---
title: "Complementary Slackness"
type: concept
source: https://www.jemoka.com/posts/kbhcomplementary_slackness/
confidence: high
status: active
---

Consider:
\begin{align} f_{0}\qty(x^{*}) &amp;= \text{inf}_{x} \qty(f_{0}\qty(x) + \sum_{i=1}^{m} \lambda_{i}f_{i}\qty(x) + \sum_{i=1}^{p} v_{i}h_{i}\qty(x)) \\ &amp;\leq f_{0}\qty(x) + \dots \\ &amp;\leq f_{0}\qty(x) \end{align}
So the inequality holds strictly
