In my problem, I have to maximize a convex function $f(x_1,x_2,\cdots,x_n)$ subject to two equality constraints $g_1=0$ and $g_2=0$. As usual, I constructed the Lagrangian $L=f+\lambda_1g_1+\lambda_2g_2$. Due to high nonlinearity, the equations $\nabla L=0$ can not be solved analytically to get all the stationary points.
However, if I find a point $x^*$ and two values $\lambda^*$ such that they satisfy $\nabla L=0$, as well as the leading principal minors of the Hessian $\frac{\partial^2 L}{\partial \lambda_i\partial\lambda_j}$ changes sign, then I know that $x^*$ is a local maximum of $f$. Since $f$ is convex (over a convex set), does it follow that $x^*$ is also a global maximum?