Support Vector Machines turn machine learning linear classification tasks into a linear optimization problems.

$$ text{minimize } J(theta,theta_0) = frac1n sum_1^n text{HingeLoss}(theta,theta_0) + frac{lambda}{2} ||theta||^2 $$

My question is, what linear programming runs on the background for the minimization of the objective function $J$. Is it Simplex?