Optimization using bonmin

In this example we are going to explore mixed-integer non linear problems optimization using OpenTURNS’ bonmin interface.

[1]:

from __future__ import print_function
import openturns as ot

[2]:

# List available algorithms
for algo in ot.Bonmin.GetAlgorithmNames():
    print(algo)

B-BB
B-OA
B-QG
B-Hyb
B-iFP

Details and references on bonmin algorithms are available here.

Setting up and solving a simple problem

The following example will demonstrate the use of bonmin “BB” algorithm to solve the following problem:

\min - x_0 - x_1 - x_2

such that:

\begin{array}{l} (x_1 - \frac{1}{2})^2 + (x_2 - \frac{1}{2})^2 \leq \frac{1}{4} \\ x_0 - x_1 \leq 0 \\ x_0 + x_2 + x_3 \leq 2\\ x_0 \in \{0,1\}^n\\ (x_1, x_2) \in \mathbb{R}^2\\ x_3 \in \mathbb{N} \end{array}

The theoretical minimum is reached for x = [1,1,0.5,0]. At this point, the objective function value is -2.5

N.B.: OpenTURNS requires equality and inequality constraints to be stated as g(x) = 0 and h(x) \geq 0, respectively. Thus the inequalities above will have to be restated to match this requirement:

\begin{array}{l} -(x_1 - \frac{1}{2})^2 - (x_2 - \frac{1}{2})^2 + \frac{1}{4} \geq 0\\ -x_0 + x_1 \geq 0 \\ -x_0 - x_2 - x_3 + 2 \geq 0\\ \end{array}

[9]:

# Definition of objective function
objectiveFunction = ot.SymbolicFunction(['x0','x1','x2','x3'], ['-x0 -x1 -x2'])

# Definition of variables bounds
bounds = ot.Interval([0,0,0,0],[1,1e99,1e99,5],[True,True,True,True],[True,False,False,True])

# Definition of constraints
# Constraints in OpenTURNS are defined as g(x) = 0 and h(x) >= 0
#    No equality constraint -> nothing to do
#    Inequality constraints:
h = ot.SymbolicFunction(['x0','x1','x2','x3'], ['-(x1-0.5)^2 - (x2-0.5)^2 + 0.25', 'x1 - x0', '-x0 - x2 - x3 + 2'])

# Definition of variables types
variablesType = [ot.OptimizationProblemImplementation.BINARY,ot.OptimizationProblemImplementation.CONTINUOUS,ot.OptimizationProblemImplementation.CONTINUOUS,ot.OptimizationProblemImplementation.INTEGER]

# Setting up Bonmin problem
problem = ot.OptimizationProblem(objectiveFunction)
problem.setBounds(bounds)
problem.setVariablesType(variablesType)
problem.setInequalityConstraint(h)

bonminAlgorithm = ot.Bonmin(problem,'B-BB')
bonminAlgorithm.setMaximumEvaluationNumber(10000)
bonminAlgorithm.setMaximumIterationNumber(1000)
bonminAlgorithm.setStartingPoint([0,0,0,0])

ot.ResourceMap.AddAsString('Bonmin-mu_oracle','loqo')
ot.ResourceMap.AddAsScalar('Bonmin-bonmin.time_limit',5)

[10]:

# Running the solver
bonminAlgorithm.run()

# Retrieving the results
result = bonminAlgorithm.getResult()
print(" -- Optimal point = " + result.getOptimalPoint().__str__())
print(" -- Optimal value = " + result.getOptimalValue().__str__())
print(" -- Evaluation number = " + result.getInputSample().getSize().__str__())


 -- Optimal point = [1,1,0.500141,0]
 -- Optimal value = [-2.50014]
 -- Evaluation number = 147