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  "body": "\n## Summary\n\nConstrained optimisation looks for the maximum or minimum of an objective function $f$ of several variables over the set of points that satisfy given constraints.\nTypical constraints are equations $g_i(x) = c_i$ and inequalities $h_j(x) \\ge d_j$.\n\n## Remarks\n\nAn example is to maximise $f(x, y) = x y$ on the circle $x^2 + y^2 = 2$. The maximum value is $1$, at $(1, 1)$ and $(-1, -1)$.\nFor equality constraints, one method solves the constraints for some variables and substitutes them into $f$. Another method uses [[Lagrange multipliers|0VKKP8X6Q6]].\nA constrained extremum need not be a critical point of $f$. In the example above, $\\nabla f(1, 1) = (1, 1) \\ne 0$.\n\n## Explanations\n\n### L1 human\n\nIn many problems you cannot choose the variables freely.\nFor example, you want the largest area for a fixed length of fence.\nConstrained optimisation finds the best value only among the choices that obey the rules.\n",
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