# Nonlinear Programming in IronPython QuickStart Sample

Illustrates solving nonlinear programs (optimization problems with linear or nonlinear constraints) using the NonlinearProgram and related classes in IronPython.

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```Python import numerics from System import Array from math import log, exp # Vectors and matrices are in the Extreme.Mathematics namespace from Extreme.Mathematics import * from Extreme.Mathematics.LinearAlgebra import * # The nonlinear programming classes reside in a namespace with # other optimization-related classes. from Extreme.Mathematics.Optimization import * # Illustrates solving nonlinear programming problems # using the classes in the Extreme.Mathematics.Optimization # namespace of Numerics.NET. # This QuickStart Sample illustrates the two ways to create a Nonlinear Program. # The first is in terms of matrices. The coefficients # are supplied as a matrix. The cost vector, right-hand side # and constraints on the variables are supplied as a vector. print "Problem with only linear constraints:" # The variables are the concentrations of each chemical compound: # H, H2, H2O, N, N2, NH, NO, O, O2, OH # The objective function is the free energy, which we want to minimize: c = Vector.Create[float](\ -6.089, -17.164, -34.054, -5.914, -24.721, \ -14.986, -24.100, -10.708, -26.662, -22.179) objectiveFunction = lambda x: x.DotProduct(c + Vector.Log(x).AddInPlace(-log(x.Sum()))) def g1(x,y): s = x.Sum() (c + Vector.Log(x).AddInPlace(1.0 - log(s)) - x / s).CopyTo(y) return y objectiveGradient = g1 # The constraints are the mass balance relationships for each element. # The rows correspond to the elements H, N, and O. # The columns are the index of the variable. # The value is the number of times the element occurs in the # compound corresponding to the variable: # H, H2, H2O, N, N2, NH, NO, O, O2, OH # All this is stored in a sparse matrix, so 0 values are omitted: A = Matrix.CreateSparse(3, 10, \ Array[int]([ 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2 ]), \ Array[int]([ 0, 1, 2, 5, 9, 3, 4, 5, 6, 2, 6, 7, 8, 9 ]), \ Array[float]([ 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1 ])) # The right-hand sides are the atomic weights of the elements # in the mixture. b = Vector.Create[float](2, 1, 1) # The number of moles for each compound must be positive. l = Vector.CreateConstant(10, 1e-6) u = Vector.CreateConstant(10, float.PositiveInfinity) # We create the nonlinear program with the specified constraints: # Because we have variable bounds, we use the constructor # that lets us do this. nlp1 = NonlinearProgram(objectiveFunction, objectiveGradient, A, b, b, l, u) # We could add more (linear or nonlinear) constraints here, # but this is all we have in our problem. nlp1.InitialGuess = Vector.CreateConstant(10, 0.1) solution = nlp1.Solve() print "Solution: {0:.3f}".format(solution) print "Optimal value: {0:.5f}".format(nlp1.OptimalValue) print "# iterations:", nlp1.SolutionReport.IterationsNeeded print # The second method is building the nonlinear program from scratch. print "Problem with nonlinear constraints:" # We start by creating a nonlinear program object. We supply # the number of variables in the constructor. nlp2 = NonlinearProgram(2) nlp2.ObjectiveFunction = lambda x: exp(x[0]) * (4 * x[0] * x[0] + 2 * x[1] * x[1] + 4 * x[0] * x[1] + 2 * x[1] + 1) def g2(x, y): exp1 = exp(x[0]) y[0] = exp1 * (4 * x[0] * x[0] + 2 * x[1] * x[1] + 4 * x[0] * x[1] + 8 * x[0] + 6 * x[1] + 1) y[1] = exp1 * (4 * x[0] + 4 * x[1] + 2) return y nlp2.ObjectiveGradient = g2 # Add constraint x0*x1 - x0 -x1 <= -1.5 def c2gradient(x, y): y[0] = x[1] - 1 y[1] = x[0] - 1 return y nlp2.AddNonlinearConstraint(lambda x: x[0] * x[1] - x[0] - x[1] + 1.5, \ ConstraintType.LessThanOrEqual, 0.0, c2gradient) # Add constraint x0*x1 >= -10 # If the gradient is omitted, it is approximated using divided differences. nlp2.AddNonlinearConstraint(lambda x: x[0] * x[1], ConstraintType.GreaterThanOrEqual, -10.0) nlp2.InitialGuess = Vector.Create[float]( -1, 1 ) solution = nlp2.Solve() print "Solution: {0:.6f}".format(solution) print "Optimal value: {0:.6f}".format(nlp2.OptimalValue) print "# iterations:", nlp2.SolutionReport.IterationsNeeded ```