/// <summary>
/// Computes the maximum descent possible from the vector x in the direction dir.
/// </summary>
/// <param name="func">Function to find it the greatest descent possible in the given direction.</param>
/// <param name="x">Current vector of the minimization Quasi-Newton algorithm.</param>
/// <param name="dir">Descent direction vector for the current vector.</param>
/// <returns>The value of the maximum descent possible.</returns>
public static double Wolfe(CompiledFunc func, Vector x, Vector dir)
{
double a = 0;
double ai = 1;
double fPrev = 0, fCurr = 0, diff = 0;
double fZero = func.Eval(x);
var normDir = dir.Normalize();
double diffZero = (func.Differentiate(x)*normDir).Sum();
while(ai < MaxAlpha)
{
fPrev = func.Eval(x + a*dir);
fCurr = func.Eval(x + ai*dir);
if (fCurr > fZero + C1*ai*diffZero || (fCurr > fPrev && ai > 1))
return Zoom(func, x, dir, a, ai, fZero, diffZero);
diff = (func.Differentiate(x + ai*dir)*normDir).Sum();
if (Math.Abs(diff) <= -C2*diffZero)
return ai;
if (diff >= 0)
return Zoom(func, x, dir, ai, a, fZero, diffZero);
a = ai;
ai *= 1.5;
}
return ai;
}
private static double Zoom(CompiledFunc func, Vector x, Vector dir, double aLow, double aHigh, double fZero, double diffZero)
{
var normDir = dir.Normalize();
double aMid = 0;
double fValue = 0;
double diff = 0;
while (Math.Abs(aLow - aHigh) > EPS)
{
aMid = aLow + (aHigh - aLow)/2;
fValue = func.Eval(x + aMid*dir);
if (fValue > fZero + C1*aMid*diffZero || fValue >= func.Eval(x + aLow*dir))
aHigh = aMid;
else
{
diff = (func.Differentiate(x + aMid*dir)*normDir).Sum();
if (Math.Abs(diff) <= -C2*diffZero)
return aMid;
if (diff*(aHigh - aLow) >= 0)
aHigh = aLow;
aLow = aMid;
}
}
return aMid;
}
protected override Vector Minimize(CompiledFunc f, Vector x = null, Tuple<Vector, Vector> bounds = null)
{
SeedPopulation(f, bounds);
if (x != null)
{
double fit = f.Eval(x);
if (fit < GlobalBestFit)
{
GlobalBestFit = fit;
GlobalBestPosition = x;
ParticlesSet[0].BestFit = GlobalBestFit;
ParticlesSet[0].BestPosition = GlobalBestPosition;
}
}
while (CurrentIteration < IterationsNumber)
{
CurrentIteration++;
foreach (var particle in ParticlesSet)
{
particle.Update();
if (particle.BestFit < GlobalBestFit)
{
GlobalBestFit = particle.BestFit;
GlobalBestPosition = particle.BestPosition;
}
}
}
return GlobalBestPosition;
}
protected override Vector Minimize(CompiledFunc f, Vector x = null, Tuple<Vector, Vector> bounds = null)
{
CurrentIteration = 0;
var b = Matrix.Identity(x.Length);
var x1 = new Vector(x);
Vector d;
double a;
while (CurrentIteration < IterationsNumber)
{
CurrentIteration++;
d = -b.Dot(f.Differentiate(x));
a = Searcher(f, x, d);
x1 = x + a*d;
b = Corrector(f, b, x, x1);
if (!Algebra.IsValid(x1) || Algebra.Norm(x1 - x) <= EPS)
break;
x = x1;
}
return x;
}
/// <summary>
/// Computes the BFGS correction formula for inverse hessiana approximation.
/// </summary>
/// <param name="func">Function to find it the hessiana approximation.</param>
/// <param name="b">Current inverse approximation of the hessiana.</param>
/// <param name="x">Current vector of the minimization Quasi-Newton algorithm.</param>
/// <param name="x1">Next vector of the minimization Quasi-Newton algorithm.</param>
/// <returns>Returns a matrix representing the next step in inverse hessiana approximation.s</returns>
public static Matrix Bfgs(CompiledFunc func, Matrix b, Vector x, Vector x1)
{
var sk = new Matrix(x1 - x);
var yk = new Matrix(func.Differentiate(x1) - func.Differentiate(x));
var t = b.Dot(sk.Transpose()).Dot(sk).Dot(b)/sk.Dot(b).Dot(sk.Transpose())[0,0];
var t1 = yk.Transpose().Dot(yk)/yk.Dot(sk.Transpose())[0,0];
return b - t + t1;
}
public Particle(CompiledFunc func, int dimensions, int neighborsNumber, Tuple<Vector, Vector> bounds, Func<double> random)
{
Func = func;
NumberOfDimensions = dimensions;
Bounds = bounds;
Neighbors = new Particle[neighborsNumber];
CurrentPosition = BestPosition = new Vector(Enumerable.Range(0, dimensions).Select(i => random()));
Velocity = new Vector(Enumerable.Range(0, dimensions).Select(i => random()));
CurrentFit = BestFit = func.Eval(CurrentPosition);
}
/// <summary>
/// Creates the particle set with non-uniform values and selects an initial best position and best fit
/// </summary>
/// <param name="f">Function to which the minimum will be looking for.</param>
/// <param name="bounds">Limits of function space search.</param>
private void SeedPopulation(CompiledFunc f, Tuple<Vector, Vector> bounds)
{
CurrentIteration = 0;
double min = bounds.Item1.Min();
double max = bounds.Item2.Max();
var dist = Distributions.ExponentialFunc(1.0 / (min * 2 + max * 2 + 1.5));
ParticlesSet =
Enumerable.Range(0, NumberOfParticles)
.Select(i => new Particle(f, f.Dimension, NumberOfNeighborsByParticle, bounds, dist)).ToArray();
int index = -1;
GlobalBestPosition = new Vector(f.Dimension);
for (int i = 0; i < GlobalBestPosition.Length; i++)
GlobalBestPosition[i] = bounds.Item1[i] + (bounds.Item2[i] - bounds.Item1[i])/2;
GlobalBestFit = f.Eval(GlobalBestPosition);
for (int i = 0; i < NumberOfParticles; i++)
{
if (ParticlesSet[i].IsFeasible() && ParticlesSet[i].BestFit < GlobalBestFit)
{
index = i;
GlobalBestFit = ParticlesSet[i].BestFit;
}
// Setting particle neighbors
for (int j = 0; j < NumberOfNeighborsByParticle; j++)
ParticlesSet[i].Neighbors[j] = ParticlesSet[(i + 1 + j) % NumberOfParticles];
}
if (index == -1)
{
ParticlesSet[0].BestPosition = GlobalBestPosition;
ParticlesSet[0].BestFit = GlobalBestFit;
return;
}
GlobalBestPosition = ParticlesSet[index].BestPosition;
GlobalBestFit = ParticlesSet[index].BestFit;
}
/// <summary> /// Algorithmic implementation of optimization model to look for the minimun function value. /// </summary> /// <param name="f">Function to which the minimum will be looking for.</param> /// <param name="x">Optional vector used as started point for the algorithm to look for.</param> /// <param name="bounds">Optional limits of function space search.</param> /// <returns>Returns the minimun vector found by optimizer.</returns> protected abstract Vector Minimize(CompiledFunc f, Vector x = null, Tuple<Vector, Vector> bounds = null);
/// <summary>
/// Looks for the vector x that minimizes function f, i.e, vector x such that f(x) <= f(y) for all
/// y values in the function search space.
/// </summary>
/// <param name="f">Function to which the minimum will be looking for.</param>
/// <param name="input">Optional vector used as started point for the algorithm to look for.</param>
/// <param name="bounds">Optional limits of function space search.</param>
/// <returns>Returns a tuple with the minimun vector found by optimizer and the value of the function in this vector.</returns>
public Tuple<Vector, double> FindMinimun(CompiledFunc f, Vector input = null, Tuple<Vector, Vector> bounds = null)
{
var res = Minimize(f, input, bounds);
return new Tuple<Vector, double>(res, f.Eval(res));
}
