Surrogate Models#

Some industrial applications require modeling complex processes that can result either in highly nonlinear functions or functions defined by a simulation process. In those contexts, optimization solvers often struggle. The reason may be that relaxations of the nonlinear functions are not good enough to make the solver prove an acceptable bound in a reasonable amount of time. Another issue may be that the solver is not able to represent the functions.

An approach that has been proposed in the literature is to approximate the problematic nonlinear functions via neural networks with ReLU activation and use MIP technology to solve the constructed approximation (see e.g. Heneao Maravelias 2011, Schweitdmann et.al. 2022). This use of neural networks can be motivated by their ability to provide a universal approximation (see e.g. Lu et.al. 2017). This use of ML models to replace complex processes is often referred to as surrogate models.

In the following example, we approximate a nonlinear function via Scikit-learn MLPRegressor and then solve an optimization problem that uses the approximation of the nonlinear function with Gurobi.

The purpose of this example is solely illustrative and doesn’t relate to any particular application.

The function we approximate is the 2D peaks function.

The function is given as

\[\begin{split} \begin{aligned} f(x) = & 3 \cdot (1-x_1)^2 \cdot \exp(-x_1^2 - (x_2+1)^2) - \\ & 10 \cdot (\frac{x_1}{5} - x_1^3 - x_2^5) \cdot \exp(-x_1^2 - x_2^2) - \\ & \frac{1}{3} \cdot \exp(-(x_1+1)^2 - x_2^2). \end{aligned}\end{split}\]

In this example, we want to find the minimum of \(f\) over the interval \([-2, 2]^2\):

\[y = \min \{f(x) : x \in [-2,2]^2\}.\]

The global minimum of this problem can be found numerically to have value \(-6.55113\) at the point \((0.2283, -1.6256)\).

Here to find this minimum of \(f\), we approximate \(f(x)\) through a neural network function \(g(x)\) to obtain a MIP and solve

\[\hat y = \min \{g(x) : x \in [-2,2]^2\} \approx y.\]

First import the necessary packages. Before applying the neural network, we do a preprocessing to extract polynomial features of degree 2. Hopefully this will help us to approximate the smooth function. Besides, gurobipy, numpy and the appropriate sklearn objects, we also use matplotlib to plot the function, and its approximation.

import gurobipy as gp
import numpy as np
from gurobipy import GRB
from matplotlib import cm
from matplotlib import pyplot as plt
from sklearn import metrics
from sklearn.neural_network import MLPRegressor
from sklearn.pipeline import make_pipeline
from sklearn.preprocessing import PolynomialFeatures

from gurobi_ml import add_predictor_constr

Define the nonlinear function of interest#

We define the 2D peak function as a python function.

def peak2d(x1, x2):
    return (
        3 * (1 - x1) ** 2.0 * np.exp(-(x1**2) - (x2 + 1) ** 2)
        - 10 * (x1 / 5 - x1**3 - x2**5) * np.exp(-(x1**2) - x2**2)
        - 1 / 3 * np.exp(-((x1 + 1) ** 2) - x2**2)
    )

To train the neural network, we make a uniform sample of the domain of the function in the region of interest using numpy’s arrange function.

We then plot the function with matplotlib.

x1, x2 = np.meshgrid(np.arange(-2, 2, 0.01), np.arange(-2, 2, 0.01))
y = peak2d(x1, x2)

fig, ax = plt.subplots(subplot_kw={"projection": "3d"})
# Plot the surface.
surf = ax.plot_surface(x1, x2, y, cmap=cm.coolwarm, linewidth=0.01, antialiased=False)
# Add a color bar which maps values to colors.
fig.colorbar(surf, shrink=0.5, aspect=5)
plt.show()
example1 2DPeakFunction

Approximate the function#

To fit a model, we need to reshape our data. We concatenate the values of x1 and x2 in an array X and make y one dimensional.

X = np.concatenate([x1.ravel().reshape(-1, 1), x2.ravel().reshape(-1, 1)], axis=1)
y = y.ravel()

To approximate the function, we use a Pipeline with polynomial features and a neural-network regressor. We do a relatively small neural-network.

# Run our regression
layers = [30] * 2
regression = MLPRegressor(hidden_layer_sizes=layers, activation="relu")
pipe = make_pipeline(PolynomialFeatures(), regression)
pipe.fit(X=X, y=y)
Pipeline(steps=[('polynomialfeatures', PolynomialFeatures()),
                ('mlpregressor', MLPRegressor(hidden_layer_sizes=[30, 30]))])
In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook.
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To test the accuracy of the approximation, we take a random sample of points, and we print the \(R^2\) value and the maximal error.

X_test = np.random.random((100, 2)) * 4 - 2

r2_score = metrics.r2_score(peak2d(X_test[:, 0], X_test[:, 1]), pipe.predict(X_test))
max_error = metrics.max_error(peak2d(X_test[:, 0], X_test[:, 1]), pipe.predict(X_test))
print("R2 error {}, maximal error {}".format(r2_score, max_error))
R2 error 0.9998444614075169, maximal error 0.08206139209028995

While the \(R^2\) value is good, the maximal error is quite high. For the purpose of this example we still deem it acceptable. We plot the function.

fig, ax = plt.subplots(subplot_kw={"projection": "3d"})
# Plot the surface.
surf = ax.plot_surface(
    x1,
    x2,
    pipe.predict(X).reshape(x1.shape),
    cmap=cm.coolwarm,
    linewidth=0.01,
    antialiased=False,
)
# Add a color bar which maps values to colors.
fig.colorbar(surf, shrink=0.5, aspect=5)
plt.show()
example1 2DPeakFunction

Visually, the approximation looks close enough to the original function.

Build and Solve the Optimization Model#

We now turn to the optimization model. For this model we want to find the minimal value of y_approx which is the approximation given by our pipeline on the interval.

Note that in this simple example, we don’t use matrix variables but regular Gurobi variables instead.

m = gp.Model()

x = m.addVars(2, lb=-2, ub=2, name="x")
y_approx = m.addVar(lb=-GRB.INFINITY, name="y")

m.setObjective(y_approx, gp.GRB.MINIMIZE)

# add "surrogate constraint"
pred_constr = add_predictor_constr(m, pipe, x, y_approx)

pred_constr.print_stats()
Restricted license - for non-production use only - expires 2025-11-24
Warning for adding constraints: zero or small (< 1e-13) coefficients, ignored
Model for pipe:
126 variables
61 constraints
6 quadratic constraints
60 general constraints
Input has shape (1, 2)
Output has shape (1, 1)

Pipeline has 2 steps:

--------------------------------------------------------------------------------
Step            Output Shape    Variables              Constraints
                                                Linear    Quadratic      General
================================================================================
poly_feat             (1, 6)            6            0            6            0

dense                (1, 30)           60           30            0           30 (relu)

dense0               (1, 30)           60           30            0           30 (relu)

dense1                (1, 1)            0            1            0            0


--------------------------------------------------------------------------------

Now call optimize. Since we use polynomial features the resulting model is a non-convex quadratic problem. In Gurobi, we need to set the parameter NonConvex to 2 to be able to solve it.

m.Params.TimeLimit = 20
m.Params.MIPGap = 0.1
m.Params.NonConvex = 2

m.optimize()
Set parameter TimeLimit to value 20
Set parameter MIPGap to value 0.1
Set parameter NonConvex to value 2
Gurobi Optimizer version 11.0.1 build v11.0.1rc0 (linux64 - "Ubuntu 20.04.6 LTS")

CPU model: Intel(R) Xeon(R) Platinum 8175M CPU @ 2.50GHz, instruction set [SSE2|AVX|AVX2|AVX512]
Thread count: 1 physical cores, 2 logical processors, using up to 2 threads

Optimize a model with 61 rows, 129 columns and 999 nonzeros
Model fingerprint: 0x00c16613
Model has 6 quadratic constraints
Model has 60 general constraints
Variable types: 129 continuous, 0 integer (0 binary)
Coefficient statistics:
  Matrix range     [6e-12, 1e+00]
  QMatrix range    [1e+00, 1e+00]
  QLMatrix range   [1e+00, 1e+00]
  Objective range  [1e+00, 1e+00]
  Bounds range     [2e+00, 2e+00]
  RHS range        [5e-03, 7e-01]
  QRHS range       [1e+00, 1e+00]
Presolve added 78 rows and 15 columns
Presolve time: 0.01s
Presolved: 149 rows, 144 columns, 1178 nonzeros
Presolved model has 3 bilinear constraint(s)

Solving non-convex MIQCP

Variable types: 102 continuous, 42 integer (42 binary)

Root relaxation: objective -5.297059e+01, 158 iterations, 0.00 seconds (0.00 work units)

    Nodes    |    Current Node    |     Objective Bounds      |     Work
 Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time

     0     0  -52.97059    0   30          -  -52.97059      -     -    0s
     0     0  -52.90717    0   29          -  -52.90717      -     -    0s
     0     0  -46.28654    0   33          -  -46.28654      -     -    0s
     0     0  -41.48312    0   33          -  -41.48312      -     -    0s
     0     0  -41.48312    0   38          -  -41.48312      -     -    0s
     0     0  -39.39217    0   40          -  -39.39217      -     -    0s
     0     0  -39.39217    0   41          -  -39.39217      -     -    0s
     0     0  -39.39217    0   40          -  -39.39217      -     -    0s
     0     0  -39.39217    0   41          -  -39.39217      -     -    0s
     0     0  -39.39217    0   41          -  -39.39217      -     -    0s
     0     0  -39.08086    0   39          -  -39.08086      -     -    0s
     0     0  -38.90309    0   41          -  -38.90309      -     -    0s
     0     0  -38.58179    0   41          -  -38.58179      -     -    0s
     0     0  -38.17761    0   41          -  -38.17761      -     -    0s
     0     0  -38.17761    0   41          -  -38.17761      -     -    0s
     0     0  -38.12087    0   41          -  -38.12087      -     -    0s
     0     0  -37.74456    0   41          -  -37.74456      -     -    0s
     0     0  -37.62619    0   41          -  -37.62619      -     -    0s
     0     0  -37.35429    0   42          -  -37.35429      -     -    0s
     0     0  -37.11395    0   41          -  -37.11395      -     -    0s
     0     0  -37.06932    0   41          -  -37.06932      -     -    0s
     0     0  -36.93682    0   42          -  -36.93682      -     -    0s
     0     0  -36.60486    0   40          -  -36.60486      -     -    0s
     0     0  -36.32490    0   41          -  -36.32490      -     -    0s
     0     0  -36.32490    0   42          -  -36.32490      -     -    0s
     0     0  -36.20410    0   42          -  -36.20410      -     -    0s
     0     0  -36.20410    0   42          -  -36.20410      -     -    0s
     0     0  -36.02202    0   42          -  -36.02202      -     -    0s
H    0     0                       2.7175205  -35.80451  1418%     -    0s
H    0     0                       2.7159490  -35.80451  1418%     -    0s
     0     2  -35.80451    0   42    2.71595  -35.80451  1418%     -    0s
H   54    42                       2.4011585  -32.46111  1452%  21.1    0s
H  156   105                       0.0027099  -29.18515      -  17.2    0s
H  162    63                      -6.2679705  -29.18515   366%  16.8    0s
H  211    81                      -6.2679714  -24.09212   284%  18.3    0s
H  266    89                      -6.2679714  -23.21997   270%  18.7    0s
H  297   112                      -6.5434808  -23.21997   255%  17.6    0s
*  311   114              37      -6.5435676  -23.21997   255%  16.9    0s
*  312   109              38      -6.5437481  -23.21997   255%  16.9    0s
*  313   108              38      -6.5437521  -23.21997   255%  16.8    0s

Cutting planes:
  Gomory: 4
  Implied bound: 26
  MIR: 99
  Flow cover: 18
  Relax-and-lift: 1

Explored 2032 nodes (39384 simplex iterations) in 2.04 seconds (1.44 work units)
Thread count was 2 (of 2 available processors)

Solution count 9: -6.54375 -6.54375 -6.54357 ... 2.71752

Optimal solution found (tolerance 1.00e-01)
Best objective -6.543752108774e+00, best bound -7.142282765730e+00, gap 9.1466%

After solving the model, we check the error in the estimate of the Gurobi solution.

print(
    "Maximum error in approximating the regression {:.6}".format(
        np.max(pred_constr.get_error())
    )
)
Maximum error in approximating the regression 3.14441e-06

Finally, we look at the solution and the objective value found.

print(
    f"solution point of the approximated problem ({x[0].X:.4}, {x[1].X:.4}), "
    + f"objective value {m.ObjVal}."
)
print(
    f"Function value at the solution point {peak2d(x[0].X, x[1].X)} error {abs(peak2d(x[0].X, x[1].X) - m.ObjVal)}."
)
solution point of the approximated problem (0.2369, -1.618), objective value -6.5437521087742265.
Function value at the solution point -6.549855528081047 error 0.00610341930682079.

The difference between the function and the approximation at the computed solution point is noticeable, but the point we found is reasonably close to the actual global minimum. Depending on the use case this might be deemed acceptable. Of course, training a larger network should result in a better approximation.

Copyright © 2023 Gurobi Optimization, LLC

Total running time of the script: (0 minutes 13.113 seconds)

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