Using log optimization function in Gurobi

Question

I'm trying to use Gurobi to model an optimization max function whose objective function is $$f(x)=\frac{r^Tx}{x^TQx}.$$

Thus maximizing this function $f(x)$ is the same as maximizing $\log f(x)$ and I want to implement it in Gurobi. My understanding is Gurobi doesn't handle log objective functions so I've made a linear maximization function $z(x) = t$ as follows: \begin{align}\max&\quad z(x)=t\\\text{s.t.}&\quad\log(r^Tx) - \log(x^TQx) - t = 0\tag{C1}\\&\quad\vdots\quad\text{(other equality constraints)}.\end{align} Does anyone know how to implement constraint C1 in Gurobi?

Answer 1

I will make the following assumptions:

1. $Q$ is symmetric positive semidefinite
2. $r^Tx \ge 0$

Note that maximizing $\frac{r^Tx}{X^TQx}$ is equivalent to minimizing $\frac{X^TQx}{r^Tx}$ .

Under the stated assumptions, this is a convex optimization problem, which can be submitted to and solved by Gurobi.

The easiest way of doing so is using a convex optimization tool such as CVX or CVXPY. The CVX code would be

cvx_begin
cvx_solver gurobi
variable x(n)
minimize(quad_over_lin(chol(A)*x,r'*x))
cvx_end

which uses the CVX function quad_over_lin. chol(A) is the upper triangular Cholesky factor of A.

CVXPY also has the same quad_ovdr_lin function as CVX, but depending on which function is called to compute the Cholesky factor, you would need to apply a transpose to the Cholesky factor if a lower triangular Cholesky factor is generated

This could instead be done using a "native" Gurobi interface. In that case, you would need to minimize a new variable, t, subject to the rotated second order constraint:

norm(chol(A)*x) <= sqrt(t*r'*x), t >= 0, r'*x >= 0

using the appropriate syntax to specify this constraint for whatever interface you are using. As before, if chol(A) is lower triangular, you need to use its transpose.