Saddle point of a Lagrangian

Summary

A saddle point of a Lagrangian is a pair made of a point and a vector of multipliers. At this pair the Lagrangian of a constrained optimization problem is minimal in the point and maximal in the multipliers. Whenever such a pair exists, the point solves the constrained problem and the multipliers satisfy complementary slackness. For convex problems that satisfy a constraint qualification, saddle points exist and they are the basis of Lagrangian duality. The definition gives the precise conditions.