Let $\mathbf{c}_i,\mathbf{s}_i$ be given entry-wise positive $n\times 1$ vectors for $i\in[1,\dots,d]$. Let $\tau, \alpha_1,\dots, \alpha_d$ be given positive constants. Consider the linear programming problem to find vectors $\mathbf{x}_1,\dots,\mathbf{x}_d$ for the joint optimization problem \begin{align} \max_{\mathbf{x}_i,\forall i\in[1,\dots,d]}~\sum_{i=1}^{d}\mathbf{c}_i^T\mathbf{x}_i \\ s.t.~~&\sum_{i=1}^{d}\left(\mathbf{s}_i-\tau\mathbf{c}_i\right)^{T}\mathbf{x}_i \leq 0,\\ &\mathbf{s}_i^T\mathbf{x}_i\leq \alpha_i~,~\forall i \in[1,\dots,d] \\ &\mathbf{1}^T\mathbf{x}_i\leq 1~,~\mathbf{x}_i\geq \mathbf{0} \end{align} where $\mathbf{1}$ is the all-ones vector.

Due to the hardware/resource limitations we have, we need to resort to (assume this limitation is non-negotiable) solving a set of $d$ linear optimization problems which is somewhat a decoupled form of the original. Essentially, we solve for each $\mathbf{x}_i$ using the optimization problem \begin{align} \max_{\mathbf{x}_i}~\mathbf{c}_i^T\mathbf{x}_i \\ s.t.~~&\left(\mathbf{s}_i-\tau\mathbf{c}_i\right)^{T}\mathbf{x}_i \leq 0,\\ &\mathbf{s}_i^T\mathbf{x}_i\leq \alpha_i \\ &\mathbf{1}^T\mathbf{x}_i\leq 1~,~\mathbf{x}_i\geq \mathbf{0} \end{align} Our engineering solution now is to use the sub-optimal solutions $\mathbf{x}_1,\dots,\mathbf{x}_d$ from the decoupled sub problems as the solution to the original one. Note that any set of such solutions from the decoupled subproblems is a feasible solution to the original. How much of a blunder are we making? Is there any study on such problems. Simulating on sample data from our system doesn't show much degradation. But, we would like to understand the technical aspect of it.