Minimize$$\sum_{i=1}^{n}\sum_{j=1}^{m_i}w_{i,j}v_{i,j}$$subject to$$\sum_{i=1}^{n}\frac{m_i}{m_i+\sum_{j=1}^{m_i}v_{i,j}} < \theta$$$$v_{i,j}\in\{0,1\}~\forall i,~j$$where $w_{i,j}$ and $m_i$ are positive integers (constants)$\theta$ is a positive real number (constant)$v_{i,j}$ are 0-1 variables is this problem NP-hard?I think this is 'harder' than knapsack problem and probably should be NP-hard. I surveyed some literature but failed to understand them. 
Complexity of a non-linear knapsack problem
complexity theory;optimization;np hard;knapsack problems;packing