# Modification of subproblem objective function

I have the following decomposed compact model with the following master problem: \begin{align} &\text{minimize} &\sum_t \sum_s \text{slack}_{ts} \\ &\text{subject to} &\sum_i \sum_r \text{motivation}_{its}^r \lambda_{ir} + \text{slack}_{ts} & = \text{demand}_{ts} &&\forall t,s &&\\ &&\sum_r \lambda_{ir} &= 1 &&\forall i &&\\ &&\lambda_{ir} &\in\mathbb{Z}^+ &&\forall i,r\\ &&\text{slack}_{ts} &\ge 0 &&\forall t,s \end{align} and subproblems($$i$$): \begin{align} &\text{minimize} &0-\sum_{t,s} \pi_{ts} \text{motivation}_{its} - \mu_i \\ &\text{subject to} &-M(1-x_{its}) \le \text{motivation}_{its} - \text{mood}_{it} &\le M(1-x_{its}) &&\forall t,s \\ &&\text{motivation}_{its} &\le x_{its} && \forall t,s \\\ &&\alpha_{it} \sum_s x_{its} + \text{mood}_{it} &= 1 &&\forall t\\ &&\text{motivation}_{its} &\in[0,1] &&\forall t,s \\ &&\text{mood}_{it} &\in[0,1] &&\forall t \\ &&x_{its}&\in \{0,1\} &&\forall t,s\\ \end{align}

Now i want to additionally impose the constraint: $$\sum_i x_{its} \geq 1 \quad \forall t, s$$ My idea would be to add this constraint to the masterproblem, which yields the following constraint: $$\sum_i \sum_r \lambda_r x_{its}^r \geq 1 \quad \forall t, s$$ How would i need to modify my subproblem objective function to additionally cope for this?

If $$\gamma_{ts}$$ is the dual variable associated with the new constraint, then the objective function of the subproblem $$i$$ becomes: $$\text{minimize} \quad -\sum_{t,s} \pi_{ts} \text{motivation}_{its} - \mu_i - \sum_{t,s} \gamma_{ts} x_{its}$$
Your original problem is: \begin{align} &\text{minimize} &\sum_t \sum_s \text{slack}_{ts} \\ &\text{subject to} &\sum_i \text{motivation}_{its} + \text{slack}_{ts} &= \text{demand}_{ts} &&\forall t,s \\ &&\sum_i x_{its} &\ge 1 &&\forall t,s \\ &&-M(1-x_{its}) \le \text{motivation}_{its} - \text{mood}_{it} &\le M(1-x_{its}) &&\forall i,t,s \\ &&\text{motivation}_{its} &\le x_{its} && \forall i,t,s \\ &&\alpha_{it} \sum_s x_{its} + \text{mood}_{it} &= 1 &&\forall i,t\\ &&\text{motivation}_{its} &\in[0,1] &&\forall i,t,s \\ &&\text{slack}_{ts} &\ge 0 &&\forall t,s \\ &&\text{mood}_{it} &\in[0,1] &&\forall i,t \\ &&x_{its}&\in \{0,1\} &&\forall i,t,s\\ \end{align}
For the Dantzig-Wolfe reformulation, introduce $$\lambda_{ir} \ge 0$$ and substitute $$\text{motivation}_{its} = \sum_r \text{motivation}_{its}^r \lambda_{ir}$$ and $$x_{its} = \sum_r x_{its}^r \lambda_{ir}$$ in the (objective and) complicating constraints to obtain the master problem: \begin{align} &\text{minimize} &\sum_t \sum_s \text{slack}_{ts} \\ &\text{subject to} &\sum_i \sum_r \text{motivation}_{its}^r \lambda_{ir} + \text{slack}_{ts} &= \text{demand}_{ts} &&\forall t,s &&\text(\text{\pi_{ts} free})\\ &&\sum_i \sum_r x_{its}^r \lambda_{ir} &\ge 1 &&\forall t,s &&\text(\text{\gamma_{ts} \ge 0})\\ &&\sum_r \lambda_{ir} &= 1 &&\forall i &&\text(\text{\mu_i free})\\ &&\lambda_{ir} &\in\mathbb{Z}^+ &&\forall i,r\\ &&\text{slack}_{ts} &\ge 0 &&\forall t,s \end{align} The reduced cost of $$\lambda_{ir}$$ is $$0-\sum_{t,s} (\pi_{ts} \text{motivation}_{its}^r + \gamma_{ts} x_{its}^r) - \mu_i$$, so the subproblem for block $$i$$ is: \begin{align} &\text{minimize} &0-\sum_{t,s} (\pi_{ts} \text{motivation}_{its} + \gamma_{ts} x_{its}) - \mu_i \\ &\text{subject to} &-M(1-x_{its}) \le \text{motivation}_{its} - \text{mood}_{it} &\le M(1-x_{its}) &&\forall t,s \\ &&\text{motivation}_{its} &\le x_{its} && \forall t,s \\\ &&\alpha_{it} \sum_s x_{its} + \text{mood}_{it} &= 1 &&\forall t\\ &&\text{motivation}_{its} &\in[0,1] &&\forall t,s \\ &&\text{mood}_{it} &\in[0,1] &&\forall t \\ &&x_{its}&\in \{0,1\} &&\forall t,s\\ \end{align}
• Note that the symbol $\alpha$ is already used in OP's subproblem May 11 at 16:53
• @RobPratt Thanks. Assuming now instead of $\sum_i x_{its}\geq 1~ \forall t, s$, i now want to enforce $\sum_i motivation_{its}\geq 0.5\cdot demand_{ts} \quad \forall t, s$. Do i again need to modify it to $\sum_i \sum_r \lambda_r motivation_{its}^r \geq 0.5\cdot Demand_{ts} \quad \forall t, s$ and add the new duals $\gamma$ or is it sufficient to just add the constraint without $\lambda$? 2 hours ago
• Yes, you need to do the substitution in terms of $\lambda$, just like in the first equality constraint. 2 hours ago