible...how can I handle this in Gurobi?

Answer 1

Use the 2nd form:

$w = F^{-1}(\alpha) \Sigma^{1/2}x$

$y = -\overline{a}^\text{T}x+b $

$w^Tw \le y^2$

$ y \ge 0$

which collectively is equivalent to $ F^{-1}(\alpha)\|\Sigma^{1/2}x\|_2\leqslant-\overline{a}^\text{T}x+b $