I have the following linear program:

\begin{equation} \min \sum_{\vec{s} \in S} \sum_{\vec{z} \in \Xi_{\vec{s}}} c(\vec{s}, \vec{z}) \times \chi(\vec{s}, \vec{z}) \label{eqADP4} \end{equation} \begin{equation} (1 - \lambda) \times \sum_{\vec{s} \in S} \sum_{\vec{z} \in \Xi_{\vec{s}}} \chi(\vec{s}, \vec{z}) = 1 \end{equation} \begin{equation} \sum_{\vec{s} \in S} \sum_{\vec{z} \in \Xi_{\vec{s}}} \pi_{koie}(\vec{s}, \vec{z}) \times \chi(\vec{s}, \vec{z}) \geq \mathbb{E}_{\eta} [p^w_{koie}] \hspace{1cm} \forall k \in \mathcal{K}; o \in \mathcal{O}; i \in \mathcal{I}_k; e \in \mathcal{E}_k \end{equation} \begin{equation} \sum_{\vec{s} \in S} \sum_{\vec{z} \in \Xi_{\vec{s}}} \delta_{ktc}(\vec{s}, \vec{z}) \times \chi(\vec{s}, \vec{z}) \geq \mathbb{E}_{\eta} [p^m_{ktc}] \hspace{1cm} \forall k \in \mathcal{K}; t \in \mathcal{T}'; c \in \mathcal{C} \end{equation} \begin{equation} \chi(\vec{s}, \vec{z}) \in \mathbb{R}_{\geq0} \end{equation}

By associating variables $y^1 \in \mathbb{R}$, $y^2_{koie} \in \mathbb{R}_{\ge0}$ and $y^3_{ktc} \in \mathbb{R}_{\ge0}$ with the first, second and third constraint sets, respectively, I specify its dual model as follows:

\begin{equation} \max y^1 + \sum_{k,o,i,e} y^2_{koie} \times \mathbb{E}_{\eta} [p^w_{koie}] + \sum_{k,t,c} y^3_{ktc} \times \mathbb{E}_{\eta} [p^m_{ktc}] \end{equation} \begin{equation} (1 - \lambda) \times y^1 \leq 0 \end{equation} \begin{equation} \sum_{k,o,i,e} \pi_{koie}(\vec{s}, \vec{z}) \times y^2_{koie} + \sum_{k,t,c} \delta_{ktc}(\vec{s}, \vec{z}) \times y^3_{ktc} \leq c(\vec{s}, \vec{z}) \hspace{0.5cm} \forall \vec{s} \in S; \vec{z} \in \Xi_{\vec{s}} \end{equation} \begin{equation} y^1 \in \mathbb{R}, y^2_{koie} \in \mathbb{R}_{\ge0}, y^3_{ktc} \in \mathbb{R}_{\ge0} \end{equation}

However, I am not getting the same result! Please let me know if you can identify any mistakes.


1 Answer 1


Your first dual constraint is wrong. The term $(1-\lambda)\times y^1$ should be included in the left side of the second dual constraint.


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