CVXPY : Minimising with parameter set to 0 and minimising without parameter gives different answers

Question

When trying to minimise an objective through CVXPY, I have two different optimisation problems. When a parameter alpha is set to 0, both these objectives should give the same minimisation results. But for me it gives two different results.

These are the two problems Problem 1 :

w = cp.Variable(shape = m)
 alpha = cp.Parameter(nonneg=True)
 w_sb = w[some_edge_indices]
 w_ob = w[other_edge_indices]
 MKw = MK @ w
 MKsbwb = MK_sb @ w_sb
 MKobwb = MK_ob @ w_ob
 MKswm = MK_some @ w_some
 MKowm = MK_other @ w_other
 alpha.value = alph
 obj1  = cp.sum_squares(MKw)
 obj2 = cp.sum_squares(MKsbwb - MKswm)
 obj3 = cp.sum_squares(MKobwb - MKowm)
 reg = obj2 + obj3
 objective = cp.Minimize(obj1 + alpha*(reg))
 constraints = [AK@w >= np.ones((n,))] 
 prob = cp.Problem(objective, constraints)
 result = prob.solve()

Consider all the unknows variables to be some given matrices. Also alph is a given value.

Problem 2:

 w = cp.Variable(shape = m)
 MKw = MK @ w
 obj1  = cp.sum_squares(MKw)
 objective = cp.Minimize(obj1)
 constraints = [AK@w >= np.ones((n,))] 
 prob = cp.Problem(objective, constraints)
 result = prob.solve()

Here, as we can see when alpha = 0, both the objectives should return the same w. But it is giving different w values. What could be the reason?