Inequality constraints of convex relaxation with McCormick envelope
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I have a nonconvex optimization problem for which I am calculating a lower bound using the [McCormick envelope.](https://optimization.mccormick.northwestern.edu/index.php/McCormick_envelopes) Each bilinear term is replaced with an auxiliary variable which has the following constraints defined:
> w_{ij} >= x_i^L * x_j + x_i * x_j^L - x_i^L * x_j^L \
> w_{ij} >= x_i^U * x_j + x_i * x_j^U - x_i^U * x_j^U \
> w_{ij} <= x_i^U * x_j + x_i * x_j^L - x_i^U * x_j^L \
> w_{ij} <= x_i^L * x_j + x_i * x_j^U - x_i^L * x_j^U \
where
> x_U <= x <= x_L
I am given a function taking in several arguments:
```
def convex_bounds(n,m,c,H,Q,A,b,lb,ub):
# n is the number of optimization variables
# m is the number of eq constraints
# H = positive, semidefinite matrix from objetcive function (n x n)
# Q is (mxn) x n
# A is m x n
# b is RHS of non linear eq constraints (m x 1)
# c,lb,ub are vectors size (n x 1)
......................................
# Create matrix B & b_ineq for inequality constraints
# where B*x <= b_ineq
B = np.eye(3)
b_ineq = np.array((10,10,10))
## these values would work in a scenario with no bilinear terms
```
My problem is that I don't know **how to specify the inequality constraints matrix B and vector b_ineq**. For this particular exercise my variables are x1, x2 and x3 with bounds 0 (x_L) and 10 (x_U). My bilinear terms are x_12 and x_23 (which will lead to auxiliary variables w_12 and w_23). How can I specify the known bounds (0 and 10) for x1,x2 and x3 and the calculated ones (as in the theory pasted above) in B and b_ineq?
I don't actually know how to proceed with this.
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My problem is that I don't know **how to specify the inequality constraints matrix B and vector b_ineq**.
I strongly recommend to have a look at our Python webinar series starting with Introduction to Modeling with Python.
This should be a good starting point when beginning to work with gurobipy.
How can I specify the known bounds (0 and 10) for x1,x2 and x3 and the calculated ones (as in the theory pasted above) in B and b_ineq?
Non-default bounds for variables can be set while adding them via the addVar(s) method or later via the corresponding attributes.
Best regards,
Jaromił0
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