void SteepestDescent (Vector & x, const MinFunction & fun,
const OptiParameters & par)
{
int it, n = x.Size();
Vector xnew(n), p(n), g(n), g2(n);
double val, alphahat;
int fail;
val = fun.FuncGrad(x, g);
alphahat = 1;
// testout << "f = ";
for (it = 0; it < 10; it++)
{
// testout << val << " ";
// p = -g;
p.Set (-1, g);
lines (x, xnew, p, val, g, fun, par, alphahat, -1e5,
0.1, 0.1, 1, 10, 0.1, 0.1, 0.6, fail);
x = xnew;
}
// testout << endl;
}
double BFGS (
Vector & x, // i: Startwert
// o: Loesung, falls IFAIL = 0
const MinFunction & fun,
const OptiParameters & par,
double eps
)
{
int i, j, n = x.Size();
long it;
char a1crit, a3acrit;
Vector d(n), g(n), p(n), temp(n), bs(n), xneu(n), y(n), s(n), x0(n);
DenseMatrix l(n);
DenseMatrix hesse(n);
double /* normg, */ alphahat, hd, fold;
double a1, a2;
const double mu1 = 0.1, sigma = 0.1, xi1 = 1, xi2 = 10;
const double tau = 0.1, tau1 = 0.1, tau2 = 0.6;
Vector typx(x.Size()); // i: typische Groessenordnung der Komponenten
double f, f0;
double typf; // i: typische Groessenordnung der Loesung
double fmin = -1e5; // i: untere Schranke fuer Funktionswert
// double eps = 1e-8; // i: Abbruchschranke fuer relativen Gradienten
double tauf = 0.1; // i: Abbruchschranke fuer die relative Aenderung der
// Funktionswerte
int ifail; // o: 0 .. Erfolg
// -1 .. Unterschreitung von fmin
// 1 .. kein Erfolg bei Liniensuche
// 2 .. Überschreitung von itmax
typx = par.typx;
typf = par.typf;
l = 0;
for (i = 1; i <= n; i++)
l.Elem(i, i) = 1;
f = fun.FuncGrad (x, g);
f0 = f;
x0 = x;
it = 0;
do
{
// Restart
if (it % (5 * n) == 0)
{
for (i = 1; i <= n; i++)
d.Elem(i) = typf/ sqr (typx.Get(i)); // 1;
for (i = 2; i <= n; i++)
for (j = 1; j < i; j++)
l.Elem(i, j) = 0;
/*
hesse = 0;
for (i = 1; i <= n; i++)
hesse.Elem(i, i) = typf / sqr (typx.Get(i));
fun.ApproximateHesse (x, hesse);
Cholesky (hesse, l, d);
*/
}
it++;
if (it > par.maxit_bfgs)
{
ifail = 2;
break;
}
// Solve with factorized B
SolveLDLt (l, d, g, p);
// (*testout) << "l " << l << endl
// << "d " << d << endl
// << "g " << g << endl
// << "p " << p << endl;
p *= -1;
y = g;
fold = f;
// line search
alphahat = 1;
lines (x, xneu, p, f, g, fun, par, alphahat, fmin,
mu1, sigma, xi1, xi2, tau, tau1, tau2, ifail);
if(ifail == 1)
(*testout) << "no success with linesearch" << endl;
/*
// if (it > par.maxit_bfgs/2)
{
(*testout) << "x = " << x << endl;
(*testout) << "xneu = " << xneu << endl;
(*testout) << "f = " << f << endl;
(*testout) << "g = " << g << endl;
}
*/
// (*testout) << "it = " << it << " f = " << f << endl;
// if (ifail != 0) break;
s.Set2 (1, xneu, -1, x);
y *= -1;
y.Add (1,g); // y += g;
x = xneu;
// BFGS Update
MultLDLt (l, d, s, bs);
a1 = y * s;
a2 = s * bs;
if (a1 > 0 && a2 > 0)
{
if (LDLtUpdate (l, d, 1 / a1, y) != 0)
{
// [JW] according to [JS] we can safely ignore this error
//cerr << "BFGS update error1" << endl;
//(*testout) << "BFGS update error1" << endl;
//(*testout) << "l " << endl << l << endl
//<< "d " << d << endl;
ifail = 1;
break;
}
if (LDLtUpdate (l, d, -1 / a2, bs) != 0)
{
// [JW] according to [JS] we can safely ignore this error
//cerr << "BFGS update error2" << endl;
//(*testout) << "BFGS update error2" << endl;
//(*testout) << "l " << endl << l << endl
// << "d " << d << endl;
ifail = 1;
break;
}
}
// Calculate stop conditions
hd = eps * max2 (typf, fabs (f));
a1crit = 1;
for (i = 1; i <= n; i++)
if ( fabs (g.Elem(i)) * max2 (typx.Elem(i), fabs (x.Elem(i))) > hd)
a1crit = 0;
a3acrit = (fold - f <= tauf * max2 (typf, fabs (f)));
// testout << "g = " << g << endl;
// testout << "a1crit, a3crit = " << int(a1crit) << ", " << int(a3acrit) << endl;
/*
// Output for tests
normg = sqrt (g * g);
testout << "it =" << setw (5) << it
<< " f =" << setw (12) << setprecision (5) << f
<< " |g| =" << setw (12) << setprecision (5) << normg;
testout << " x = (" << setw (12) << setprecision (5) << x.Elem(1);
for (i = 2; i <= n; i++)
testout << "," << setw (12) << setprecision (5) << x.Elem(i);
testout << ")" << endl;
*/
//(*testout) << "it = " << it << " f = " << f << " x = " << x << endl
// << " g = " << g << " p = " << p << endl << endl;
// (*testout) << "|g| = " << g.L2Norm() << endl;
if (g.L2Norm() < fun.GradStopping (x)) break;
}
while (!a1crit || !a3acrit);
/*
(*testout) << "it = " << it << " g = " << g << " f = " << f
<< " fail = " << ifail << endl;
*/
if (f0 < f || (ifail == 1))
{
(*testout) << "fail, f = " << f << " f0 = " << f0 << endl;
f = f0;
x = x0;
}
// (*testout) << "x = " << x << ", x0 = " << x0 << endl;
return f;
}
/// Line search, modified Mangasarien conditions
void lines (Vector & x, // i: initial point of line-search
Vector & xneu, // o: solution, if successful
Vector & p, // i: search direction
double & f, // i: function-value at x
// o: function-value at xneu, iff ifail = 0
Vector & g, // i: gradient at x
// o: gradient at xneu, iff ifail = 0
const MinFunction & fun, // function to minimize
const OptiParameters & par,
double & alphahat, // i: initial value for alpha_hat
// o: solution alpha iff ifail = 0
double fmin, // i: lower bound for f
double mu1, // i: Parameter mu_1 of Alg.2.1
double sigma, // i: Parameter sigma of Alg.2.1
double xi1, // i: Parameter xi_1 of Alg.2.1
double xi2, // i: Parameter xi_1 of Alg.2.1
double tau, // i: Parameter tau of Alg.2.1
double tau1, // i: Parameter tau_1 of Alg.2.1
double tau2, // i: Parameter tau_2 of Alg.2.1
int & ifail) // o: 0 on success
// -1 bei termination because lower limit fmin
// 1 bei illegal termination due to different reasons
{
double phi0, phi0prime, phi1, phi1prime, phihatprime;
double alpha1, alpha2, alphaincr, c;
char flag = 1;
long it;
alpha1 = 0;
alpha2 = 1e50;
phi0 = phi1 = f;
phi0prime = g * p;
if (phi0prime > 0)
{
ifail = 1;
return;
}
ifail = 1;
phi1prime = phi0prime;
// (*testout) << "phi0prime = " << phi0prime << endl;
// it = 100000l;
it = 0;
while (it++ <= par.maxit_linsearch)
{
xneu.Set2 (1, x, alphahat, p);
// f = fun.FuncGrad (xneu, g);
// f = fun.Func (xneu);
f = fun.FuncDeriv (xneu, p, phihatprime);
// (*testout) << "lines, f = " << f << " phip = " << phihatprime << endl;
if (f < fmin)
{
ifail = -1;
break;
}
if (alpha2 - alpha1 < eps0 * alpha2)
{
ifail = 0;
break;
}
// (*testout) << "i = " << it << " al = " << alphahat << " f = " << f << " fprime " << phihatprime << endl;;
if (f - phi0 > mu1 * alphahat * phi1prime + eps0 * fabs (phi0))
{
flag = 0;
alpha2 = alphahat;
c =
(f - phi1 - phi1prime * (alphahat-alpha1)) /
sqr (alphahat-alpha1);
alphahat = alpha1 - 0.5 * phi1prime / c;
if (alphahat > alpha2)
alphahat = alpha1 + 1/(4*c) *
( (sigma+mu1) * phi0prime - 2*phi1prime
+ sqrt (sqr(phi1prime - mu1 * phi0prime) -
4 * (phi1 - phi0 - mu1 * alpha1 * phi0prime) * c));
alphahat = max2 (alphahat, alpha1 + tau * (alpha2 - alpha1));
alphahat = min2 (alphahat, alpha2 - tau * (alpha2 - alpha1));
// (*testout) << " if-branch" << endl;
}
else
{
/*
f = fun.FuncGrad (xneu, g);
phihatprime = g * p;
*/
f = fun.FuncDeriv (xneu, p, phihatprime);
if (phihatprime < sigma * phi0prime * (1 + eps0))
{
if (phi1prime < phihatprime)
// Approximationsfunktion ist konvex
alphaincr = (alphahat - alpha1) * phihatprime /
(phi1prime - phihatprime);
else
alphaincr = 1e99; // MAXDOUBLE;
if (flag)
{
alphaincr = max2 (alphaincr, xi1 * (alphahat-alpha1));
alphaincr = min2 (alphaincr, xi2 * (alphahat-alpha1));
}
else
{
alphaincr = max2 (alphaincr, tau1 * (alpha2 - alphahat));
alphaincr = min2 (alphaincr, tau2 * (alpha2 - alphahat));
}
alpha1 = alphahat;
alphahat += alphaincr;
phi1 = f;
phi1prime = phihatprime;
}
else
{
ifail = 0; // Erfolg !!
break;
}
// (*testout) << " else, " << endl;
}
}
// (*testout) << "linsearch: it = " << it << " ifail = " << ifail << endl;
fun.FuncGrad (xneu, g);
if (it < 0)
ifail = 1;
// (*testout) << "fail = " << ifail << endl;
}
double BFGS (
Vector & x, // i: Startwert
// o: Loesung, falls IFAIL = 0
const MinFunction & fun,
const OptiParameters & par,
double eps
)
{
int n = x.Size();
long it;
char a1crit, a3acrit;
Vector d(n), g(n), p(n), temp(n), bs(n), xneu(n), y(n), s(n), x0(n);
DenseMatrix l(n);
DenseMatrix hesse(n);
double /* normg, */ alphahat, hd, fold;
double a1, a2;
const double mu1 = 0.1, sigma = 0.1, xi1 = 1, xi2 = 10;
const double tau = 0.1, tau1 = 0.1, tau2 = 0.6;
Vector typx(x.Size()); // i: typische Groessenordnung der Komponenten
double f, f0;
double typf; // i: typische Groessenordnung der Loesung
double fmin = -1e5; // i: untere Schranke fuer Funktionswert
// double eps = 1e-8; // i: Abbruchschranke fuer relativen Gradienten
double tauf = 0.1; // i: Abbruchschranke fuer die relative Aenderung der
// Funktionswerte
int ifail; // o: 0 .. Erfolg
// -1 .. Unterschreitung von fmin
// 1 .. kein Erfolg bei Liniensuche
// 2 .. Überschreitung von itmax
typx = par.typx;
typf = par.typf;
l = 0;
for (int i = 1; i <= n; i++)
l.Elem(i, i) = 1;
f = fun.FuncGrad (x, g);
f0 = f;
x0 = x;
it = 0;
do
{
// Restart
if (it % (5 * n) == 0)
{
for (int i = 1; i <= n; i++)
d(i-1) = typf/ sqr (typx(i-1)); // 1;
for (int i = 2; i <= n; i++)
for (int j = 1; j < i; j++)
l.Elem(i, j) = 0;
}
it++;
if (it > par.maxit_bfgs)
{
ifail = 2;
break;
}
// Solve with factorized B
SolveLDLt (l, d, g, p);
p *= -1;
y = g;
fold = f;
// line search
alphahat = 1;
lines (x, xneu, p, f, g, fun, par, alphahat, fmin,
mu1, sigma, xi1, xi2, tau, tau1, tau2, ifail);
if(ifail == 1)
cout << "no success with linesearch" << endl;
s.Set2 (1, xneu, -1, x);
y *= -1;
y.Add (1,g); // y += g;
x = xneu;
// BFGS Update
MultLDLt (l, d, s, bs);
a1 = y * s;
a2 = s * bs;
if (a1 > 0 && a2 > 0)
{
if (LDLtUpdate (l, d, 1 / a1, y) != 0)
{
cerr << "BFGS update error1" << endl;
cout << "BFGS update error1" << endl;
cout << "l " << endl << l << endl
<< "d " << d << endl;
ifail = 1;
break;
}
if (LDLtUpdate (l, d, -1 / a2, bs) != 0)
{
cerr << "BFGS update error2" << endl;
cout << "BFGS update error2" << endl;
cout << "l " << endl << l << endl
<< "d " << d << endl;
ifail = 1;
break;
}
}
// Calculate stop conditions
hd = eps * max2 (typf, fabs (f));
a1crit = 1;
for (int i = 1; i <= n; i++)
if ( fabs (g(i-1)) * max2 (typx(i-1), fabs (x(i-1))) > hd)
a1crit = 0;
a3acrit = (fold - f <= tauf * max2 (typf, fabs (f)));
if (g.L2Norm() < fun.GradStopping (x)) break;
}
while (!a1crit || !a3acrit);
if (f0 < f || (ifail == 1))
{
cout << "fail, f = " << f << " f0 = " << f0 << endl;
f = f0;
x = x0;
}
return f;
}
