496 lines
13 KiB
C++
496 lines
13 KiB
C++
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#include <basetypes.h>
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#include <float.h>
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#include "simplex.h"
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// a nice tutorial on simplex method: http://math.uww.edu/~mcfarlat/ism.htm
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CSimplex::CSimplex():
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m_numVariables(0),m_numConstraints(0),m_pTableau(0),m_pInitialTableau(0), m_pSolution(0), m_pBasis(0)
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{
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}
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CSimplex::CSimplex(int numVariables, int numConstraints):
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m_numVariables(0),m_numConstraints(0),m_pTableau(0),m_pInitialTableau(0), m_pSolution(0), m_pBasis(0)
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{
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Init(numVariables, numConstraints);
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}
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void CSimplex::Init(int numVariables, int numConstraints)
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{
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Destruct();
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m_numVariables = numVariables; m_numConstraints = numConstraints;
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m_pTableau = new float[(NumRows()+1) * NumColumns()];
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m_pInitialTableau = new float[(NumRows()+1) * NumColumns()];
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m_pSolution = m_pTableau + NumRows() * NumColumns();
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// allocating basis and non-basis indices in one call
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m_pBasis = new int[m_numConstraints + m_numVariables];
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m_pNonBasis = m_pBasis + m_numConstraints;
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m_state = kUnknown;
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}
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void CSimplex::PrintTableau()const
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{
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Msg("problem.Init(%d,%d);\nfloat test[%d]={", m_numVariables, m_numConstraints, (m_numVariables+1)*(m_numConstraints+1));
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for(int i = 0; i < NumRows(); ++i)
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{
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for(int j = 0;j < NumColumns(); ++j)
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{
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Msg(" %g,",Tableau(i,j));
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}
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Msg("\n");
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}
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Msg("}");
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}
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void CSimplex::InitTableau(const float *pTableau)
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{
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const float *p = pTableau;
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for(int nRow = 0; nRow <= m_numConstraints; ++nRow)
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{
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for(int nColumn = 0; nColumn < m_numVariables; ++nColumn)
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{
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Tableau(nRow, nColumn) = *(p++);
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}
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Tableau(nRow, NumColumns()-1) = *(p++);
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}
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}
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CSimplex::~CSimplex()
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{
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Destruct();
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}
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void CSimplex::Destruct()
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{
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delete[]m_pInitialTableau;
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m_pInitialTableau = NULL;
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delete[]m_pTableau;
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m_pTableau = NULL;
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delete[]m_pBasis;
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m_pBasis = NULL;
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}
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CSimplex::StateEnum CSimplex::Solve(float flThreshold, int maxStallIterations)
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{
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m_state = kUnknown;
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PrepareTableau();
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if(SolvePhase1(flThreshold, maxStallIterations) == kUnknown)
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SolvePhase2(flThreshold, maxStallIterations);
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GatherSolution();
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return m_state;
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}
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///////////////////////////////////////////////////////////////////////////
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// bring constraints to b>=0 form for phase-2 full solution
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CSimplex::StateEnum CSimplex::SolvePhase1(float flThreshold, int maxStallIterations)
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{
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for(int nPotentiallyInfiniteLoop = 0; nPotentiallyInfiniteLoop < maxStallIterations; ++nPotentiallyInfiniteLoop)
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{
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if(!IteratePhase1())
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break;
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}
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return m_state;
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}
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//////////////////////////////////////////////////////////////////////////
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// Solve the linear problem ;
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// \param flThreshold - this is how much we need to improve objective every step that's not considered lost
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// \param maxStallIterations - this is how many "lost" (see flThreshold) steps we may take before we bail
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//
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CSimplex::StateEnum CSimplex::SolvePhase2(float flThreshold, int maxStallIterations)
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{
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for(int nPotentiallyInfiniteLoop = 0; nPotentiallyInfiniteLoop < maxStallIterations; ++nPotentiallyInfiniteLoop)
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{
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if(!IteratePhase2())
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break;
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}
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Validate();
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return m_state;
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}
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// fill out m-pSolution array (primal solution)
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void CSimplex::GatherSolution()
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{
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// Notes:
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// PRIMAL SOLUTION is indicated by the rightmost column of the tableau;
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// there are at most m_numConstraint basic variables that participate in the solution.
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// The original problem PRIMAL unknowns are numbered 0..m_numVariables; the rest (m_numVariables+1..m_numVariables+m_numConstraints) are the PRIMAL SLACK variables
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// DUAL SOLUTION is in the row [m_numConstraints], and it's basic variables are indicated by m_pNonBasic array and are reversed:
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// first the DUAL SLACK variables are numbered 0..m_numVariables; the rest (m_numVariables+1..m_numVariables+m_numConstraints) are the DUAL variables
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memset(m_pSolution, 0, sizeof(*m_pSolution) * NumColumns()); // initial value of all X's are 0's
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for(int nRow = 0; nRow < m_numConstraints; ++nRow)
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{
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int nBasisVariable = m_pBasis[nRow];
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m_pSolution[nBasisVariable] = Tableau(nRow, NumColumns()-1);
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}
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m_pSolution[m_numVariables+m_numConstraints] = Tableau(m_numConstraints, NumColumns()-1);
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}
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///////////////////////////////////////////////////////////////////////////
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// Find and pivot a row with negative constraint const (right side)
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// return false - if can't find such constraint or can't pivot
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//
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bool CSimplex::IteratePhase1()
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{
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int nFixRow = FindLastNegConstrRow();
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if(nFixRow < 0)
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return false; // phase 1 complete: no rows to fix
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int nPivotColumn = ChooseNegativeElementInRow(nFixRow);
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if(nPivotColumn < 0)
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{
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m_state = kInfeasible;
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return false;
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}
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int nPivotRow = nFixRow;
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float flMinimizer = Tableau (nPivotRow, NumColumns()-1)/Tableau(nPivotRow, nPivotColumn); // minimize this
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// UNTESTED! What's the rule to choose pivot in phase1?
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for(int nCandidatePivotRow = nPivotRow + 1; nCandidatePivotRow < m_numConstraints; ++nCandidatePivotRow)
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{
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float flCandidateConst = Tableau (nCandidatePivotRow,NumColumns()-1), flCandidatePivot = Tableau (nCandidatePivotRow, nPivotColumn);
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if ( flCandidateConst < 0 && flCandidatePivot > 1e-6f )
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{
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float flCandidateMinimizer = flCandidateConst / flCandidatePivot;
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if(flCandidateMinimizer < flMinimizer)
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{
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flCandidateMinimizer = flMinimizer;
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nPivotRow = nCandidatePivotRow; // UNTESTED!
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}
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}
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}
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return Pivot(nPivotRow, nPivotColumn);
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}
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//////////////////////////////////////////////////////////////////////////
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// Return the index of the last row with negative Constraint Const (b[i] in A.x<=b formulation)
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int CSimplex::FindLastNegConstrRow()
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{
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int nFixRow = -1;
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for(int nRow = 0; nRow < m_numConstraints; ++nRow)
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{
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if(Tableau(nRow, NumColumns()-1) < 0)
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{
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nFixRow = nRow;
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}
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}
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return nFixRow;
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}
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///////////////////////////////////////////////////////////////////////////
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// Choose some (e.g. the most negative) negative number in the row
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int CSimplex::ChooseNegativeElementInRow(int nFixRow)
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{
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int indexNegElement = -1;
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float flMinElement = 0;
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for(int nColumn = 0; nColumn < m_numVariables; ++nColumn)
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{
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float flElement = Tableau(nFixRow, nColumn);
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if(flElement < flMinElement)
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{
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indexNegElement = nColumn;
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flMinElement = flElement;
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}
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}
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return indexNegElement;
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}
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bool CSimplex::IteratePhase2()
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{
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int nPivotColumn = FindPivotColumn();
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if(nPivotColumn < 0)
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{
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m_state = kOptimal;
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return false;
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}
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int nPivotRow = FindPivotRow(nPivotColumn);
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if(nPivotRow < 0)
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{
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m_state = kUnbound;
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return false;
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}
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bool ok = Pivot(nPivotRow, nPivotColumn);
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// since we replaced the basis variable, we have to replace its corresponding column
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return ok;
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}
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//////////////////////////////////////////////////////////////////////////
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// Self-explanatory, isn't it?
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bool CSimplex::Pivot(int nPivotRow, int nPivotColumn)
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{
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if(fabs(Tableau(nPivotRow, nPivotColumn)) < 1e-8f)
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{
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m_state = kCannotPivot;
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return false; // Can NOT pivot on zero :( choose another (ie. fancier) pivot rule
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}
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/// get the 1/Tij, then replace the multiplied element with it
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float flFactor = 1.0f / Tableau(nPivotRow, nPivotColumn);
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MultiplyRow(nPivotRow, flFactor);
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for(int i = 0; i <= m_numConstraints; ++i)
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{
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if(i != nPivotRow)
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{
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float flFactorOther = -Tableau(i,nPivotColumn);
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AddRowMulFactor(i, nPivotRow, flFactorOther);
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Tableau(i,nPivotColumn) = flFactorOther * flFactor; // replace the column with original column / -pivot
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}
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}
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Tableau(nPivotRow, nPivotColumn) = flFactor;
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int nEnteringVariable = m_pNonBasis[nPivotColumn];
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int nExitingVariable = m_pBasis[nPivotRow];
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// remember the index of the entering new basis var
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m_pBasis[nPivotRow] = nEnteringVariable;
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m_pNonBasis[nPivotColumn] = nExitingVariable;
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Validate();
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return true;
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}
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//////////////////////////////////////////////////////////////////////////
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// find the column with the most negative number in the last (objective) row
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int CSimplex::FindPivotColumn()
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{
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int nBest = -1;
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float flBest = 0;
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for(int i = 0; i < m_numVariables; ++i)
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{
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float flElement = Tableau(m_numConstraints, i);
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if(flElement > flBest)
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{
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flBest = flElement;
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nBest = i;
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}
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}
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if(nBest < 0)
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{
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m_state = kOptimal;
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return -1;
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}
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else
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return nBest;
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};
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int CSimplex::FindPivotRow(int nColumn)
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{
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float flBest = FLT_MAX;
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int nBest = -1;
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for(int nRow = 0; nRow < m_numConstraints; ++nRow)
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{
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float flPivotCandidate = Tableau(nRow, nColumn);
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if(flPivotCandidate > 1e-6f)
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{
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// don't perform any tests unless flTest is finite
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float flTest = Tableau(nRow, NumColumns()-1) / flPivotCandidate;
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if(flTest < flBest)
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{
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// flBest is either Infinity or is worse; it's worse in any case, so replace it
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flBest = flTest;
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nBest = nRow;
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}
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}
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}
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return nBest;
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}
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void CSimplex::MultiplyRow(int nRow, float flFactor)
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{
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for(int nColumn = 0; nColumn < NumColumns(); ++nColumn)
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{
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Tableau(nRow, nColumn) *= flFactor;
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}
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}
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void CSimplex::AddRowMulFactor(int nTargetRow, int nPivotRow, float fFactor)
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{
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for(int nColumn = 0; nColumn < NumColumns(); ++nColumn)
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{
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Tableau(nTargetRow, nColumn) += Tableau(nPivotRow, nColumn) * fFactor;
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}
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}
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// set the I matrix in the slack columns of the tableau
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void CSimplex::PrepareTableau()
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{
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/*
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for(int nRow = 0; nRow < m_numConstraints + 1; ++nRow)
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{
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for(int nColumn = 0; nColumn < m_numConstraints; ++nColumn)
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Tableau(nRow, nColumn + m_numVariables) = 0;
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}
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*/
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for(int nonBasis = 0; nonBasis < m_numVariables; ++nonBasis)
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{
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m_pNonBasis[nonBasis] = nonBasis;
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}
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for(int nConstraint = 0; nConstraint < m_numConstraints; ++nConstraint)
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{
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m_pBasis[nConstraint] = m_numVariables + nConstraint; // slack variables
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//Tableau(nConstraint, nConstraint + m_numVariables) = 1.0f;
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}
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//m_pSolution[m_numVariables+m_numConstraints] =
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Tableau(m_numConstraints, NumColumns()-1) = 0.0f; // starting with "0" objective, and all "0" variables
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memcpy(m_pInitialTableau,m_pTableau,(NumRows()+1) * NumColumns() * sizeof(float));
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}
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void CSimplex::SetConstraintConst(int nConstraint, float fConst)
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{
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m_pSolution[m_numVariables + nConstraint] = Tableau(nConstraint, NumColumns()-1) = fConst;
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}
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void CSimplex::SetConstraintFactor(int nConstraint, int nConstant, float fFactor)
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{
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Tableau(nConstraint, nConstant) = fFactor;
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}
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void CSimplex::SetObjectiveFactor(int nConstant, float fFactor)
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{
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// the objective factor is negated because for the objective P = cx , we write it as -c x + P -> max
|
||
|
|
Tableau(m_numConstraints, nConstant) = fFactor;
|
||
|
|
}
|
||
|
|
|
||
|
|
void CSimplex::SetObjectiveFactors(int numFactors, const float *pFactors)
|
||
|
|
{
|
||
|
|
Assert(numFactors == m_numVariables);
|
||
|
|
for(int i =0; i < m_numVariables && i < numFactors; ++i)
|
||
|
|
SetObjectiveFactor(i,pFactors[i]);
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
float CSimplex::GetSolution(int nVariable)const
|
||
|
|
{
|
||
|
|
Assert(nVariable < m_numVariables);
|
||
|
|
return m_pSolution[nVariable];
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
float CSimplex::GetSlack(int nConstraint)const
|
||
|
|
{
|
||
|
|
Assert(nConstraint < m_numConstraints);
|
||
|
|
return m_pSolution[m_numVariables + nConstraint];
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
float CSimplex::GetObjective()const
|
||
|
|
{
|
||
|
|
/*
|
||
|
|
float flResult = 0;
|
||
|
|
for(int i = 0; i < m_numVariables + m_numConstraints; ++i)
|
||
|
|
flResult -= m_pSolution[i] * Tableau(m_numConstraints,i);
|
||
|
|
return flResult;
|
||
|
|
*/
|
||
|
|
return Tableau(m_numConstraints, NumColumns()-1);
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
void CSimplex::Validate()
|
||
|
|
{
|
||
|
|
#if defined(_DEBUG) && 0
|
||
|
|
GatherSolution();
|
||
|
|
for(int i = 0; i <= m_numConstraints; ++i)
|
||
|
|
{
|
||
|
|
float flRes = 0;
|
||
|
|
for(int j = 0; j < m_numVariables; ++j)
|
||
|
|
flRes += GetInitialTableau(i,j) * m_pSolution[j];
|
||
|
|
if(i == m_numConstraints)
|
||
|
|
{
|
||
|
|
Msg("Objective = %g; basis:",flRes);
|
||
|
|
for (int j = 0; j < m_numVariables; ++j)
|
||
|
|
Msg(" %g", m_pSolution[j]);
|
||
|
|
Msg(" |slacks:");
|
||
|
|
for(int j = 0; j < m_numConstraints; ++j)
|
||
|
|
Msg(" %g", m_pSolution[j+m_numVariables]);
|
||
|
|
Msg("\n");
|
||
|
|
}
|
||
|
|
else
|
||
|
|
Msg("%g\t<= %g\n", flRes, GetInitialTableau(i,NumColumns()-1));
|
||
|
|
}
|
||
|
|
#endif
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
class CSimplexTestUnit
|
||
|
|
{
|
||
|
|
public:
|
||
|
|
CSimplexTestUnit()
|
||
|
|
{
|
||
|
|
CSimplex test(3,2);
|
||
|
|
test.SetObjectiveFactor(0, 12);
|
||
|
|
test.SetObjectiveFactor(1, 8);
|
||
|
|
test.SetObjectiveFactor(2, 24);
|
||
|
|
test.SetConstraintFactor(0, 0, 6);
|
||
|
|
test.SetConstraintFactor(0, 1, 2);
|
||
|
|
test.SetConstraintFactor(0, 2, 4);
|
||
|
|
test.SetConstraintConst(0, 200);
|
||
|
|
test.SetConstraintFactor(1, 0, 2);
|
||
|
|
test.SetConstraintFactor(1, 1, 2);
|
||
|
|
test.SetConstraintFactor(1, 2, 12);
|
||
|
|
test.SetConstraintConst(1, 160);
|
||
|
|
|
||
|
|
test.Solve();
|
||
|
|
|
||
|
|
test.Init(2,2);
|
||
|
|
float test2[] = {2,1,3, 3,1,4, 17,5,0};
|
||
|
|
test.InitTableau(test2);
|
||
|
|
test.Solve();
|
||
|
|
// m_pSolution (test.m_pSolution) should be : 30 40 | 0 0 | 4100
|
||
|
|
|
||
|
|
//////////////////////////////////////////////////////////////////////////
|
||
|
|
// unbound-solution problem: x1-x2<=1 && x2-x1<=1, maximize x1+x2; if x1==x2, we can go unbound x1==x2 -> +inf
|
||
|
|
// the dual formulation is infeasible in this case: v2-v1 >= 1 && v1-v2 >= 1, which are self-contradictory
|
||
|
|
test.Init(2,2);
|
||
|
|
float testUnsolvable[] = {-1,1,1, 1,-1,1, 1,1,0};
|
||
|
|
test.InitTableau(testUnsolvable);
|
||
|
|
test.Solve();
|
||
|
|
|
||
|
|
|
||
|
|
//////////////////////////////////////////////////////////////////////////
|
||
|
|
// General Simplex problem: equality constraint
|
||
|
|
test.Init(2, 3);
|
||
|
|
float testGenSimplex[] = {1,1,20, 1,2,30, -1,-2,-30, 2,1,0};
|
||
|
|
test.InitTableau(testGenSimplex);
|
||
|
|
test.Solve();
|
||
|
|
|
||
|
|
test.Init(7,6);
|
||
|
|
float testA[56]={ -1, 1, 0, -0, -0, 0, 1, 13.0048,
|
||
|
|
1, -1, 0, -0, -0, 0, 1, 13.0048,
|
||
|
|
0, -0, -1, 1, -0, 0, 1, 13.0048,
|
||
|
|
0, -0, 1, -1, -0, 0, 1, 13.0048,
|
||
|
|
0, -0, 0, -0, 1, -1, 1, 0.00100005,
|
||
|
|
0, -0, 0, -0, -1, 1, 1, 0.405401,
|
||
|
|
0, 0, 0, 0, 0, 0, 1, 0
|
||
|
|
};
|
||
|
|
test.InitTableau(testA);
|
||
|
|
test.Solve();
|
||
|
|
}
|
||
|
|
};
|
||
|
|
|
||
|
|
|
||
|
|
// this is for debugging and unit-testing
|
||
|
|
//static CSimplexTestUnit s_test;
|