I am learning optimization through a course on Youtube. I have one confusion in solving the following problem. As per my understanding, the objective function is not convex. The problem is a constraint optimization problem.
Please see the following problem
$$\begin{array}{ll} \text{minimize} & \frac{1}{3} x^3 - \frac{3}{2} y^2 + 2x\\ \text{subject to} & x - y = 0\end{array}$$
As per my understanding, the given function should not be a convex function because
When I draw the graph, I saw that the line joining the two points on the convex set sometimes lies above, while other times lie below the graph itself. (See the attached figure I plotted in Python).
When I found the eigenvalue for the function using the Hessian matrix, it came out to be negative. Again, as per my knowledge about convex programming, a convex function has its eigenvalue as positive.
It can be seen from the graph clearly, that the point where gradient comes out to be zero is a saddle point and neither local minima, nor maxima.
Still, the instructor solved this problem as a convex problem using Lagrange function. Is there anything that I am missing or did the instructor made a mistake (by the way, the instructor taught very well generally)
