internal static QuadraticInterpolationModel Construct(MultiFunctor f, IReadOnlyList <double> x, double s)
{
QuadraticInterpolationModel model = new QuadraticInterpolationModel();
model.Initialize(f, x, s);
return(model);
}
public static QuadraticInterpolationModel Construct(double[][] points, double[] values)
{
int m = points.Length;
int d = points[0].Length;
// find the minimum point, use it as the origin
int iMin = 0; double fMin = values[0];
for (int i = 1; i < values.Length; i++)
{
if (values[i] < fMin)
{
iMin = i; fMin = values[i];
}
}
SquareMatrix A = new SquareMatrix(m);
int c = 0;
for (int r = 0; r < m; r++)
{
A[r, 0] = 1.0;
}
for (int i = 0; i < d; i++)
{
c++;
for (int r = 0; r < m; r++)
{
A[r, c] = points[r][i] - points[iMin][i];
}
}
for (int i = 0; i < d; i++)
{
for (int j = 0; j <= i; j++)
{
c++;
for (int r = 0; r < m; r++)
{
A[r, c] = (points[r][i] - points[iMin][i]) * (points[r][j] - points[iMin][j]);
}
}
}
ColumnVector b = new ColumnVector(values);
SquareQRDecomposition QR = A.QRDecomposition();
ColumnVector a = QR.Solve(b);
QuadraticInterpolationModel model = new QuadraticInterpolationModel();
model.d = d;
model.origin = points[iMin];
model.f0 = a[0];
model.g = new double[d];
c = 0;
for (int i = 0; i < d; i++)
{
c++;
model.g[i] = a[c];
}
model.h = new double[d][];
for (int i = 0; i < d; i++)
{
model.h[i] = new double[d];
}
for (int i = 0; i < d; i++)
{
for (int j = 0; j <= i; j++)
{
c++;
if (i == j)
{
model.h[i][j] = 2.0 * a[c];
}
else
{
model.h[i][j] = a[c];
model.h[j][i] = a[c];
}
}
}
return(model);
}
public static QuadraticInterpolationModel Construct(double[][] points, double[] values)
{
int m = points.Length;
int d = points[0].Length;
// find the minimum point, use it as the origin
int iMin = 0; double fMin = values[0];
for (int i = 1; i < values.Length; i++) {
if (values[i] < fMin) { iMin = i; fMin = values[i]; }
}
SquareMatrix A = new SquareMatrix(m);
int c = 0;
for (int r = 0; r < m; r++) {
A[r, 0] = 1.0;
}
for (int i = 0; i < d; i++) {
c++;
for (int r = 0; r < m; r++) {
A[r, c] = points[r][i] - points[iMin][i];
}
}
for (int i = 0; i < d; i++) {
for (int j = 0; j <= i; j++) {
c++;
for (int r = 0; r < m; r++) {
A[r, c] = (points[r][i] - points[iMin][i]) * (points[r][j] - points[iMin][j]);
}
}
}
ColumnVector b = new ColumnVector(values);
SquareQRDecomposition QR = A.QRDecomposition();
ColumnVector a = QR.Solve(b);
QuadraticInterpolationModel model = new QuadraticInterpolationModel();
model.d = d;
model.origin = points[iMin];
model.f0 = a[0];
model.g = new double[d];
c = 0;
for (int i = 0; i < d; i++) {
c++;
model.g[i] = a[c];
}
model.h = new double[d][];
for (int i = 0; i < d; i++) {
model.h[i] = new double[d];
}
for (int i = 0; i < d; i++) {
for (int j = 0; j <= i; j++) {
c++;
if (i == j) {
model.h[i][j] = 2.0 * a[c];
} else {
model.h[i][j] = a[c];
model.h[j][i] = a[c];
}
}
}
return (model);
}
// This method is due to Powell (http://en.wikipedia.org/wiki/Michael_J._D._Powell), but it is not what
// is usually called Powell's Method (http://en.wikipedia.org/wiki/Powell%27s_method); Powell
// developed that method in the 1960s, it was included in Numerical Recipes and is very popular.
// This is a model trust algorithm developed by Powell in the 2000s. It typically uses many
// fewer function evaluations, but does more intensive calculations between each evaluation.
// This is basically the UOBYQA variant of Powell's new methods. It maintains a quadratic model
// that interpolates between (d + 1) (d + 2) / 2 points. The model is trusted
// within a given radius. At each step, it moves to the minimum of the model (or the boundary of
// the trust region in that direction) and evaluates the function. The new value is incorporated
// into the model and the trust region expanded or contracted depending on how accurate its
// prediction of the function value was.
// Papers on these methods are collected at http://mat.uc.pt/~zhang/software.html#powell_software.
// The UOBYQA paper is here: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.28.1756.
// The NEWUOA paper is here: http://www.damtp.cam.ac.uk/user/na/NA_papers/NA2004_08.pdf.
// The CONDOR system (http://www.applied-mathematics.net/optimization/CONDORdownload.html) is based on these same ideas.
// The thesis of CONDOR's author (http://www.applied-mathematics.net/mythesis/index.html) was also helpful.
// It should be very easy to extend this method to constrained optimization, either by incorporating the bounds into
// the step limits or by mapping hyper-space into a hyper-cube.
private static MultiExtremum FindMinimum_ModelTrust(MultiFunctor f, IReadOnlyList <double> x, double s, MultiExtremumSettings settings)
{
// Construct an initial model.
QuadraticInterpolationModel model = QuadraticInterpolationModel.Construct(f, x, s);
double trustRadius = s;
while (f.EvaluationCount < settings.EvaluationBudget)
{
// Find the minimum point of the model within the trust radius
double[] z = model.FindMinimum(trustRadius);
double expectedValue = model.Evaluate(z);
double deltaExpected = model.MinimumValue - expectedValue;
// Evaluate the function at the suggested minimum
double[] point = model.ConvertPoint(z);
double value = f.Evaluate(point);
double delta = model.MinimumValue - value;
double tol = settings.ComputePrecision(Math.Min(model.MinimumValue, value));
// Note value can be way off, so use better of old best and new value to compute tol.
// When we didn't do this before, we got value = infinity, so tol = infinity, and thus terminated!
if (delta > 0.0 && settings.Listener != null)
{
MultiExtremum report = new MultiExtremum(f.EvaluationCount, settings, point, value, Math.Max(Math.Abs(delta), 0.75 * tol), model.GetHessian());
settings.Listener(report);
}
// To terminate, we demand: a reduction, that the reduction be small, that the reduction be in line with
// its expected value, that we have not run up against the trust boundary, and that the gradient is small.
// I had wanted to demand delta > 0, but we run into some cases where delta keeps being very slightly
// negative, typically orders of magnitude less than tol, causing the trust radius to shrink in an
// endless cycle that causes our approximation to ultimately go sour, even though terminating on the original
// very slightly negative delta would have produced an accurate estimate. So we tolerate this case for now.
if ((delta <= tol) && (-0.25 * tol <= delta))
{
// We demand that the model be decent, i.e. that the expected delta was within tol of the measured delta.
if (Math.Abs(delta - deltaExpected) <= tol)
{
// We demand that the step not just be small because it ran up against the trust radius.
// If it ran up against the trust radius, there is probably more to be hand by continuing.
double zm = Blas1.dNrm2(z, 0, 1, z.Length);
if (zm < trustRadius)
{
// Finally, we demand that the gradient be small. You might think this was obvious since
// z was small, but if the Hessian is not positive definite
// the interplay of the Hessian and the gradient can produce a small z even if the model looks nothing like a quadratic minimum.
double gm = Blas1.dNrm2(model.GetGradient(), 0, 1, z.Length);
if (gm * zm <= tol)
{
if (f.IsNegated)
{
value = -value;
}
return(new MultiExtremum(f.EvaluationCount, settings, point, value, Math.Max(Math.Abs(delta), 0.75 * tol), model.GetHessian()));
}
}
}
}
// There are now three decisions to be made:
// 1. How to change the trust radius
// 2. Whether to accept the new point
// 3. Which existing point to replace
// If the actual change was very far from the expected change, reduce the trust radius.
// If the expected change did a good job of predicting the actual change, increase the trust radius.
if ((delta < 0.25 * deltaExpected) /*|| (8.0 * deltaExpected < delta)*/)
{
trustRadius = 0.5 * trustRadius;
}
else if ((0.75 * deltaExpected <= delta) /*&& (delta <= 2.0 * deltaExpected)*/)
{
trustRadius = 2.0 * trustRadius;
}
// It appears that the limits on delta being too large don't help, and even hurt if made too stringent.
// Replace an old point with the new point.
int iMax = 0; double fMax = model.values[0];
int iBad = 0; double fBad = model.ComputeBadness(0, z, point, value);
for (int i = 1; i < model.values.Length; i++)
{
if (model.values[i] > fMax)
{
iMax = i; fMax = model.values[i];
}
double bad = model.ComputeBadness(i, z, point, value);
if (bad > fBad)
{
iBad = i; fBad = bad;
}
}
// Use the new point as long as it is better than our worst existing point.
if (value < fMax)
{
Debug.Assert(!Double.IsPositiveInfinity(value) && !Double.IsNaN(value));
model.ReplacePoint(iBad, point, z, value);
}
// There is some question about how best to choose which point to replace.
// The largest value? The furthest away? The one closest to new min?
}
throw new NonconvergenceException();
}
internal static QuadraticInterpolationModel Construct(MultiFunctor f, IList<double> x, double s)
{
QuadraticInterpolationModel model = new QuadraticInterpolationModel();
model.Initialize(f, x, s);
return (model);
}
