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# Linear Programming QuickStart Sample (F#)

Illustrates solving linear programming (LP) problems using classes in the Extreme.Mathematics.Optimization.LinearProgramming namespace in F#.

```// Illustrates solving linear programming problems
// using the classes in the Extreme.Mathematics.Optimization
// namespace of the Extreme Optimization Numerical Libraries for .NET.

#light

open System

// Vectors and matrices are in the Extreme.Mathematics
// namespace
open Extreme.Mathematics
// The linear programming classes reside in their own namespace.
open Extreme.Mathematics.Optimization

// This QuickStart sample illustrates the three ways to create a Linear 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.

// The cost vector:
let c = Vector.Create(-1.0, -3.0, 0.0, 0.0, 0.0, 0.0)
// The coefficients of the constraints:
let A =
Matrix.Create(4, 6,
[|
1.0; 1.0; 1.0; 0.0; 0.0; 0.0;
1.0; 1.0; 0.0; -1.0; 0.0; 0.0;
1.0; 0.0; 0.0; 0.0; 1.0; 0.0;
0.0; 1.0; 0.0; 0.0; 0.0; 1.0
|], MatrixElementOrder.RowMajor)
// The right-hand sides of the constraints:
let b = Vector.Create(1.5, 0.5, 1.0, 1.0)

// We're now ready to call the constructor.
// The last parameter specifies the number of equality
// constraints.
let lp1 = LinearProgram(c, A, b, 4)

// Now we can call the Solve method to run the Revised
// Simplex algorithm:
let x = lp1.Solve()
// The GetDualSolution method returns the dual solution:
let y = lp1.GetDualSolution()
printfn "Primal: %O" (x.ToString("F1"))
printfn "Dual:   %O" (y.ToString("F1"))
// The optimal value is returned by the OptimalValue property:
printfn "Optimal value:   %O" (lp1.OptimalValue.ToString("F1"))

// The second way to create a Linear Program is by constructing
let lp2 = LinearProgram()

// Next, we add two variables: we specify the name, the cost,
// and optionally the lower and upper bound.
// The method returns the new variable, which we ignore here.
lp2.AddVariable("X1", -1.0, 0.0, 1.0) |> ignore
lp2.AddVariable("X2", -3.0, 0.0, 1.0) |> ignore

// Next, we add constraints. Constraints also have a name.
// We also specify the coefficients of the variables,
// the lower bound and the upper bound.
lp2.AddLinearConstraint("C1", Vector.Create(1.0, 1.0), 0.5, 1.5) |> ignore
// If a constraint is a simple equality or inequality constraint,
// you can supply a LinearProgramConstraintType value and the
// right-hand side of the constraint.

// We can now solve the linear program:
let x2 = lp2.Solve()
let y2 = lp2.GetDualSolution()
printfn "Primal: %O" (x2.ToString("F1"))
printfn "Dual:   %O" (y2.ToString("F1"))
printfn "Optimal value:   %O" (lp2.OptimalValue.ToString("F1"))

// Finally, we can create a linear program from an MPS file.
// The MPS format is a standard format.