private void quadraticBacktrack(
final DoubleMatrix1D p,
final Function1D<DoubleMatrix1D, Double> function,
final DataBundle data) {
final double lambda0 = data.getLambda0();
final double g0 = data.getG0();
final double lambda =
Math.max(0.01 * lambda0, g0 * lambda0 * lambda0 / (data.getG1() + g0 * (2 * lambda0 - 1)));
data.swapLambdaAndReplace(lambda);
updatePosition(p, function, data);
}
private boolean isConverged(final DataBundle data) {
final DoubleMatrix1D deltaX = data.getDeltaX();
final DoubleMatrix1D x = data.getX();
final int n = deltaX.getNumberOfElements();
double diff, scale;
for (int i = 0; i < n; i++) {
diff = Math.abs(deltaX.getEntry(i));
scale = Math.abs(x.getEntry(i));
if (diff > _absoluteTol + scale * _relativeTol) {
return false;
}
}
return (MA.getNorm2(data.getGrad()) < _absoluteTol);
}
@Override
public DoubleMatrix1D minimize(
final Function1D<DoubleMatrix1D, Double> function,
final Function1D<DoubleMatrix1D, DoubleMatrix1D> grad,
final DoubleMatrix1D startPosition) {
final DataBundle data = new DataBundle();
final double y = function.evaluate(startPosition);
data.setX(startPosition);
data.setG0(y * y);
data.setGrad(grad.evaluate(startPosition));
data.setInverseHessianEsimate(getInitializedMatrix(startPosition));
if (!getNextPosition(function, grad, data)) {
throw new MathException(
"Cannot work with this starting position. Please choose another point");
}
int count = 0;
int resetCount = 1;
while (!isConverged(data)) {
if ((resetCount) % RESET_FREQ == 0) {
data.setInverseHessianEsimate(getInitializedMatrix(startPosition));
resetCount = 1;
} else {
_hessainUpdater.update(data);
}
if (!getNextPosition(function, grad, data)) {
data.setInverseHessianEsimate(getInitializedMatrix(startPosition));
resetCount = 1;
if (!getNextPosition(function, grad, data)) {
throw new MathException("Failed to converge in backtracking");
}
}
count++;
resetCount++;
if (count > _maxSteps) {
throw new MathException(
"Failed to converge after "
+ _maxSteps
+ " iterations. Final point reached: "
+ data.getX().toString());
}
}
return data.getX();
}
private void bisectBacktrack(
final DoubleMatrix1D p,
final Function1D<DoubleMatrix1D, Double> function,
final DataBundle data) {
do {
data.setLambda0(data.getLambda0() * 0.1);
updatePosition(p, function, data);
} while (Double.isNaN(data.getG1())
|| Double.isInfinite(data.getG1())
|| Double.isNaN(data.getG2())
|| Double.isInfinite(data.getG2()));
}
protected void updatePosition(
final DoubleMatrix1D p,
final Function1D<DoubleMatrix1D, Double> function,
final DataBundle data) {
final double lambda0 = data.getLambda0();
final DoubleMatrix1D deltaX = (DoubleMatrix1D) MA.scale(p, lambda0);
final DoubleMatrix1D xNew = (DoubleMatrix1D) MA.add(data.getX(), deltaX);
data.setDeltaX(deltaX);
data.setG2(data.getG1());
final double y = function.evaluate(xNew);
data.setG1(y * y);
}
private void cubicBacktrack(
final DoubleMatrix1D p,
final Function1D<DoubleMatrix1D, Double> function,
final DataBundle data) {
double temp1, temp2, temp3, temp4, temp5;
final double lambda0 = data.getLambda0();
final double lambda1 = data.getLambda1();
final double g0 = data.getG0();
temp1 = 1.0 / lambda0 / lambda0;
temp2 = 1.0 / lambda1 / lambda1;
temp3 = data.getG1() + g0 * (2 * lambda0 - 1.0);
temp4 = data.getG2() + g0 * (2 * lambda1 - 1.0);
temp5 = 1.0 / (lambda0 - lambda1);
final double a = temp5 * (temp1 * temp3 - temp2 * temp4);
final double b = temp5 * (-lambda1 * temp1 * temp3 + lambda0 * temp2 * temp4);
double lambda = (-b + Math.sqrt(b * b + 6 * a * g0)) / 3 / a;
lambda =
Math.min(
Math.max(lambda, 0.01 * lambda0),
0.75 * lambda1); // make sure new lambda is between 1% & 75% of old value
data.swapLambdaAndReplace(lambda);
updatePosition(p, function, data);
}
private boolean getNextPosition(
final Function1D<DoubleMatrix1D, Double> function,
final Function1D<DoubleMatrix1D, DoubleMatrix1D> grad,
final DataBundle data) {
final DoubleMatrix1D p = getDirection(data);
if (data.getLambda0() < 1.0) {
data.setLambda0(1.0);
} else {
data.setLambda0(data.getLambda0() * BETA);
}
updatePosition(p, function, data);
final double g1 = data.getG1();
// the function is invalid at the new position, try to recover
if (Double.isInfinite(g1) || Double.isNaN(g1)) {
bisectBacktrack(p, function, data);
}
if (data.getG1() > data.getG0() / (1 + ALPHA * data.getLambda0())) {
quadraticBacktrack(p, function, data);
int count = 0;
while (data.getG1() > data.getG0() / (1 + ALPHA * data.getLambda0())) {
if (count > 5) {
return false;
}
cubicBacktrack(p, function, data);
count++;
}
}
final DoubleMatrix1D deltaX = data.getDeltaX();
data.setX((DoubleMatrix1D) MA.add(data.getX(), deltaX));
data.setG0(data.getG1());
final DoubleMatrix1D gradNew = grad.evaluate(data.getX());
data.setDeltaGrad((DoubleMatrix1D) MA.subtract(gradNew, data.getGrad()));
data.setGrad(gradNew);
return true;
}
private DoubleMatrix1D getDirection(final DataBundle data) {
return (DoubleMatrix1D)
MA.multiply(data.getInverseHessianEsimate(), MA.scale(data.getGrad(), -1.0));
}
