Minimum under an equality constraint

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Problem

Minimize $x^2+y^2$ subject to $x+y=1$ using a Lagrange multiplier. Report the optimizer, multiplier for $L=x^2+y^2+\lambda(x+y-1)$ , and minimum value.

Reveal answer or reference solution

Stationarity gives $2x+\lambda=0$ , $2y+\lambda=0$ , and $x+y=1$ . Hence $x=y=1/2$ and $\lambda=-1$ . The minimum value is

x^2+y^2=\frac14+\frac14=\frac12

The objective is strictly convex and the constraint is affine, so this feasible stationary point is the unique global minimum.

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Minimum under an equality constraint

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