含乘积项非线性转线性规划

假设有如下规划问题，其中变量有\(\mu , w_i, x_{ij}\)，含了两变量乘积项，属于非线性规划。

\begin{align} &\min \mu\\ \text{Such that: }& w_0U_{ij}+w_jZ_i+w_iZ_j\leq \mu Bx_{ij},\text{ for any }i,j=1,2,...,N \\ &\sum_{i=0}^N w_i=1 \\ & w_i\geq 0, i=0, 1, 2,...,N \\ & \sum_{j=1}^N x_{ij}\leq R_i, \sum_{i=1}^N x_{ij}\leq R_j,\\ & x_{ij}\text{ are non-negative integers and }x_{ii}=0 \end{align}

直接解，会报错类似：gurobipy.GurobiError: Q matrix is not positive semi-definite (PSD)

可以令 \(\alpha = \frac{1}{\mu}\) 后，转换为线性规划。此时有新变量 \(y_{i} = \alpha w_{i}\)。

\begin{eqnarray} &\max \alpha\\ \text{Such that: }& y_0U_{ij}+y_jZ_i+y_iZ_j\leq Bx_{ij},\text{ for any }i,j=1,2,...,N \\ &\sum_{i=0}^N y_i=\alpha \\ & y_i\geq 0, i=0, 1, 2,...,N \\ & \sum_{j=1}^N x_{ij}\leq R_i, \sum_{i=1}^N x_{ij}\leq R_j,\\ & x_{ij}\text{ are non-negative integers and }x_{ii}=0 \end{eqnarray}