dotnet-mathematical-program

This library allows you to build and solve linear programs and integer linear programs in a very handy way. The implementation uses Google´s GLOP linear solver for linear programs and optionally, the Coin-OR CBC branch and cut solver or the Gurobi solver for integer linear programs via the Google OR-Tools API.

Installation

• Install the latest version of Anexia.MathematicalProgram package via nuget

Description

This library works for any linear program (LP) or integer linear program (ILP).

Anexia.MathematicalProgram.Model

• To build the objective function of your LP/ILP you can use the class Terms. Each Term is defined by a Coefficient and a variable of type Google.OrTools.LinearSolver.Variable. Moreover, there is the possibility to have an additional Constant.

• To build your constraints you can use the class Constraints. Each Constraint is defined by Terms and an Interval that has a lower and an upper bound of type double. For binary intervals simply use Interval.BinaryInterval. Another implementation is the class Point for the case of lower bound equals upper bound.

Anexia.MathematicalProgram.Solve

Linear Programming

For solving an LP you may initialize the LinearProgramSolver which uses the GLOP solver in the background.

• Configuration: Via LinearProgramSolver.SetSolverConfigurations() you can set SolverParameter containing a TimeLimitInMilliseconds, the NumberOfThreads that should be maximally used, a EnableSolverOutput flag to determine if the solver output should be printed on the console and the RelativeGap to specify the gap where the solver terminates.
• Variables: Via LinearProgramSolver.AddContinuousVariable() your continuous LP variables can be added using an IInterval and a variable name of type string. The outparameter of this method is of type Google.OrTools.LinearSolver.Variable.
• Constraints: Via LinearProgramSolver.AddConstraints() you can simply add your beforehand initialized contraints to the solver.
• Objective: Via LinearProgramSolver.SetObjective() you can add your objective in form of the Terms and a Constant to the solver. With a bool you can choose if the LP should be minimized or maximized.
• Solve: Via LinearProgramSolver.Solve() you either
  • obtain a SolverResult (explained below) or
  • a MathematicalProgramException with a detailed message on the occured problem is thrown.

Integer Linear Programming

For solving an ILP you may initialize the IntegerLinearProgramSolver which uses per default the CBC solver in the background. Using the highly performant Gurobi solver requires a valid license. Creating a solver using Gurobi can be done in two ways. Either by passing the argument IntegerLinearProgramSolver(ILPSolverType.GurobiMixedIntegerProgramming), or using the static method IntegerLinearProgramSolver.Create(ILPSolverType.GurobiMixedIntegerProgramming, our var message). In both ways, the solver checks if the given type is supported, e.g., a valid licence is present, or otherwise, creates the solver of type CBC. The Create method additionally returns a warning message via out parameter. If the solver has been created as expected, this message is null, otherwise, it contains information that the solver type switched to CBC.

The main difference to linear program solving is that in this case just integer variables can be added. The rest of the methods work similar to the LinearProgramSolver.

• Variables: Via IntegerLinearProgramSolver.AddIntegerVariable() your integer ILP variables can be added using an IInterval and a variable name of type string. The outparameter of this method is of type Google.OrTools.LinearSolver.Variable which are strictly integer.
• Configuration, Constraints, Objective, and Solve as above.

Anexia.MathematicalProgram.Result

After solving the LP/ILP you get a SolverResult according to the Google.OrTools.LinearSolver.Solver.ResultStatus. The SolverResult containts following information:

• Solver: This is the already solved Google.OrTools.LinearSolver.Solver.
  • You have the opportunity to log the LP/ILP model in a human readable format by Solver.ExportModelAsLpFormat().
  • You can read out the actual values of the variables via Google.OrTools.LinearSolver.Variable.SolutionValue() to transform the result correctly.
  • As soon as the solved solver is not needed any more, it should be removed via Solver.Dispose().
• ObjectiveValue: Actual objective value. This value can be either the optimum, a deviation of the optimum if the LP/ILP was not entirely solved, or double.NaN if the LP/ILP is infeasible.
• IsFeasible: Information whether the LP/ILP is generally feasible.
• IsOptimal: Information wheter the LP/ILP was solved to optimality.
• OptimalityGap: The deviation to the optimum calculated by Math.Abs(objective.BestBound() - objectiveValue) / objectiveValue). This value is 0 if the optimum was reached, and double.NaN if the model is infeasible.

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Examples for using this library

Example 1 (Build and solve LP)

• Feasible model: max x, s.t. x <= 2, x >= 1, continuous variable x <= 5
• Result: x = 2, objective value = 2

var solver = new LinearProgramSolver();

solver = solver.AddContinuousVariable(new Interval(double.NegativeInfinity, 5), "TestVariable", out var testVariable);

solver = solver.SetObjective(new Terms(new Term(new Coefficient(1), testVariable)), false);

var constraints = new Constraints(
            new Constraint(new Terms(new Term(new Coefficient(1), testVariable)),
                new Interval(double.NegativeInfinity, 2)),
            new Constraint(new Terms(new Term(new Coefficient(1), testVariable)),
                new Interval(1, double.PositiveInfinity)));

solver = solver.AddConstraints(constraints);

var result = solver.Solve();

Logger.Information(result.SolvedSolver.ExportModelAsLpFormat(false));

Example 2 (Build and solve ILP)

• Feasible model: min 2x + y, s.t. x >= y, integer variables x in [1,3], y binary
• Result: x = 1, y = 0, objective value = 2

var solver = new IntegerLinearProgramSolver()
            .AddIntegerVariable(new Interval(1, 3), "VariableX", out var variableX)
            .AddIntegerVariable(new Interval(0, 1), "VariableY", out var variableY)
            .SetObjective(
                new Terms(new Term(new Coefficient(2), variableX), new Term(new Coefficient(1), variableY)), true)
            .AddConstraints(new Constraints(new Constraint(
                    new Terms(new Term(new Coefficient(1), variableX), new Term(new Coefficient(-1), variableY)),
                    new Interval(0, double.PositiveInfinity))));

 var result = solver.Solve(new SolverParameter(true, RelativeGap.EMinus7,new TimeLimitInMilliseconds(10), new NumberOfThreads(2)));

Example 3 (Build and solve ILP)

• Infeasible model: max 2x, s.t. x = 3, variable x binary

var solver = new IntegerLinearProgramSolver()
            .AddIntegerVariable(Interval.BinaryInterval, "TestVariable", out var testVariable)
            .SetObjective(new Terms(new Term(new Coefficient(2), testVariable)), false)
            .AddConstraints(
                new Constraints(new Constraint(new Terms(new Term(new Coefficient(1), testVariable)), new Point(3))));

 var result = solver.Solve(new SolverParameter(true, RelativeGap.EMinus7));

 Logger.Information(result.SolvedSolver.ExportModelAsLpFormat(false));

Contributing

Contributions are welcomed! Read the Contributing Guide for more information.

Licensing

This project is licensed under MIT License. See LICENSE for more information.