public void MinimizePerturbedQuadratic2D()
{
int count = 0;
MultiExtremumSettings s = new MultiExtremumSettings()
{
EvaluationBudget = 100,
RelativePrecision = 1.0E-10,
Listener = r => { count++; }
};
Func <IReadOnlyList <double>, double> f = (IReadOnlyList <double> x) =>
1.0 + 2.0 * MoreMath.Sqr(x[0] - 3.0) + 4.0 * (x[0] - 3.0) * (x[1] - 5.0) + 6.0 * MoreMath.Sqr(x[1] - 5.0) +
7.0 * MoreMath.Pow(x[0] - 3.0, 4) + 8.0 * MoreMath.Pow(x[1] - 5.0, 4);
MultiExtremum m = MultiFunctionMath.FindLocalMinimum(f, new double[] { 1.0, 1.0 }, s);
Assert.IsTrue(m.EvaluationCount <= s.EvaluationBudget);
Assert.IsTrue(m.Dimension == 2);
Assert.IsTrue(TestUtilities.IsNearlyEqual(m.Value, 1.0, s));
Assert.IsTrue(TestUtilities.IsNearlyEqual(m.Location, new ColumnVector(3.0, 5.0), new EvaluationSettings()
{
RelativePrecision = Math.Sqrt(s.RelativePrecision)
}));
Assert.IsTrue(count > 0);
}
public void Bukin()
{
// Burkin has a narrow valley, not aligned with any axis, punctuated with many tiny "wells" along its bottom.
// The deepest well is at (-10,1)-> 0.
Func <IReadOnlyList <double>, double> function = (IReadOnlyList <double> x) => 100.0 * Math.Sqrt(Math.Abs(x[1] - 0.01 * x[0] * x[0])) + 0.01 * Math.Abs(x[0] + 10.0);
IReadOnlyList <Interval> box = new Interval[] { Interval.FromEndpoints(-15.0, 0.0), Interval.FromEndpoints(-3.0, 3.0) };
int count = 0;
MultiExtremumSettings settings = new MultiExtremumSettings()
{
RelativePrecision = 0.0,
AbsolutePrecision = 1.0E-4,
EvaluationBudget = 1000000,
Listener = r => { count++; }
};
/*
* settings.Update += (object result) => {
* MultiExtremum e = (MultiExtremum) result;
* Console.WriteLine("After {0} evaluations, best value {1}", e.EvaluationCount, e.Value);
* };
*/
MultiExtremum minimum = MultiFunctionMath.FindGlobalMinimum(function, box, settings);
Console.WriteLine(minimum.EvaluationCount);
Console.WriteLine("{0} ({1})", minimum.Value, minimum.Precision);
Console.WriteLine("{0} {1}", minimum.Location[0], minimum.Location[1]);
// We do not end up finding the global minimum.
Assert.IsTrue(count > 0);
}
public void Ackley()
{
// Ackley's function has many local minima, and a global minimum at (0, 0) -> 0.
Func <IReadOnlyList <double>, double> function = (IReadOnlyList <double> x) => {
double s = 0.0;
double c = 0.0;
for (int i = 0; i < x.Count; i++)
{
s += x[i] * x[i];
c += Math.Cos(2.0 * Math.PI * x[i]);
}
return(-20.0 * Math.Exp(-0.2 * Math.Sqrt(s / x.Count)) - Math.Exp(c / x.Count) + 20.0 + Math.E);
};
MultiExtremumSettings settings = new MultiExtremumSettings()
{
AbsolutePrecision = 1.0E-8, EvaluationBudget = 10000000
};
for (int n = 2; n < 16; n = (int)Math.Round(AdvancedMath.GoldenRatio * n))
{
Console.WriteLine("n={0}", n);
Interval[] box = new Interval[n];
for (int i = 0; i < box.Length; i++)
{
box[i] = Interval.FromEndpoints(-32.0, 32.0);
}
MultiExtremum minimum = MultiFunctionMath.FindGlobalMinimum(function, box, settings);
Console.WriteLine(minimum.EvaluationCount);
Console.WriteLine(minimum.Value);
foreach (double coordinate in minimum.Location)
{
Console.WriteLine(coordinate);
}
Assert.IsTrue(minimum.Dimension == n);
Assert.IsTrue(TestUtilities.IsNearlyEqual(minimum.Value, 0.0, new EvaluationSettings()
{
AbsolutePrecision = 2.0 * minimum.Precision
}));
ColumnVector solution = new ColumnVector(n);
Assert.IsTrue(TestUtilities.IsNearlyEqual(minimum.Location, solution, new EvaluationSettings()
{
AbsolutePrecision = 2.0 * Math.Sqrt(minimum.Precision)
}));
}
}
public void SumOfPowers()
{
// This test is difficult because the minimum is emphatically not quadratic.
// We do get close to the minimum but we massivly underestimate our error.
Func <IReadOnlyList <double>, double> function = (IReadOnlyList <double> x) => {
double s = 0.0;
for (int i = 0; i < x.Count; i++)
{
s += MoreMath.Pow(Math.Abs(x[i]), i + 2);
}
return(s);
};
for (int n = 2; n < 8; n++)
{
Console.WriteLine(n);
ColumnVector start = new ColumnVector(n);
for (int i = 0; i < n; i++)
{
start[i] = 1.0;
}
MultiExtremumSettings settings = new MultiExtremumSettings()
{
AbsolutePrecision = 1.0E-8, EvaluationBudget = 32 * n * n * n
};
MultiExtremum minimum = MultiFunctionMath.FindLocalMinimum(function, start, settings);
Console.WriteLine(minimum.EvaluationCount);
Console.WriteLine("{0} {1}", minimum.Value, minimum.Precision);
Console.WriteLine("|| {0} {1} ... || = {2}", minimum.Location[0], minimum.Location[1], minimum.Location.FrobeniusNorm());
Assert.IsTrue(TestUtilities.IsNearlyEqual(minimum.Value, 0.0, new EvaluationSettings()
{
AbsolutePrecision = 1.0E-4
}));
//Assert.IsTrue(TestUtilities.IsNearlyEqual(minimum.Location, new ColumnVector(n), new EvaluationSettings() { AbsolutePrecision = 1.0E-2 }));
}
}
public void Griewank()
{
// See http://mathworld.wolfram.com/GriewankFunction.html
for (int n = 2; n < 8; n++)
{
Console.WriteLine(n);
Func <IReadOnlyList <double>, double> function = (IReadOnlyList <double> x) => {
double s = 0.0;
double p = 1.0;
for (int i = 0; i < x.Count; i++)
{
s += MoreMath.Sqr(x[i]);
p *= Math.Cos(x[i] / Math.Sqrt(i + 1.0));
}
return(1.0 + s / 4000.0 - p);
};
Interval[] box = new Interval[n];
for (int i = 0; i < n; i++)
{
box[i] = Interval.FromEndpoints(-100.0, 100.0);
}
MultiExtremumSettings settings = new MultiExtremumSettings()
{
AbsolutePrecision = 1.0E-6, EvaluationBudget = 1000000
};
MultiExtremum minimum = MultiFunctionMath.FindGlobalMinimum(function, box, settings);
Console.WriteLine(minimum.Dimension);
Console.WriteLine(minimum.EvaluationCount);
Console.WriteLine("{0} ({1}) ?= 0.0", minimum.Value, minimum.Precision);
Console.WriteLine("{0} {1} ...", minimum.Location[0], minimum.Location[1]);
// We usually find the minimum at 0, but sometimes land a valley or two over.
}
}
public void ThomsonLocal()
{
for (int n = 2; n < 8; n++)
{
// define the Thompson metric
Func <IReadOnlyList <double>, double> f = GetThompsonFunction(n);
// random distribution to start
// using antipodal pairs gives us a better starting configuration
Random r = new Random(1001110000);
double[] start = new double[2 * (n - 1)];
for (int i = 0; i < (n - 1) / 2; i++)
{
int j = 4 * i;
start[j] = -Math.PI + 2.0 * r.NextDouble() * Math.PI;
start[j + 1] = Math.Asin(2.0 * r.NextDouble() - 1.0);
start[j + 2] = -(Math.PI - start[j]);
start[j + 3] = -start[j + 1];
}
// add one more point if necessary
if (n % 2 == 0)
{
start[2 * n - 4] = -Math.PI + 2.0 * r.NextDouble() * Math.PI;
start[2 * n - 3] = Math.Asin(2.0 * r.NextDouble() - 1.0);
}
MultiExtremumSettings set = new MultiExtremumSettings()
{
RelativePrecision = 1.0E-9, AbsolutePrecision = 0.0
};
MultiExtremum min = MultiFunctionMath.FindLocalMinimum(f, start, set);
Assert.IsTrue(min.Dimension == 2 * (n - 1));
Assert.IsTrue(TestUtilities.IsNearlyEqual(min.Value, thompsonSolutions[n], new EvaluationSettings()
{
AbsolutePrecision = 10.0 * min.Precision
}));
}
}
public void PackCirclesInSquare()
{
// Put n points into the unit square. Place them so as to maximize the minimum distance between them.
// See http://en.wikipedia.org/wiki/Circle_packing_in_a_square, http://hydra.nat.uni-magdeburg.de/packing/csq/csq.html
// This is a great test problem because it is simple to understand, hard to solve,
// solutions are known/proven for many n, and it covers many dimensions.
double[] solutions = new double[] {
Double.NaN,
Double.NaN,
Math.Sqrt(2.0), /* at opposite corners */
Math.Sqrt(6.0) - Math.Sqrt(2.0),
1.0, /* at each corner */
Math.Sqrt(2.0) / 2.0 /* at each corner and in center */,
Math.Sqrt(13.0) / 6.0,
4.0 - 2.0 * Math.Sqrt(3.0),
(Math.Sqrt(6.0) - Math.Sqrt(2.0)) / 2.0,
1.0 / 2.0, /* 3 X 3 grid */
0.421279543983903432768821760651,
0.398207310236844165221512929748,
Math.Sqrt(34.0) / 15.0,
0.366096007696425085295389370603,
2.0 / (4.0 + Math.Sqrt(3.0)),
2.0 / (Math.Sqrt(6.0) + 2.0 + Math.Sqrt(2.0)),
1.0 / 3.0
};
//int n = 11;
for (int n = 2; n < 8; n++)
{
Console.WriteLine("n={0}", n);
Func <IReadOnlyList <double>, double> function = (IReadOnlyList <double> x) => {
// Interpret coordinates as (x_1, y_1, x_2, y_2, \cdots, x_n, y_n)
// Iterate over all pairs of points, finding smallest distance between any pair of points.
double sMin = Double.MaxValue;
for (int i = 0; i < n; i++)
{
for (int j = 0; j < i; j++)
{
double s = MoreMath.Hypot(x[2 * i] - x[2 * j], x[2 * i + 1] - x[2 * j + 1]);
if (s < sMin)
{
sMin = s;
}
}
}
return(sMin);
};
Interval[] box = new Interval[2 * n];
for (int i = 0; i < box.Length; i++)
{
box[i] = Interval.FromEndpoints(0.0, 1.0);
}
MultiExtremumSettings settings = new MultiExtremumSettings()
{
RelativePrecision = 1.0E-4, AbsolutePrecision = 1.0E-6, EvaluationBudget = 10000000
};
MultiExtremum maximum = MultiFunctionMath.FindGlobalMaximum(function, box, settings);
Console.WriteLine(maximum.EvaluationCount);
Console.WriteLine("{0} {1}", solutions[n], maximum.Value);
Assert.IsTrue(maximum.Dimension == 2 * n);
Assert.IsTrue(TestUtilities.IsNearlyEqual(maximum.Value, solutions[n], new EvaluationSettings()
{
AbsolutePrecision = 2.0 * maximum.Precision
}));
}
}
public static void Optimization()
{
Interval range = Interval.FromEndpoints(0.0, 6.28);
Extremum min = FunctionMath.FindMinimum(x => Math.Sin(x), range);
Console.WriteLine($"Found minimum of {min.Value} at {min.Location}.");
Console.WriteLine($"Required {min.EvaluationCount} evaluations.");
MultiExtremum rosenbrock = MultiFunctionMath.FindLocalMinimum(
x => MoreMath.Sqr(2.0 - x[0]) + 100.0 * MoreMath.Sqr(x[1] - x[0] * x[0]),
new ColumnVector(0.0, 0.0)
);
ColumnVector xm = rosenbrock.Location;
Console.WriteLine($"Found minimum of {rosenbrock.Value} at ({xm[0]},{xm[1]}).");
Console.WriteLine($"Required {rosenbrock.EvaluationCount} evaluations.");
Func <IReadOnlyList <double>, double> leviFunction = z => {
double x = z[0];
double y = z[1];
return(
MoreMath.Sqr(MoreMath.Sin(3.0 * Math.PI * x)) +
MoreMath.Sqr(x - 1.0) * (1.0 + MoreMath.Sqr(MoreMath.Sin(3.0 * Math.PI * y))) +
MoreMath.Sqr(y - 1.0) * (1.0 + MoreMath.Sqr(MoreMath.Sin(2.0 * Math.PI * y)))
);
};
Interval[] leviRegion = new Interval[] {
Interval.FromEndpoints(-10.0, +10.0),
Interval.FromEndpoints(-10.0, +10.0)
};
MultiExtremum levi = MultiFunctionMath.FindGlobalMinimum(leviFunction, leviRegion);
ColumnVector zm = levi.Location;
Console.WriteLine($"Found minimum of {levi.Value} at ({zm[0]},{zm[1]}).");
Console.WriteLine($"Required {levi.EvaluationCount} evaluations.");
// Select a dimension
int n = 6;
// Define a function that takes 2n polar coordinates in the form
// phi_1, theta_1, phi_2, theta_2, ..., phi_n, theta_n, computes
// the sum of the potential energy 1/d for all pairs.
Func <IReadOnlyList <double>, double> thompson = (IReadOnlyList <double> v) => {
double e = 0.0;
// iterate over all distinct pairs of points
for (int i = 0; i < n; i++)
{
for (int j = 0; j < i; j++)
{
// compute the chord length between points i and j
double dx = Math.Cos(v[2 * j]) * Math.Cos(v[2 * j + 1]) - Math.Cos(v[2 * i]) * Math.Cos(v[2 * i + 1]);
double dy = Math.Cos(v[2 * j]) * Math.Sin(v[2 * j + 1]) - Math.Cos(v[2 * i]) * Math.Sin(v[2 * i + 1]);
double dz = Math.Sin(v[2 * j]) - Math.Sin(v[2 * i]);
double d = Math.Sqrt(dx * dx + dy * dy + dz * dz);
e += 1.0 / d;
}
}
return(e);
};
// Limit all polar coordinates to their standard ranges.
Interval[] box = new Interval[2 * n];
for (int i = 0; i < n; i++)
{
box[2 * i] = Interval.FromEndpoints(-Math.PI, Math.PI);
box[2 * i + 1] = Interval.FromEndpoints(-Math.PI / 2.0, Math.PI / 2.0);
}
// Use settings to monitor proress toward a rough solution.
MultiExtremumSettings roughSettings = new MultiExtremumSettings()
{
RelativePrecision = 0.05, AbsolutePrecision = 0.0,
Listener = r => {
Console.WriteLine($"Minimum {r.Value} after {r.EvaluationCount} evaluations.");
}
};
MultiExtremum roughThompson = MultiFunctionMath.FindGlobalMinimum(thompson, box, roughSettings);
// Use settings to monitor proress toward a refined solution.
MultiExtremumSettings refinedSettings = new MultiExtremumSettings()
{
RelativePrecision = 1.0E-5, AbsolutePrecision = 0.0,
Listener = r => {
Console.WriteLine($"Minimum {r.Value} after {r.EvaluationCount} evaluations.");
}
};
MultiExtremum refinedThompson = MultiFunctionMath.FindLocalMinimum(thompson, roughThompson.Location, refinedSettings);
Console.WriteLine($"Minimum potential energy {refinedThompson.Value}.");
Console.WriteLine($"Required {roughThompson.EvaluationCount} + {refinedThompson.EvaluationCount} evaluations.");
/*
* // Define a function that takes 2n coordinates x1, y1, x2, y2, ... xn, yn
* // and finds the smallest distance between two coordinate pairs.
* Func<IReadOnlyList<double>, double> function = (IReadOnlyList<double> x) => {
* double sMin = Double.MaxValue;
* for (int i = 0; i < n; i++) {
* for (int j = 0; j < i; j++) {
* double s = MoreMath.Hypot(x[2 * i] - x[2 * j], x[2 * i + 1] - x[2 * j + 1]);
* if (s < sMin) sMin = s;
* }
* }
* return (sMin);
* };
*
* // Limit all coordinates to the unit box.
* Interval[] box = new Interval[2 * n];
* for (int i = 0; i < box.Length; i++) box[i] = Interval.FromEndpoints(0.0, 1.0);
*
* // Use settings to monitor proress toward a rough solution.
* MultiExtremumSettings roughSettings = new MultiExtremumSettings() {
* RelativePrecision = 1.0E-2, AbsolutePrecision = 0.0,
* Listener = r => {
* Console.WriteLine($"Minimum {r.Value} after {r.EvaluationCount} evaluations.");
* }
* };
* MultiExtremum roughMaximum = MultiFunctionMath.FindGlobalMaximum(function, box, roughSettings);
*
* // Use settings to monitor proress toward a rough solution.
* MultiExtremumSettings refinedSettings = new MultiExtremumSettings() {
* RelativePrecision = 1.0E-8, AbsolutePrecision = 0.0,
* Listener = r => {
* Console.WriteLine($"Minimum {r.Value} after {r.EvaluationCount} evaluations.");
* }
* };
* MultiExtremum refinedMaximum = MultiFunctionMath.FindLocalMaximum(function, roughMaximum.Location, refinedSettings);
*/
}