1.2.2 Constrained optimization

Now suppose that $D$ is a surface in $\mathbb{R}^n$ defined by the equality constraints

$$h_1(x)=h_2(x)=\dots=h_m(x)=0$$ (1.18)

where $h_i$ , $i=1,\dots,m$ are $\mathcal C^1$ functions from $\mathbb{R}^n$ to $\mathbb{R}$ . We assume that $f$ is a $\mathcal C^1$ function and study its minima over $D$ .