> For the complete documentation index, see [llms.txt](https://theshank.gitbook.io/ai/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://theshank.gitbook.io/ai/optimization/lagrange-multiplier.md).

# Lagrange Multiplier

Nice was to look at lagrange, which might also be related to implicit function theorem:

$$
\min\_x f(x) \quad \text{s.t.} \quad C(x)=0
$$

To solve the above minimization problem, we can use lagrange multiplier as follows

$$
\mathcal{L}(x,\lambda) = f(x) + \lambda C(x) \\
g(\lambda) = \mathcal L(x^*(\lambda), \lambda) \quad \text{where} \quad x^* = \arg \min\_x \mathcal L(x, \lambda)
$$