/// <inheritdoc /> protected override PointValuePair <T> DoOptimize() { // reset the tableau to indicate a non-feasible solution in case // we do not pass phase 1 successfully if (solutionCallback != null) { solutionCallback.Tableau = null; } var tableau = new SimplexTableau <T, TPolicy>(Function, Constraints, GoalType, IsRestrictedToNonNegative, epsilon, maxUlps); SolvePhase1(tableau); tableau.DropPhase1Objective(); // after phase 1, we are sure to have a feasible solution if (solutionCallback != null) { solutionCallback.Tableau = tableau; } while (!tableau.IsOptimal()) { DoIteration(tableau); } // check that the solution respects the nonNegative restriction in case // the epsilon/cutOff values are too large for the actual linear problem // (e.g. with very small constraint coefficients), the solver might actually // find a non-valid solution (with negative coefficients). PointValuePair <T> solution = tableau.GetSolution(); if (IsRestrictedToNonNegative) { T[] coeff = solution.Point; for (int i = 0; i < coeff.Length; i++) { if (Precision <T, TPolicy> .CompareTo(coeff[i], Policy.Zero(), epsilon) < 0) { throw new NoFeasibleSolutionException(); } } } return(solution); }
/// <summary> /// Runs one iteration of the Simplex method on the given model. /// </summary> /// <param name="tableau"></param> protected void DoIteration(SimplexTableau <T, TPolicy> tableau) { IncrementIterationCount(); int?pivotCol = GetPivotColumn(tableau); int?pivotRow = GetPivotRow(tableau, pivotCol.Value); if (pivotRow == null) { throw new UnboundedSolutionException(); } tableau.PerformRowOperations(pivotCol.Value, pivotRow.Value); }
/// <summary> /// Checks whether the given column is valid pivot column, i.e. will result /// in a valid pivot row. /// When applying Bland's rule to select the pivot column, it may happen that /// there is no corresponding pivot row. This method will check if the selected /// pivot column will return a valid pivot row. /// </summary> /// <param name="tableau"></param> /// <param name="col"></param> /// <returns></returns> private bool IsValidPivotColumn(SimplexTableau <T, TPolicy> tableau, int col) { for (int i = tableau.NumObjectiveFunctions; i < tableau.Height; i++) { T entry = tableau.GetEntry(i, col); // do the same check as in getPivotRow if (Precision <T, TPolicy> .CompareTo(entry, Policy.Zero(), cutOff) > 0) { return(true); } } return(false); }
/// <summary> /// Solves Phase 1 of the Simplex method. /// </summary> /// <param name="tableau"></param> protected void SolvePhase1(SimplexTableau <T, TPolicy> tableau) { // make sure we're in Phase 1 if (tableau.NumArtificialVariables == 0) { return; } while (!tableau.IsOptimal()) { DoIteration(tableau); } // if W is not zero then we have no feasible solution if (!Precision <T, TPolicy> .Equals(tableau.GetEntry(0, tableau.RhsOffset), Policy.Zero(), epsilon)) { throw new NoFeasibleSolutionException(); } }
/// <summary> /// Returns the column with the most negative coefficient in the objective function row. /// </summary> /// <param name="tableau"></param> /// <returns></returns> private int?GetPivotColumn(SimplexTableau <T, TPolicy> tableau) { T minValue = Policy.Zero(); int?minPos = null; for (int i = tableau.NumObjectiveFunctions; i < tableau.Width - 1; i++) { T entry = tableau.GetEntry(0, i); // check if the entry is strictly smaller than the current minimum // do not use a ulp/epsilon check if (Policy.IsBelowZero(Policy.Sub(entry, minValue))) { minValue = entry; minPos = i; // Bland's rule: chose the entering column with the lowest index if (pivotSelection == PivotSelectionRule.BLAND && IsValidPivotColumn(tableau, i)) { break; } } } return(minPos); }
/// <summary> /// Returns the row with the minimum ratio as given by the minimum ratio test (MRT). /// </summary> /// <param name="tableau"></param> /// <param name="col"></param> /// <returns></returns> private int?GetPivotRow(SimplexTableau <T, TPolicy> tableau, int col) { // create a list of all the rows that tie for the lowest score in the minimum ratio test List <int?> minRatioPositions = new List <int?>(); T minRatio = default(T); bool minRationUnassigned = true; for (int i = tableau.NumObjectiveFunctions; i < tableau.Height; i++) { T rhs = tableau.GetEntry(i, tableau.Width - 1); T entry = tableau.GetEntry(i, col); // only consider pivot elements larger than the cutOff threshold // selecting others may lead to degeneracy or numerical instabilities if (Precision <T, TPolicy> .CompareTo(entry, Policy.Zero(), cutOff) > 0) { T ratio = Policy.Abs(Policy.Div(rhs, entry)); // check if the entry is strictly equal to the current min ratio // do not use a ulp/epsilon check int cmp; if (minRationUnassigned) { cmp = -1; } else { cmp = Policy.Compare(ratio, minRatio); } if (cmp == 0) { minRatioPositions.Add(i); } else if (cmp < 0) { minRatio = ratio; minRationUnassigned = false; minRatioPositions.Clear(); minRatioPositions.Add(i); } } } if (minRatioPositions.Count == 0) { return(null); } else if (minRatioPositions.Count > 1) { // there's a degeneracy as indicated by a tie in the minimum ratio test // 1. check if there's an artificial variable that can be forced out of the basis if (tableau.NumArtificialVariables > 0) { foreach (int?row in minRatioPositions) { for (int i = 0; i < tableau.NumArtificialVariables; i++) { int column = i + tableau.ArtificialVariableOffset; T entry = tableau.GetEntry(row.Value, column); if (Precision <T, TPolicy> .Equals(entry, Policy.One(), epsilon) && row.Equals(tableau.GetBasicRow(column))) { return(row); } } } } // 2. apply Bland's rule to prevent cycling: // take the row for which the corresponding basic variable has the smallest index // // see http://www.stanford.edu/class/msande310/blandrule.pdf // see http://en.wikipedia.org/wiki/Bland%27s_rule (not equivalent to the above paper) int?minRow = null; int minIndex = tableau.Width; foreach (int?row in minRatioPositions) { int basicVar = tableau.GetBasicVariable(row.Value); if (basicVar < minIndex) { minIndex = basicVar; minRow = row; } } return(minRow); } return(minRatioPositions[0]); }