# KKT conditions (equations) for Generalized Assignment Problem or Binary integer programming problem

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.

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

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

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

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

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

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

• using the gradient (lagragian or kkt conditions) requires continuous variables, and with $x_{ij} \in [0;1]$ it becomes a linear programming problem. so what ? (note this is the trick of branch and cut : relaxing the integer constraint of integer programming problems to get linear programming from which we get bounds on the solution and allow us to cut the search tree) – reuns Jan 9 '16 at 5:11
• So as I understand from you, using KKT conditions is not applicable because all variables x are binary? In other meaning, Variables x are not continuous so we cannot apply KKT conditions? thank you for answering me, I appreciate it a lot. – mohamad zalghout Jan 10 '16 at 22:38
• I have read in a book that finding optimal or near optimal multipliers for this problem can be done using sub-gradient method. However I am not sure that I have understand It correctly. So my question is: can we find the lagrangian multipliers for this problem using sub-gradient method? – mohamad zalghout Jan 10 '16 at 22:44
• you are right, there seem to be possibilities by replacing the binary constraint $x_i \in \{0,1\}$ by the continuous constraint $x_i(x_i-1) = 0$ usc.edu/dept/ise/caie/Checked%20Papers%20[ruhi%2012th%20sept]/… – reuns Jan 10 '16 at 23:12
• yes, but actually after converting the binary constraint to a continuous one, while deriving the kkt conditions, we get a system of n equations with n unkowns, however some of these equations are quadratic due to x(x-1)=0, i can't seem to have solution for that. Althogh the paper you have cited says that it can be solved using kkt. do you have any idea? – mohamad zalghout Jan 17 '16 at 10:41