Suggest edit — perturbation analysis

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---
title: "perturbation analysis"
type: concept
source: https://www.jemoka.com/posts/kbhpreturbation_analysis/
confidence: high
status: active
---

Let the unpreturb problem be:
\begin{align} \min_{x}\quad &amp; f_{0}\qty(x) \\ \textrm{s.t.} \quad &amp; f_{i}\qty(x) \leq 0, i = 1 \dots m \\ &amp; h_{i}\qty(x) = 0, i = 1 \dots p \end{align}
Preturbed on is just:
\begin{align} \min_{x}\quad &amp; f_{0}\qty(x) \\ \textrm{s.t.} \quad &amp; f_{i}\qty(x) \leq u_{i} \\ &amp; h_{i}\qty(x) = v_{i} \end{align}
Global Tightness So we can get a lower bound:
\begin{equation} p^{*}\qty(u,v) \geq g\qty(\lambda^{*}, v^{*}) - u^{T} \lambda^{*} - v^{T}\lambda^{*} \end{equation}
by subtracting the original strictly feasible.
if \(\lambda_{i}\) is large, if \(u\) decreases (i.e. \(u &lt; 0\)), then \(p\) increases greatly if \(\lambda_{i}\) is small, if \(u\) increases (i.e. \(u &gt;0\)), then we can&rsquo;t say anything about \(p\) Local Sensitivity \begin{equation} \lambda_{i}^{*} = - \pdv{p^{*}\qty(0,0)}{u_{i}} \end{equation}
\begin{equation} v_{i}^{*} = - \pdv{p^{*}\qty(0,0)}{v_{i}} \end{equation}