# VB Constrained Least Squares Example

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```Imports System

Imports CenterSpace.NMath.Core

Namespace CenterSpace.NMath.Examples.VisualBasic

<summary>
A .NET example in Visual Basic showing how to use ConstrainedLeastSquares class to
solve the constrained least squares problem
Cx = d, subject to the constraints
Ax &lt b
</summary>
Public Module ConstrainedLeastSquaresExample

Sub Main()

Solve, in the least squares sense, Cx = d, subject to Ax <= b
and -0.1 <= x[i] <= 2.0
Dim C As New DoubleMatrix("5x4 [0.9501    0.7620    0.6153    0.4057 " &
"0.2311    0.4564    0.7919    0.9354 " &
"0.6068    0.0185    0.9218    0.9169 " &
"0.4859    0.8214    0.7382    0.4102 " &
"0.8912    0.4447    0.1762    0.8936]")

Dim D As New DoubleVector(0.0578, 0.3528, 0.8131, 0.0098, 0.1388)

Constraint coefficient matrix
Dim A As New DoubleMatrix("3x4[0.2027    0.2721    0.7467    0.4659 " &
"0.1987    0.1988    0.4450    0.4186 " &
"0.6037    0.0152    0.9318    0.8462]")

Constraints right hand sides.
Dim B As New DoubleVector(0.5251, 0.2026, 0.6721)

Create the constrained least squares problem for minimizing
|| Cx - d||^2 subject to Ax <= b and -0.1 <= x[i] <= 2.0
We first construct the problem object from the matrix C and the
vector d, we then add the constraints.
Dim Problem = New ConstrainedLeastSquaresProblem(C, D)

Add the inequality constraints Ax <= b using a constraint tolerance
of 0.00001. This allows for small violations of the constraints.
Specifically the constraints will be considered satisfied for a
vector x if
Ax <= b + 0.00001
Dim ConstraintTolerance As Double = 0.00001

Dim I As Integer
For I = 0 To A.Rows - 1
Next

All variable values for the solution must satisfy the bounds
-0.1 <= x[i] <= 2.0
Dim LB As New DoubleVector(Problem.NumVariables, -0.1)
Dim UB As New DoubleVector(Problem.NumVariables, 2.0)
For I = 0 To Problem.NumVariables - 1
Next

Create the solver instance.
Dim Solver As New ConstrainedLeastSquares()

The ConstrainedLeastSquares solver uses a QP (Quadratic Programming) solver
to solve the constrained least squares problem.
The current default QP solver is the NMath active set quadratic programming
solver with default options.
Dim Success As Boolean = Solver.Solve(Problem)
Console.WriteLine("Default solver success = " & Success)
Console.WriteLine("Default solver solution x = " & Solver.X.ToString("0.0000"))
Console.WriteLine("Default solver residual norm = " & Solver.ResidualNorm.ToString("0.0000"))
Console.WriteLine("Default solver performed {0} iterations", solver.Iterations)

You can pass in an instance of a quadratic programming solver for the
constrained least squares class to use. This allows you to set
option on the QP solver and inspect results of the QP
solver.
Dim InteriorPointQP As New InteriorPointQPSolver()
Dim SolverParams As New InteriorPointQPSolverParams()
SolverParams.MaxIterations = 10000
SolverParams.PresolveLevel = InteriorPointQPSolverParams.PresolveLevelOption.None
Solver.Solve(Problem, InteriorPointQP, SolverParams)
Console.WriteLine()
Console.WriteLine("Interior point solver success = " & Success)
Console.WriteLine("Interior point QP result = " & InteriorPointQP.Result)
Console.WriteLine("Interior point QP solver iteration count = " & Solver.Iterations)
Console.WriteLine("Interior point solver solution x = " & Solver.X.ToString("0.0000"))
Console.WriteLine("Interior point solver residual norm = " & Solver.ResidualNorm.ToString("0.0000"))

If you use the active set QP solver you can determine which constraints
are active in the solution by accessing the active set QP solvers
Lagrange multiplier property. A constraint is active if its corresponding
Lagrange multiplier is nonzero.
Dim ActiveSetQP As New ActiveSetQPSolver()
Success = Solver.Solve(Problem, ActiveSetQP)
Console.WriteLine()
Console.WriteLine("Active set solver success = " & Success)
Console.WriteLine("Active set solver solution x = " & Solver.X.ToString("0.0000"))
Console.WriteLine("Active set solver residual norm = " & Solver.ResidualNorm.ToString("0.0000"))
Print out the active constraints.
For I = 0 To ActiveSetQP.LagrangeMultiplier.Length - 1
If (ActiveSetQP.LagrangeMultiplier(I) <> 0.0) Then
Console.WriteLine("Constraint {0} = {1} is active", I, Problem.Constraints(I).ToString())
End If
Next

Console.WriteLine()
Console.WriteLine("Press Enter Key")