what if the hessian of the Lagrangian is null?

Question

In a non-linear optimization problem, suppose given a candidate solution $x*$, we want to verify second order sufficient conditions of optimality but the Hessian of the Lagrangian of the problem at $x*$ is null.

What could we conclude in this case?

Answer 1

You conclude nothing. For instance, if

• $f(x,y)=x^4+y^4$;
• $g(x,y)=x^3+y^3$;
• $h(x,y)=-x^4-y^2$

then, in each case, $(0,0)$ is a critical point and the Hessian at $(0,0)$ is the null matrix. But $(0,0)$ is

• the point at which $f$ attains its absolute minimum;
• a saddle point of $g$;
• the point at which $h$ attains its absolute maximum.