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--- title: "perturbation analysis" source: https://www.jemoka.com/posts/kbhpreturbation_analysis/ --- Let the unpreturb problem be: \begin{align} \min_{x}\quad & f_{0}\qty(x) \\ \textrm{s.t.} \quad & f_{i}\qty(x) \leq 0, i = 1 \dots m \\ & h_{i}\qty(x) = 0, i = 1 \dots p \end{align} Preturbed on is just: \begin{align} \min_{x}\quad & f_{0}\qty(x) \\ \textrm{s.t.} \quad & f_{i}\qty(x) \leq u_{i} \\ & 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. \(u < 0\)), then \(p\) increases greatly if \(\lambda_{i}\) is small, if \(u\) increases (i.e. \(u >0\)), then we can’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}