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}