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Suppose I have a Master Problem (MP) with several inequality constraints for the decision variables, e.g. $$\min c^Tx \quad \text{s.t.} \quad Ax \leq b, \quad \Vert x\Vert_1 \leq r, \quad x\geq 0.$$ Suppose the columns of $A$ are the vectors of the current basis matrix for the problem and we want to add a new column.

How should I calculate the reduced cost vector?

Attempt, calculate optimal $x^*$ for the basis $A$ and then define $\overline{c}_j = c_j - \pi_0 a_j - \pi_1$. Where $\pi_0$ corresponds to the constraint $Ax \leq b$ and $\pi_1$ corresponds to $\Vert x\Vert_1 \leq r$.

M....
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1 Answers1

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First rewrite in standard form as \begin{align} &\text{minimize} &\sum_j c_j x_j \\ &\text{subject to} &-\sum_j A_{ij} x_j &\ge - b_i && \text{for all $i$} && (\pi_i \ge 0) \\ &&-\sum_j x_j &\ge -r && && (\alpha \ge 0) \\ &&x_j &\ge 0 &&\text{for all $j$} \end{align} The reduced cost of $x_j$ is $c_j + \sum_i \pi_i A_{ij} + \alpha$.

RobPratt
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  • Thank you very much! Could you elaborate on how to get the reduced cost from standard form? Is there any resource that explains this type of stuff? – M.... Mar 14 '25 at 01:48
  • In general, it is $c-\pi A$, so you just read off the coefficients of $x_j$ and multiply by the dual variables. Chvatal's Linear Programming (1983) is a standard reference. – RobPratt Mar 14 '25 at 02:01