internal StartingBasis InitializeSimplex(SimplexTuple tuple)
{
var negativeCount = tuple.FreeTerms.Count(x => x < 0);
int varCount = tuple.ObjFuncCoeffs.Count;
var startingEq = new List<int>(varCount);
for (int i = 0; i < tuple.EqualityCoeffs.RowCount; i++)
{
var row = tuple.EqualityCoeffs.Row(i);
if (row.Count(x => x.FloatEquals(-1)) == 1
&& row.Count(x => x.FloatEquals(0)) == (varCount - 1)
&& startingEq.Count < varCount)
startingEq.Add(i);
}
//If min index > 0 -> all free terms are positive.
//This means that basis just zero variables and indexes of that inequalities
if (negativeCount == 0)
{
return new StartingBasis
{
InequalityIndexes = startingEq.ToArray(),
FeasibleBasis = new double[varCount]
};
}
//Create auxilary problem
var auxObjFunc = new List<double>(varCount + negativeCount);
auxObjFunc.AddRange(tuple.ObjFuncCoeffs.Select(ofc => 0).Select(dummy => (double) dummy));
var auxA = tuple.EqualityCoeffs.Clone();
var auxb = tuple.FreeTerms.Clone();
int rowCount = tuple.EqualityCoeffs.RowCount;
//for each negative index we should go with added slack variable
//TODO: work with Dense/Sparse vectors
for (int i = 0; i < tuple.EqualityCoeffs.RowCount; i++)
{
if (tuple.FreeTerms[i] >= 0) continue;
//add a new slack variable to obj function with coeff -1
auxObjFunc.Add(-1);
//add new column for that variable
var zeroVector = new DenseVector(rowCount);
zeroVector[i] = -1;
auxA = auxA.InsertColumn(varCount++, zeroVector);
//add new row for this variable
var nonNegativeRow = new DenseVector(varCount);
nonNegativeRow[varCount - 1] = -1;
auxA = auxA.InsertRow(rowCount, nonNegativeRow);
var zeroFreeTerm = new DenseVector(new[] {0.0});
auxb = auxb.ToColumnMatrix().InsertRow(rowCount, zeroFreeTerm).Column(0);
rowCount++;
//add top constraint to this variable
var freeTermRow = new DenseVector(varCount);
freeTermRow[varCount - 1] = 1;
auxA = auxA.InsertRow(rowCount, freeTermRow);
var newFreeTerm = new DenseVector(new[] {-tuple.FreeTerms[i]});
auxb = auxb.ToColumnMatrix().InsertRow(rowCount, newFreeTerm).Column(0);
startingEq.Add(rowCount);
rowCount++;
}
var auxBasis = new StartingBasis
{
InequalityIndexes = startingEq.ToArray(),
FeasibleBasis = new double[varCount]
};
iteration = 0;
var columns = Enumerable.Range(0, varCount).ToArray();
//TODO: work with Dense/Sparse vectors
var auxAb = ((DenseMatrix) auxA).Extract(startingEq.ToArray(), columns);
var auxbB = ((DenseVector) auxb).Extract(startingEq.ToArray());
Vector vector = SolveInternal((DenseMatrix) auxA, (DenseVector) auxb, new DenseVector(auxObjFunc.ToArray()),
auxBasis, auxAb, auxbB);
for (int k = tuple.ObjFuncCoeffs.Count; k < varCount; k++)
if (!vector[k].FloatEquals(0))
throw new SimplexException("Problem is unsolvable");
var basis = vector.Take(tuple.ObjFuncCoeffs.Count).ToArray();
var indexes =
auxBasis.InequalityIndexes.Where(x => x < tuple.EqualityCoeffs.RowCount).Take(tuple.ObjFuncCoeffs.Count).ToArray();
return new StartingBasis
{
InequalityIndexes = indexes,
FeasibleBasis = basis
};
}
internal Vector SolveInternal(Matrix A, Vector b, Vector c, StartingBasis B, Matrix AB, Vector bB)
{
//Init simplex
var m = A.RowCount;
var ABi = AB.Inverse();
//X is a starting vector
var x = AB.LU().Solve(bB);
while (true)
{
iteration++;
//Compute lambda (cT*AB.inv())T
var lambda = (c.ToRowMatrix()*ABi).Transpose().ToRowWiseArray();
//check for optimality
if (lambda.All(l => l >= 0)) return (Vector) x;
//Find leaving index r (first index where component < 0)
var r = lambda.Select((i, index) => new {i, index})
.Where((i, index) => i.i < 0)
.First().index;
//compute direction to move int - take r-th column
var d = ABi.Column(r)*(-1);
//Determine the set K (all indexes of positive values of lambda)
//all k that a(k).T*d>0, 1 <= i <=m
var K = new List<int>();
for (int k = 0; k < m; k++)
{
var val = A.Row(k)*d;
if (val > 0 && !val.FloatEquals(0))
K.Add(k);
}
if (K.Count == 0)
throw new SimplexException("Problem is unbounded") {Iteration = iteration};
//Find entering index e
int e = 0;
var v = double.MaxValue;
foreach (var k in K)
{
var w = (b[k] - A.Row(k)*x)/(A.Row(k)*d);
if (!(w < v)) continue;
v = w;
e = k;
}
//Update basis
B.InequalityIndexes[r] = e;
AB.SetRow(r, A.Row(e));
bB[r] = b[e];
//Trick - lets update inverse AB in a smart way - sinse there is only one new inequality we only need to
//compute new inversed row (should drop complexity of whole algo to n*n)
var f = AB.Row(r)*ABi;
var g = -f;
g[r] = 1;
g /= f[r];
g[r] -= 1;
ABi = ABi.Add(ABi.Column(r).ToColumnMatrix()*g.ToRowMatrix());
//Compute new x
x = x + v*d;
}
}
