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Consider the standard LP with value function \begin{equation} \Omega(\mathbf{A}, \mathbf{b}, \mathbf{c}) = \max_{\mathbf{x}} \left\{\mathbf{c} \cdot \mathbf{x} \,|\, \mathbf{A}\mathbf{x} \leq \mathbf{b}, \mathbf{x} \geq \mathbf{0}\right\} \end{equation} and given parameters of suitable dimensions. Supposing primal non-degeneracy, we know the $k$-th Lagrange multiplier $\lambda_k = \partial \Omega / \partial b_k$ has the shadow price interpretation.

Are there any general results about $\partial \lambda_k / \partial c_\ell$, perhaps under additional assumptions about the parameters?

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  • $\begingroup$ Let $z=\textbf{cx}$ the profit of $n$ products with $m$ constraints, where $1\leq k\leq m$. If you raise the bound $b_k$ by one unit, then the optimal value $z^*$ raises by $\lambda_k$. I can post an example if you want. $\endgroup$ Jan 31, 2023 at 18:37
  • $\begingroup$ Thank you for the comment, but I have already noted this in the question. $\endgroup$
    – bodhi
    Jan 31, 2023 at 18:42
  • $\begingroup$ Sorry . I don't see that any additional assumption has to be made. $\endgroup$ Jan 31, 2023 at 18:44
  • $\begingroup$ You seem to answer a different question than I asked: I am interested in the derivative of $\lambda_k$ in the $\ell$-th coefficient in $\mathbf{c}$ of the value function $\Omega$. $\endgroup$
    – bodhi
    Jan 31, 2023 at 18:46

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