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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2001, 2002, 2003 Sadruddin Rejeb
Copyright (C) 2006 Ferdinando Ametrano
Copyright (C) 2007 Marco Bianchetti
Copyright (C) 2007 François du Vignaud
This file is part of QuantLib, a free-software/open-source library
for financial quantitative analysts and developers - http://quantlib.org/
QuantLib is free software: you can redistribute it and/or modify it
under the terms of the QuantLib license. You should have received a
copy of the license along with this program; if not, please email
<quantlib-dev@lists.sf.net>. The license is also available online at
<http://quantlib.org/license.shtml>.
This program is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
FOR A PARTICULAR PURPOSE. See the license for more details.
*/
/* The implementation of the algorithm was highly inspired by
* "Numerical Recipes in C", 2nd edition, Press, Teukolsky, Vetterling,
* Flannery, chapter 10.
* Modified may 2007: end criteria set on x instead on fx,
* inspired by bad behaviour found with test function fx=x*x+x+1,
* xStart = -100, lambda = 1.0, ftol = 1.e-16
* (it reports x=0 as the minimum!)
* and by GSL implementation, v. 1.9 (http://www.gnu.org/software/gsl/)
*/
#include <ql/math/optimization/simplex.hpp>
#include <ql/math/optimization/constraint.hpp>
#if !defined(__GNUC__) || __GNUC__ > 3 || __GNUC_MINOR__ > 4
#define QL_ARRAY_EXPRESSIONS
#endif
namespace QuantLib {
namespace {
// Computes the size of the simplex
Real computeSimplexSize (const std::vector<Array>& vertices) {
Array center(vertices.front().size(),0);
for (const auto& vertice : vertices)
center += vertice;
center *=1/Real(vertices.size());
Real result = 0;
for (const auto& vertice : vertices) {
Array temp = vertice - center;
result += Norm2(temp);
}
return result/Real(vertices.size());
}
}
Real Simplex::extrapolate(Problem& P,
Size iHighest,
Real &factor) const {
Array pTry;
do {
Size dimensions = values_.size() - 1;
Real factor1 = (1.0 - factor)/dimensions;
Real factor2 = factor1 - factor;
#if defined(QL_ARRAY_EXPRESSIONS)
pTry = sum_*factor1 - vertices_[iHighest]*factor2;
#else
// composite expressions fail to compile with gcc 3.4 on windows
pTry = sum_*factor1;
pTry -= vertices_[iHighest]*factor2;
#endif
factor *= 0.5;
} while (!P.constraint().test(pTry) && std::fabs(factor) > QL_EPSILON);
if (std::fabs(factor) <= QL_EPSILON) {
return values_[iHighest];
}
factor *= 2.0;
Real vTry = P.value(pTry);
if (vTry < values_[iHighest]) {
values_[iHighest] = vTry;
#if defined(QL_ARRAY_EXPRESSIONS)
sum_ += pTry - vertices_[iHighest];
#else
sum_ += pTry;
sum_ -= vertices_[iHighest];
#endif
vertices_[iHighest] = pTry;
}
return vTry;
}
EndCriteria::Type Simplex::minimize(Problem& P,
const EndCriteria& endCriteria) {
// set up of the problem
//Real ftol = endCriteria.functionEpsilon(); // end criteria on f(x) (see Numerical Recipes in C++, p.410)
Real xtol = endCriteria.rootEpsilon(); // end criteria on x (see GSL v. 1.9, http://www.gnu.org/software/gsl/)
Size maxStationaryStateIterations_
= endCriteria.maxStationaryStateIterations();
EndCriteria::Type ecType = EndCriteria::None;
P.reset();
Array x_ = P.currentValue();
if (!P.constraint().test(x_))
QL_FAIL("Initial guess " << x_ << " is not in the feasible region.");
Integer iterationNumber_=0;
// Initialize vertices of the simplex
Size n = x_.size();
vertices_ = std::vector<Array>(n+1, x_);
for (Size i=0; i<n; ++i) {
Array direction(n, 0.0);
direction[i] = 1.0;
P.constraint().update(vertices_[i+1], direction, lambda_);
}
// Initialize function values at the vertices of the simplex
values_ = Array(n+1, 0.0);
for (Size i=0; i<=n; ++i)
values_[i] = P.value(vertices_[i]);
// Loop looking for minimum
do {
sum_ = Array(n, 0.0);
Size i;
for (i=0; i<=n; i++)
sum_ += vertices_[i];
// Determine the best (iLowest), worst (iHighest)
// and 2nd worst (iNextHighest) vertices
Size iLowest = 0;
Size iHighest, iNextHighest;
if (values_[0]<values_[1]) {
iHighest = 1;
iNextHighest = 0;
} else {
iHighest = 0;
iNextHighest = 1;
}
for (i=1;i<=n; i++) {
if (values_[i]>values_[iHighest]) {
iNextHighest = iHighest;
iHighest = i;
} else {
if ((values_[i]>values_[iNextHighest]) && i!=iHighest)
iNextHighest = i;
}
if (values_[i]<values_[iLowest])
iLowest = i;
}
// Now compute accuracy, update iteration number and check end criteria
//// Numerical Recipes exit strategy on fx (see NR in C++, p.410)
//Real low = values_[iLowest];
//Real high = values_[iHighest];
//Real rtol = 2.0*std::fabs(high - low)/
// (std::fabs(high) + std::fabs(low) + QL_EPSILON);
//++iterationNumber_;
//if (rtol < ftol ||
// endCriteria.checkMaxIterations(iterationNumber_, ecType)) {
// GSL exit strategy on x (see GSL v. 1.9, http://www.gnu.org/software/gsl
Real simplexSize = computeSimplexSize(vertices_);
++iterationNumber_;
if (simplexSize < xtol ||
endCriteria.checkMaxIterations(iterationNumber_, ecType)) {
endCriteria.checkStationaryPoint(0.0, 0.0,
maxStationaryStateIterations_, ecType);
endCriteria.checkMaxIterations(iterationNumber_, ecType);
x_ = vertices_[iLowest];
Real low = values_[iLowest];
P.setFunctionValue(low);
P.setCurrentValue(x_);
return ecType;
}
// If end criteria is not met, continue
Real factor = -1.0;
Real vTry = extrapolate(P, iHighest, factor);
if ((vTry <= values_[iLowest]) && (factor == -1.0)) {
factor = 2.0;
extrapolate(P, iHighest, factor);
} else if (std::fabs(factor) > QL_EPSILON) {
if (vTry >= values_[iNextHighest]) {
Real vSave = values_[iHighest];
factor = 0.5;
vTry = extrapolate(P, iHighest, factor);
if (vTry >= vSave && std::fabs(factor) > QL_EPSILON) {
for (Size i=0; i<=n; i++) {
if (i!=iLowest) {
#if defined(QL_ARRAY_EXPRESSIONS)
vertices_[i] =
0.5*(vertices_[i] + vertices_[iLowest]);
#else
vertices_[i] += vertices_[iLowest];
vertices_[i] *= 0.5;
#endif
values_[i] = P.value(vertices_[i]);
}
}
}
}
}
// If can't extrapolate given the constraints, exit
if (std::fabs(factor) <= QL_EPSILON) {
x_ = vertices_[iLowest];
Real low = values_[iLowest];
P.setFunctionValue(low);
P.setCurrentValue(x_);
return EndCriteria::StationaryFunctionValue;
}
} while (true);
QL_FAIL("optimization failed: unexpected behaviour");
}
}
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