I do not understand the connection between complementarity problems and optimization problems. I have tried to look at other definitions for complementarity problem to see if that would help me with the connection but it did not.
My question is:
What excatly is a complementarity problem? - This is where my main problem is because I do not understand the definition of a complementarity problem. Is it possible to show/explain the definition graphically? Maybe using cones and dual cones?
Why (and when) are complementarity problems useful for optimization problems?
Best Husky
Complementarity problems is simply optimization problems with a special kind of constraints. Essentially orthogonality constraints between two non-negative vectors, $x\geq 0, y\geq 0, x^Ty = 0$.
This arise, for example, in purely geometric applications (orthogonality constraints), or in situations where you want to encode either-or conditions ($x_i$ is zero or $y_i$ is zero, such as when you want to encode optimality conditions where either the dual variable is zero, or the slack is zero, or both)