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  "body": "\n## Definition\n\nLet $C \\subseteq \\R^n$ be a set, and let $f, g_1, \\dots, g_m \\colon C \\to \\R$ be functions.\nConsider the problem of minimizing $f(x)$ over $x \\in C$ subject to $g_i(x) \\le 0$ for $i = 1, \\dots, m$.\nIts *Lagrangian* is the function $L \\colon C \\times [0, \\infty)^m \\to \\R$ given by\n$$L(x, \\lambda) = f(x) + \\sum_{i=1}^m \\lambda_i g_i(x).$$\nA pair $(x^*, \\lambda^*) \\in C \\times [0, \\infty)^m$ is a *saddle point of the Lagrangian* if\n$$L(x^*, \\lambda) \\le L(x^*, \\lambda^*) \\le L(x, \\lambda^*) \\quad \\text{for all } x \\in C \\text{ and } \\lambda \\in [0, \\infty)^m.$$\n\n## Remarks\n\nThe function $L$ is minimal in the first variable and maximal in the second at $(x^*, \\lambda^*)$, as for a saddle point of any function on a product.\nThe numbers $\\lambda^*_i$ play the role of [[Lagrange multipliers|0VKKP8X6Q6]].\nIf $(x^*, \\lambda^*)$ is a saddle point, the left inequality forces $g_i(x^*) \\le 0$ for all $i$ and $\\lambda^*_i g_i(x^*) = 0$ for all $i$.\nThe right inequality then gives $f(x^*) = L(x^*, \\lambda^*) \\le L(x, \\lambda^*) \\le f(x)$ for every feasible $x$, so $x^*$ is optimal.\nThis needs no convexity; the converse needs convexity of $f$, $g_i$ and $C$ and a constraint qualification such as Slater's condition.\n\nExample: minimize $x^2$ over $C = \\R$ subject to $g(x) = 1 - x \\le 0$, with $L(x, \\lambda) = x^2 + \\lambda (1 - x)$.\nThe pair $(1, 2)$ is a saddle point.\nFor every $\\lambda \\ge 0$, $L(1, \\lambda) = 1 = L(1, 2)$.\nFor every $x$, $L(x, 2) = (x - 1)^2 + 1 \\ge 1$.\n\nNon-example: the pair $(1, 1)$ for the same problem.\nThe right inequality fails, since $L(\\tfrac12, 1) = \\tfrac34 \u003c 1 = L(1, 1)$.\n",
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