# Nature of stationary points of a Lagrangian fuction

I would like to extremize a certain function $$f$$ with respect to a parameter $$x$$, under constraints $$g_1(x) = 0, ..., g_m(x)=0$$. In order to achieve this, I consider the Lagrangian function $$L(x, \lambda) = f(x) - \sum_{i=1}^m\lambda_i g_i(x)$$ and look for a stationary point.

In my particular case, this amounts to looking for a saddle point, which is not convenient for me because I have to perform several extremizations before I find a satisfactory point (for example by using Uzawa's algorithm). I'd rather have to look for a minimum or a maximum, that I can reach with a single gradient descent/BFGS run/whatever.

So, I was wondering: is there any way I can "rephrase" my optimization problem so that the stationary point of the Lagrangian is an extremum, or are there some cases where a saddle point is the best I can hope for?

• Welcome to SciComp.SE! What do you consider a "stationary point" of a Lagrangian? (Also, are you assuming any convexity?) – Christian Clason Nov 2 '18 at 15:47