Algebraic Formulation - Note: All parameter values are subject to change, I have just used the numbers you see here as place holders for the time being. Additionally, there are a few other equations but I am highly confident that they are not causing the infeasibility problems that I am having. \begin{align}\min\quad&\sum_{i=1}^I\sum_{j=1}^J\sum_{t=1}^T(2+5 L_{t}^{ij}) I_{t,post}^{ij}\\\text{s.t.}\quad&I_{t,post}^{ij} = \frac{I_{t-1,treated}^2} {I_{t-1,treated}^2 + a^2} S_{t-1}^{ij}\\\quad&I_{t,treated} = (1 - 0.5L_{t}^{ij})I_{t,post}^{ij}\\\quad&S_{t}^{ij} = S_{t}^{ij} + sj_{t-1,k=2}^{ij} - I_{t-1,treated}\\\quad&\sum_{t=1}^T\sum_{i=1}^I\sum_{j=1}^J2L_{t}^{ij} \le B \\\quad&0 \le I_{t,post}^{ij},I_{t,treated}^{ij},S_{t}^{ij} \\ \quad&0 \le I_{0,treated}^{ij} \,\ S_{0}^{ij} \,\ j_{0,k=2}^{ij}\\\quad&L_{t}^{ij}=\,\text{ binary decision variable}\end{align}
I am running a MINLP model of invasive species control and I would ideally want all of my variables (except the binary variable L) to be in the NonNegativeReals domain, however, when I set this specification on any of the variables it forces some of my constraints to become infeasible. I can run the model with all variables in the Real domain and everything works out okay but it provides me with negative numbers which is not realistic in regards to the problems physical interpretation.
Consider the case where all variables are set to real and we look at the first few solutions for Constraint 4: Susceptible Recruitment under Output 1, we can see that at (1,1,3) that the variable begins taking negative values, and the culprit in the constraint is model.inf_treated[i,j,t-1].
# Constraint 4: Susceptible recruitment
def susceptible_advance_rule(model, i, j, t):
if t == 0:
return Constraint.Skip
else:
return model.susceptible[i, j, t ] == model.susceptible[i, j, t-1] + model.juvsurv * model.juvenille[i, j, t-1, Kmax] - model.inf_treated[i,j,t-1]
model.susceptible_advancement = Constraint(model.Iset, model.Jset, model.Tset, rule=susceptible_advance_rule)
Output 1: model.susceptible in the 'Reals' domain
susceptible : Size=20, Index=susceptible_index
Key : Lower : Value : Upper : Fixed : Stale : Domain
(1, 1, 0) : None : 50.0 : None : True : True : Reals
(1, 1, 1) : None : 53.2 : None : False : False : Reals
(1, 1, 2) : None : 19.28555555556563 : None : False : False : Reals
(1, 1, 3) : None : -32.15963392758387 : None : False : False : Reals
(1, 1, 4) : None : -50.31374332256311 : None : False : False : Reals
(1, 2, 0) : None : 30.0 : None : True : True : Reals
(1, 2, 1) : None : 30.700000000000003 : None : False : False : Reals
(1, 2, 2) : None : 14.750000000000497 : None : False : False : Reals
(1, 2, 3) : None : -15.149263681592133 : None : False : False : Reals
(1, 2, 4) : None : -29.438417445000606 : None : False : False : Reals
In the case, I would want to try and restrict the domain of model.susceptible to the NonNegativeReals and upon doing so I would expect that all of my model variables become greater than 0, but I get infeasibility in my constraints as a result, and my binary variables become non-binary as a symptom. The same infeasibility comes about if I then turn every one of my variables to exist in the domain of the NonNegativeReals, but I think the infeasibility comes from the conflict that arises from Constraint 4 when I restrict model.susceptible to be positive, but then I subtract away model.inf_treated.
Output 2: model.susceptible in the 'NonNegativeReals' domain
susceptible : Size=20, Index=susceptible_index
Key : Lower : Value : Upper : Fixed : Stale : Domain
(1, 1, 0) : 0 : 50.0 : None : True : True : NonNegativeReals
(1, 1, 1) : 0 : 53.2 : None : False : False : NonNegativeReals
(1, 1, 2) : 0 : 63.222379594592645 : None : False : False : NonNegativeReals
(1, 1, 3) : 0 : 58.85143663732718 : None : False : False : NonNegativeReals
(1, 1, 4) : 0 : 1.4845965610247883e-08 : None : False : False : NonNegativeReals
(1, 2, 0) : 0 : 30.0 : None : True : True : NonNegativeReals
(1, 2, 1) : 0 : 30.700000000000003 : None : False : False : NonNegativeReals
(1, 2, 2) : 0 : 14.750000001550784 : None : False : False : NonNegativeReals
(1, 2, 3) : 0 : 6.753462538302196 : None : False : False : NonNegativeReals
(1, 2, 4) : 0 : 3.620342424247316e-10 : None : False : False : NonNegativeReals
Infeasible Constraints When Variable Boundaries Appied
#Constraint 9: Treated Infestation
def treatment_rule(model, i, j, t):
if t == 0:
return Constraint.Skip
else:
return model.inf_treated[i, j, t] == model.inf_b4treat[i, j, t] * (1 - 0.5*model.level1[i, j, t] )
model.treated_population = Constraint(model.Iset, model.Jset, model.Tset, rule=treatment_rule)
INFO: CONSTR treated_population[1,1,1]: -21.71460181746589 != 0.0
INFO: CONSTR treated_population[1,2,2]: -6.629094367820306 != 0.0
INFO: CONSTR treated_population[2,1,2]: -7.140628683877534 != 0.0
INFO: CONSTR treated_population[2,2,1]: -15.008019076531863 != 0.0
Output From Solver With NonNegativeReals boundaries applied + Code 413 Description from Knitro
Problem:
- Lower bound: -inf
Upper bound: inf
Number of objectives: 1
Number of constraints: 180
Number of variables: 164
Sense: unknown
Solver:
- Status: warning
Message: Knitro 11.1.0\x3a MIP\x3a All nodes have been explored. No integer feasible point found.; objective 1398.5939509578325; integrality gap 1.8e+308; 3 nodes; 104 subproblem solves
Termination condition: maxIterations
Id: 413
Solution:
- number of solutions: 0
number of solutions displayed: 0
Code 413 Description: All nodes have been explored. No integer feasible point was found. The MIP optimality gap has not been reduced below the specified threshold, but there are no more nodes to explore in the branch and bound tree. If the problem is convex, this could occur if the gap tolerance is difficult to meet because of bad scaling or roundoff errors, or there was a failure at one or more of the subproblem nodes. This might also occur if the problem is nonconvex.
In theory, none of my variables should be negative when the model gets solutions. However, the question becomes the following: Why are the constraints becoming infeasible when I specify that all of my variables exist in the NonNegativeReals but are feasible when all domains are Real? Perhaps it has to do with the initial input values? Alternatively, perhaps there is a way that I can tell the model that if the model.susceptible variable goes negative, that instead it will just floor it to zero instead.
