I have the following optimization problem:
\begin{align} \min_{x,s,\lambda} \quad J(x)\\ f(x)=0\\ \lambda^\top x \leq s\\ \lambda^\top e=1\\ \lambda\geq 0 \end{align}
where $x \in \mathcal{X} \subset \mathbb{R}^n$, $s \in \mathbb{R}$, $\mathcal{X}$ is a convex set, $f(x)$ and $J(x)$ are affine and quadratic functions respectively. $e$ is a vector consisting of all ones.
I know that the second constraint is a bilinear one. Is there any method to reformulate this problem into a convex program?
I have tried solving this using fmincon and other nonlinear solvers, but either I get an error, or the solution is not the desired one.