private static MultiExtremum FindGlobalExtremum(Func <IReadOnlyList <double>, double> function, IReadOnlyList <Interval> volume, MultiExtremumSettings settings, bool negate)
{
if (function == null)
{
throw new ArgumentNullException(nameof(function));
}
if (volume == null)
{
throw new ArgumentNullException(nameof(volume));
}
MultiFunctor f = new MultiFunctor(function, negate);
DifferentialEvolutionSettings deSettings = GetDefaultSettings(settings, volume.Count);
MultiExtremum extremum = FindGlobalExtremum(f, volume, deSettings);
return(extremum);
}
// 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();
}
// Differential evolution is a global optimization algorithm over continuous inputs that is adapted from genetic algorithms for finite inputs.
// The idea is to maintain a population of input vectors ("agents") and to vary that population over cycles ("generations") according to rules that incorporate
// random mutation but on average tend to bring them closer to optima ("fitter").
private static MultiExtremum FindGlobalExtremum(MultiFunctor f, IReadOnlyList <Interval> volume, DifferentialEvolutionSettings settings)
{
int d = volume.Count;
// Choose a number of agents that increases with dimension and required precision.
int m = settings.Population;
Debug.WriteLine("d={0} m={1}", d, m);
Random rng = new Random(3);
// Start with random points in the allowed region.
double[][] points = new double[m][];
double[] values = new double[m];
for (int i = 0; i < m; i++)
{
points[i] = new double[d];
for (int j = 0; j < d; j++)
{
points[i][j] = volume[j].LeftEndpoint + rng.NextDouble() * volume[j].Width;
}
values[i] = f.Evaluate(points[i]);
}
while (f.EvaluationCount < settings.EvaluationBudget)
{
double[][] newPoints = new double[m][];
double[] newValues = new double[m];
for (int i = 0; i < m; i++)
{
// Mutation
// construct donor vector
int a = i; while (a == i)
{
a = rng.Next(m);
}
int b = i; while ((b == i) || (b == a))
{
b = rng.Next(m);
}
int c = i; while ((c == i) || (c == b) || (c == a))
{
c = rng.Next(m);
}
double[] donor = new double[d];
for (int j = 0; j < d; j++)
{
donor[j] = points[a][j] + mutationFactor * (points[b][j] - points[c][j]);
if (donor[j] < volume[j].LeftEndpoint)
{
donor[j] = volume[j].LeftEndpoint;
}
if (donor[j] > volume[j].RightEndpoint)
{
donor[j] = volume[j].RightEndpoint;
}
}
// Recombination
double[] trial = new double[d];
int k = rng.Next(d);
for (int j = 0; j < d; j++)
{
if ((j == k) || (rng.NextDouble() < settings.CrossoverProbability))
{
trial[j] = donor[j];
}
else
{
trial[j] = points[i][j];
}
}
// Selection
double value = f.Evaluate(trial);
if (value <= values[i])
{
newPoints[i] = trial;
newValues[i] = value;
}
else
{
newPoints[i] = points[i];
newValues[i] = values[i];
}
}
points = newPoints;
values = newValues;
// Check termination criteria
int minIndex = -1;
double minValue = Double.MaxValue;
double maxValue = Double.MinValue;
for (int i = 0; i < m; i++)
{
if (values[i] < minValue)
{
minValue = values[i];
minIndex = i;
}
if (values[i] > maxValue)
{
maxValue = values[i];
}
}
double range = maxValue - minValue;
double tol = settings.ComputePrecision(minValue);
if (range <= tol)
{
MultiExtremum result = new MultiExtremum(f.EvaluationCount, settings, points[minIndex], f.IsNegated ? -values[minIndex] : values[minIndex], Math.Max(range, 0.75 * tol), null);
return(result);
}
else if (settings.Listener != null)
{
MultiExtremum report = new MultiExtremum(f.EvaluationCount, settings, points[minIndex], f.IsNegated ? -values[minIndex] : values[minIndex], Math.Max(range, 0.75 * tol), null);
settings.Listener(report);
}
}
throw new NonconvergenceException();
}
// Differential evoluation is a global optimization algorithm over continuous inputs that is adapted from genetic algorithms for finite inputs.
// The idea is maintain a population of input vectors ("agents") and to vary that population over cycles ("generations") according to rules that incorporate
// random mutation but on average tend to bring them closer to optima ("fitter").
private static MultiExtremum FindGlobalExtremum(MultiFunctor f, IList<Interval> volume, DifferentialEvolutionSettings settings)
{
int d = volume.Count;
// Choose a number of agents that increases with dimension and required precision.
int m = settings.Population;
Debug.WriteLine("d={0} m={1}", d, m);
Random rng = new Random(3);
//Random rng = new Random(1001110000);
// Start with random points in the allowed region.
double[][] points = new double[m][];
double[] values = new double[m];
for (int i = 0; i < m; i++) {
points[i] = new double[d];
for (int j = 0; j < d; j++) {
points[i][j] = volume[j].LeftEndpoint + rng.NextDouble() * volume[j].Width;
}
values[i] = f.Evaluate(points[i]);
}
while (f.EvaluationCount < settings.EvaluationBudget) {
//double mutationFactor = 0.5 + 0.5 * rng.NextDouble();
double[][] newPoints = new double[m][];
double[] newValues = new double[m];
for (int i = 0; i < m; i++) {
// Mutation
// construct donor vector
int a = i; while (a == i) a = rng.Next(m);
int b = i; while ((b == i) || (b == a)) b = rng.Next(m);
int c = i; while ((c == i) || (c == b) || (c == a)) c = rng.Next(m);
double[] donor = new double[d];
for (int j = 0; j < d; j++) {
donor[j] = points[a][j] + mutationFactor * (points[b][j] - points[c][j]);
if (donor[j] < volume[j].LeftEndpoint) donor[j] = volume[j].LeftEndpoint;
if (donor[j] > volume[j].RightEndpoint) donor[j] = volume[j].RightEndpoint;
}
// Recombination
double[] trial = new double[d];
int k = rng.Next(d);
for (int j = 0; j < d; j++) {
if ((j == k) || (rng.NextDouble() < settings.CrossoverProbability)) {
trial[j] = donor[j];
} else {
trial[j] = points[i][j];
}
}
// Selection
double value = f.Evaluate(trial);
if (value <= values[i]) {
newPoints[i] = trial;
newValues[i] = value;
} else {
newPoints[i] = points[i];
newValues[i] = values[i];
}
}
points = newPoints;
values = newValues;
// Check termination criteria
int minIndex = -1;
double minValue = Double.MaxValue;
double maxValue = Double.MinValue;
for (int i = 0; i < m; i++) {
if (values[i] < minValue) {
minValue = values[i];
minIndex = i;
}
if (values[i] > maxValue) maxValue = values[i];
}
double range = maxValue - minValue;
double tol = settings.ComputePrecision(minValue);
if (range <= tol) {
MultiExtremum result = new MultiExtremum(f.EvaluationCount, settings, points[minIndex], f.IsNegated ? -values[minIndex] : values[minIndex], Math.Max(range, 0.75 * tol), null);
return (result);
}
//settings.OnUpdate(new MultiExtremum(points[minIndex], values[minIndex], null, f.EvaluationCount));
}
throw new NonconvergenceException();
}