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;
}
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;
}