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btLemkeAlgorithm.cpp
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btLemkeAlgorithm.cpp
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/* Copyright (C) 2004-2013 MBSim Development Team
Code was converted for the Bullet Continuous Collision Detection and Physics Library
This software is provided 'as-is', without any express or implied warranty.
In no event will the authors be held liable for any damages arising from the use of this software.
Permission is granted to anyone to use this software for any purpose,
including commercial applications, and to alter it and redistribute it freely,
subject to the following restrictions:
1. The origin of this software must not be misrepresented; you must not claim that you wrote the original software. If you use this software in a product, an acknowledgment in the product documentation would be appreciated but is not required.
2. Altered source versions must be plainly marked as such, and must not be misrepresented as being the original software.
3. This notice may not be removed or altered from any source distribution.
*/
//The original version is here
//https://code.google.com/p/mbsim-env/source/browse/trunk/kernel/mbsim/numerics/linear_complementarity_problem/lemke_algorithm.cc
//This file is re-distributed under the ZLib license, with permission of the original author
//Math library was replaced from fmatvec to a the file src/LinearMath/btMatrixX.h
//STL/std::vector replaced by btAlignedObjectArray
#include <iostream>
#include "btLemkeAlgorithm.h"
#undef BT_DEBUG_OSTREAM
#ifdef BT_DEBUG_OSTREAM
using namespace std;
#endif //BT_DEBUG_OSTREAM
#define LEXICO_MINIMUM_EPSILON 1.0e-15
btScalar btMachEps()
{
static bool calculated = false;
static btScalar machEps = btScalar(1.);
if (!calculated)
{
do
{
machEps /= btScalar(2.0);
// If next epsilon yields 1, then break, because current
// epsilon is the machine epsilon.
} while ((btScalar)(1.0 + (machEps / btScalar(2.0))) != btScalar(1.0));
// printf( "\nCalculated Machine epsilon: %G\n", machEps );
calculated = true;
}
// return machEps;
return LEXICO_MINIMUM_EPSILON; // the logic above essentially calculates the number of mantissa bits, which may be too big to be used as numerical tolerance threshold.
}
btScalar btEpsRoot()
{
static btScalar epsroot = 0.;
static bool alreadyCalculated = false;
if (!alreadyCalculated)
{
epsroot = btSqrt(btMachEps());
alreadyCalculated = true;
}
return epsroot;
}
btVectorXu btLemkeAlgorithm::solve(unsigned int maxloops /* = 0*/, bool useImprovedLexicoMimimum )
{
steps = 0;
int dim = m_q.size();
#ifdef BT_DEBUG_OSTREAM
if (DEBUGLEVEL >= 1)
{
cout << "Dimension = " << dim << endl;
}
#endif //BT_DEBUG_OSTREAM
btVectorXu solutionVector(2 * dim);
solutionVector.setZero();
//, INIT, 0.);
btMatrixXu ident(dim, dim);
ident.setIdentity();
#ifdef BT_DEBUG_OSTREAM
cout << m_M << std::endl;
#endif
btMatrixXu mNeg = m_M.negative();
btMatrixXu A(dim, 2 * dim + 2);
//
A.setSubMatrix(0, 0, dim - 1, dim - 1, ident);
A.setSubMatrix(0, dim, dim - 1, 2 * dim - 1, mNeg);
A.setSubMatrix(0, 2 * dim, dim - 1, 2 * dim, -1.f);
A.setSubMatrix(0, 2 * dim + 1, dim - 1, 2 * dim + 1, m_q);
#ifdef BT_DEBUG_OSTREAM
cout << A << std::endl;
#endif //BT_DEBUG_OSTREAM
// btVectorXu q_;
// q_ >> A(0, 2 * dim + 1, dim - 1, 2 * dim + 1);
btAlignedObjectArray<int> basis;
//At first, all w-values are in the basis
for (int i = 0; i < dim; i++)
basis.push_back(i);
int pivotRowIndex = -1;
btScalar minValue = 1e30f;
bool greaterZero = true;
for (int i = 0; i < dim; i++)
{
btScalar v = A(i, 2 * dim + 1);
if (v < minValue)
{
minValue = v;
pivotRowIndex = i;
}
if (v < 0)
greaterZero = false;
}
// int pivotRowIndex = q_.minIndex();//minIndex(q_); // first row is that with lowest q-value
int z0Row = pivotRowIndex; // remember the col of z0 for ending algorithm afterwards
int pivotColIndex = 2 * dim; // first col is that of z0
#ifdef BT_DEBUG_OSTREAM
if (DEBUGLEVEL >= 3)
{
// cout << "A: " << A << endl;
cout << "pivotRowIndex " << pivotRowIndex << endl;
cout << "pivotColIndex " << pivotColIndex << endl;
cout << "Basis: ";
for (int i = 0; i < basis.size(); i++)
cout << basis[i] << " ";
cout << endl;
}
#endif //BT_DEBUG_OSTREAM
if (!greaterZero)
{
if (maxloops == 0)
{
maxloops = 100;
// maxloops = UINT_MAX; //TODO: not a really nice way, problem is: maxloops should be 2^dim (=1<<dim), but this could exceed UINT_MAX and thus the result would be 0 and therefore the lemke algorithm wouldn't start but probably would find a solution within less then UINT_MAX steps. Therefore this constant is used as a upper border right now...
}
/*start looping*/
for (steps = 0; steps < maxloops; steps++)
{
GaussJordanEliminationStep(A, pivotRowIndex, pivotColIndex, basis);
#ifdef BT_DEBUG_OSTREAM
if (DEBUGLEVEL >= 3)
{
// cout << "A: " << A << endl;
cout << "pivotRowIndex " << pivotRowIndex << endl;
cout << "pivotColIndex " << pivotColIndex << endl;
cout << "Basis: ";
for (int i = 0; i < basis.size(); i++)
cout << basis[i] << " ";
cout << endl;
}
#endif //BT_DEBUG_OSTREAM
int pivotColIndexOld = pivotColIndex;
/*find new column index */
if (basis[pivotRowIndex] < dim) //if a w-value left the basis get in the correspondent z-value
pivotColIndex = basis[pivotRowIndex] + dim;
else
//else do it the other way round and get in the corresponding w-value
pivotColIndex = basis[pivotRowIndex] - dim;
/*the column becomes part of the basis*/
basis[pivotRowIndex] = pivotColIndexOld;
if (useImprovedLexicoMimimum)
pivotRowIndex = findLexicographicMinimum2(A, pivotColIndex, z0Row);
else
pivotRowIndex = findLexicographicMinimum(A, pivotColIndex, z0Row);
if (z0Row == pivotRowIndex)
{ //if z0 leaves the basis the solution is found --> one last elimination step is necessary
GaussJordanEliminationStep(A, pivotRowIndex, pivotColIndex, basis);
basis[pivotRowIndex] = pivotColIndex; //update basis
break;
}
}
#ifdef BT_DEBUG_OSTREAM
if (DEBUGLEVEL >= 1)
{
cout << "Number of loops: " << steps << endl;
cout << "Number of maximal loops: " << maxloops << endl;
}
#endif //BT_DEBUG_OSTREAM
// if (!validBasis(basis))
if (pivotRowIndex == -1)
{
info = -1;
#ifdef BT_DEBUG_OSTREAM
if (DEBUGLEVEL >= 1)
cerr << "Lemke-Algorithm ended with Ray-Termination (no valid solution)." << endl;
#endif //BT_DEBUG_OSTREAM
std::cerr << "Lemke-Algorithm ended with Ray-Termination (no valid solution)." << std::endl;
return solutionVector;
}
}
#ifdef BT_DEBUG_OSTREAM
if (DEBUGLEVEL >= 2)
{
// cout << "A: " << A << endl;
cout << "pivotRowIndex " << pivotRowIndex << endl;
cout << "pivotColIndex " << pivotColIndex << endl;
}
#endif //BT_DEBUG_OSTREAM
for (int i = 0; i < basis.size(); i++)
{
solutionVector[basis[i]] = A(i, 2 * dim + 1); //q_[i];
}
info = 0;
return solutionVector;
}
// Logic issues:
// 1. A(*, 2*dim), the z_0 column, is used for lexico minimum test, which must not be used.
// 2. If the check on the 1st column 'q' produces multiple minimums, and if it contains z_0, the tie must be broken in favor of z_0 and return.
// Efficiency issues:
// 4. The 1st part to construct Rows is unnecessary and inefficient.
// 5. The 2nd part has three nested loops with brute-force search, which is inefficient.
int btLemkeAlgorithm::findLexicographicMinimum(const btMatrixXu& A, const int& pivotColIndex, const int& z0Row)
{
/*
btScalar currentMin = 0;
int currentMinIndex = 0;
int numTies = 0;
for (int row = 0; row < A.rows(); row++)
{
const btScalar denom = A(row, pivotColIndex);
if (denom > btMachEps())
{
const btScalar ratio = A(row, 2 * A.rows() + 1)/ A(row, pivotColIndex);
if (numTies == 0)
{
currentMin = ratio;
currentMinIndex = row;
numTies = 1;
}
else if (fabs(currentMin - ratio) < btMachEps() && currentMinIndex != z0Row)
{
currentMin = ratio;
currentMinIndex = row;
numTies++;
}
else if (currentMin > ratio)
{
currentMin = ratio;
currentMinIndex = row;
numTies = 1;
}
}
}
if (numTies == 1 || currentMinIndex == z0Row)
{
return currentMinIndex;
}
*/
int RowIndex = 0;
int dim = A.rows();
btAlignedObjectArray<btVectorXu> Rows;
for (int row = 0; row < dim; row++)
{
btVectorXu vec(dim + 1);
vec.setZero(); //, INIT, 0.)
Rows.push_back(vec);
btScalar a = A(row, pivotColIndex);
if (a > btMachEps())
{
Rows[row][0] = A(row, 2 * dim + 1) / a;
Rows[row][1] = A(row, 2 * dim) / a; // This is a column for z_0, which must not be used here.
for (int j = 2; j < dim + 2; j++)
Rows[row][j] = A(row, j - 2) / a;
#ifdef BT_DEBUG_OSTREAM
// if (DEBUGLEVEL) {
// cout << "Rows(" << row << ") = " << Rows[row] << endl;
// }
#endif
}
else
{ // Setting 0.0 to the invalid rows whose denominators are non-positive may produce wrong lexico ordering.
for (int j = 0; j < dim + 1; j++)
Rows[row][j] = FLT_MAX;
}
}
for (int i = 0; i < Rows.size(); i++)
{
if (Rows[i].nrm2() > 0.)
{
int j = 0;
for (; j < Rows.size(); j++)
{
if (i != j)
{
if (Rows[j].nrm2() > 0.)
{
btVectorXu test(dim + 1);
for (int ii = 0; ii < dim + 1; ii++)
{
test[ii] = Rows[j][ii] - Rows[i][ii];
}
//=Rows[j] - Rows[i]
if (!LexicographicPositive(test))
{
break;
}
}
}
}
if (j == Rows.size())
{
RowIndex += i;
break;
}
}
}
return RowIndex;
}
bool btLemkeAlgorithm::LexicographicPositive(const btVectorXu& v)
{
int i = 0;
// if (DEBUGLEVEL)
// cout << "v " << v << endl;
while (i < v.size() - 1 && fabs(v[i]) < btMachEps())
i++;
if (v[i] > 0)
return true;
return false;
}
void btLemkeAlgorithm::GaussJordanEliminationStep(btMatrixXu& A, int pivotRowIndex, int pivotColumnIndex, const btAlignedObjectArray<int>& basis)
{
btScalar a = -1 / A(pivotRowIndex, pivotColumnIndex);
#ifdef BT_DEBUG_OSTREAM
cout << A << std::endl;
#endif
for (int i = 0; i < A.rows(); i++)
{
if (i != pivotRowIndex)
{
for (int j = 0; j < A.cols(); j++)
{
if (j != pivotColumnIndex)
{
btScalar v = A(i, j);
v += A(pivotRowIndex, j) * A(i, pivotColumnIndex) * a;
A.setElem(i, j, v);
}
}
}
}
#ifdef BT_DEBUG_OSTREAM
cout << A << std::endl;
#endif //BT_DEBUG_OSTREAM
for (int i = 0; i < A.cols(); i++)
{
A.mulElem(pivotRowIndex, i, -a);
}
#ifdef BT_DEBUG_OSTREAM
cout << A << std::endl;
#endif //#ifdef BT_DEBUG_OSTREAM
for (int i = 0; i < A.rows(); i++)
{
if (i != pivotRowIndex)
{
A.setElem(i, pivotColumnIndex, 0);
}
else
{
A.setElem(i, pivotColumnIndex, 1.0);
}
}
#ifdef BT_DEBUG_OSTREAM
cout << A << std::endl;
#endif //#ifdef BT_DEBUG_OSTREAM
}
bool btLemkeAlgorithm::greaterZero(const btVectorXu& vector)
{
bool isGreater = true;
for (int i = 0; i < vector.size(); i++)
{
if (vector[i] < 0)
{
isGreater = false;
break;
}
}
return isGreater;
}
bool btLemkeAlgorithm::validBasis(const btAlignedObjectArray<int>& basis)
{
bool isValid = true;
for (int i = 0; i < basis.size(); i++)
{
if (basis[i] >= basis.size() * 2)
{ //then z0 is in the base
isValid = false;
break;
}
}
return isValid;
}
int btLemkeAlgorithm::findLexicographicMinimum2(const btMatrixXu& A, const int& pivotColIndex, const int& z0Row)
{
btAlignedObjectArray<int> activeRows;
btScalar currentMin = std::numeric_limits<float>::max();
int dim = A.rows();
for (int row = 0; row < dim; row++)
{
const float denom = A(row, pivotColIndex);
if (denom > btMachEps())
{
const float q = A(row, dim + dim + 1) / denom;
if (fabs(currentMin - q) < btMachEps())
{
activeRows.push_back(row);
}
else if (currentMin > q)
{
currentMin = q;
activeRows.clear();
activeRows.push_back(row);
}
}
}
if (activeRows.size() == 0)
{
// ray termination.
return -1;
}
else if (activeRows.size() == 1)
{
return activeRows[0];
}
// If there are multiple rows, check if they contain the row for z_0.
for (int i = 0; i < activeRows.size(); i++)
{
if (activeRows[i] == z0Row)
{
return z0Row;
}
}
// look through the columns of the inverse of the basic matrix from left to right until the tie is broken.
for (int col = 0; col < dim ; col++)
{
btAlignedObjectArray<int> activeRowsCopy(activeRows);
activeRows.clear();
currentMin = std::numeric_limits<float>::max();
for (int i = 0; i<activeRowsCopy.size();i++)
{
const int row = activeRowsCopy[i];
// denom must be positive here as an invariant.
const btScalar denom = A(row, pivotColIndex);
const btScalar ratio = A(row, col) / denom;
if (fabs(currentMin - ratio) < btMachEps())
{
activeRows.push_back(row);
}
else if (currentMin > ratio)
{
currentMin = ratio;
activeRows.clear();
activeRows.push_back(row);
}
}
if (activeRows.size() == 1)
{
return activeRows[0];
}
}
// must not reach here.
return -1;
}