I have this formulated Generalized Assignment Problem (GAP) or it can also be considered as Binary integer programming problem. Solving this problem can be achieved through Branch and Bound Technique.

$max \text{ } \sum_{i=1}^{M}\sum_{j=1}^{N} f_{ij}x_{ij}$

$subject \text{ }to\text{ } $

$\sum_{i=1}^{M} B_{i}x_{ij}\leqslant C_j $

$\sum_{j=1}^{N}x_{ij}\leqslant 1 $

$x_{ij}\in\{0,1\} $

However, I am interested in the Lagrangian relaxation of the problem. What I really need is to relax all the constraints. However, I can't manage to find the KKT conditions for the problem. The optimal (or near optimal) solution for the multipliers can be estimated through sub gradient method, however I need an analytic way to get calculate the multipliers (Maybe through KKT conditions). At least I need the KKT conditions of the problem, and its relaxed version. Thank you in advance