I have a binary programing problem of the form:
The input variable of this optimization problem is matrix $X$ that its entries are just $0$ or $1$, it is a $N \times M$ matrix. $t$ is the objective function that is connected to input variable through constraint (1). $\beta_{ij}$ and $B_{ij}$ are two $N \times M$ constant matrices. $B_{min}$ is a constant scalar.
In order to solve this problem, I use Lagrangian multipliers to get the dual problem of this primal problem.
The final solution of the primal violates the constraint (2), the result of nested sum is much higher than $B_{min}$. It seems that the solver completely ignores constraint (2) of the primal.
I want to know if there are mathematical problems with my approach that I get this result.