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  "body": "\n## Summary\n\nA saddle point of a Lagrangian is a pair made of a point and a vector of multipliers.\nAt this pair the Lagrangian of a constrained optimization problem is minimal in the point and maximal in the multipliers.\nWhenever such a pair exists, the point solves the constrained problem and the multipliers satisfy complementary slackness.\nFor convex problems that satisfy a constraint qualification, saddle points exist and they are the basis of Lagrangian duality.\nThe [[definition|QQMQGXS0XE]] gives the precise conditions.\n\n## Explanations\n\n### L1 human\n\nA constrained problem asks for the smallest value of $f$ among points that obey some constraints.\nThe Lagrangian adds to $f$ a penalty for each constraint, with a price $\\lambda_i \\ge 0$ per unit of violation.\nAt a saddle point, the chooser of the point cannot gain by moving and the chooser of the prices cannot gain by changing them.\nFor example, to minimize $x^2$ subject to $x \\ge 1$, the point $x = 1$ with price $\\lambda = 2$ is a saddle point.\nUnlike the [[saddle point|0JW6MDHAZD]] of a surface in calculus, it is defined by inequalities, not by derivatives.\n",
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