Non-linear optimization problem with more constraints than variables

Hello all,

For the proof of a game-theoretic proposition I need to solve a non-convex non-linear optimization problem. The problem has both equality and inequality constraints (but can be turned into a problem with all inequality constraints). I wanted to solve this problem using the Lagrange / KKT approach, but ran into the following difficulty.

The original problem has 3*n variables and 4*n + 1 constraints. Therefore, when I use the Lagrange / KKT approach I end up with a system of 3*n + 4*n + 1 variables and 3*n equations.

Although I can eliminate some Lagrangian multipliers, I still have a system with more variables than equations - and hence, a system which I cannot solve.

Any ideas on how to tackle this problem?

Thanks for your consideration.

Kris