optimization/downhill_simplex.h

#ifndef TOON_DOWNHILL_SIMPLEX_H
#define TOON_DOWNHILL_SIMPLEX_H
#include <TooN/TooN.h>
#include <TooN/helpers.h>
#include <algorithm>
#include <cstdlib>

namespace TooN
{

/** This is an implementation of the Downhill Simplex (Nelder & Mead, 1965)
    algorithm. This particular instance will minimize a given function.
    
    The function maintains \f$N+1\f$ points for a $N$ dimensional function, \f$f\f$
    
    At each iteration, the following algorithm is performed:
    - Find the worst (largest) point, \f$x_w\f$.
    - Find the centroid of the remaining points, \f$x_0\f$.
    - Let \f$v = x_0 - x_w\f$
    - Compute a reflected point, \f$ x_r = x_0 + \alpha v\f$
    - If \f$f(x_r)\f$ is better than the best point
      - Expand the simplex by extending the reflection to \f$x_e = x_0 + \rho \alpha v \f$
      - Replace \f$x_w\f$ with the best point of \f$x_e\f$,  and \f$x_r\f$.
    - Else, if  \f$f(x_r)\f$ is between the best and second-worst point
      - Replace \f$x_w\f$ with \f$x_r\f$.
    - Else, if  \f$f(x_r)\f$ is better than \f$x_w\f$
      - Contract the simplex by computing \f$x_c = x_0 + \gamma v\f$
      - If \f$f(x_c) < f(x_r)\f$
        - Replace \f$x_w\f$ with \f$x_c\f$.
    - If \f$x_w\f$ has not been replaced, then shrink the simplex by a factor of \f$\sigma\f$ around the best point.

    This implementation uses:
    - \f$\alpha = 1\f$
    - \f$\rho = 2\f$
    - \f$\gamma = 1/2\f$
    - \f$\sigma = 1/2\f$
    
    Example usage:
    @code
#include <TooN/optimization/downhill_simplex.h>
using namespace std;
using namespace TooN;

double sq(double x)
{
    return x*x;
}

double Rosenbrock(const Vector<2>& v)
{
        return sq(1 - v[0]) + 100 * sq(v[1] - sq(v[0]));
}

int main()
{
        Vector<2> starting_point = makeVector( -1, 1);

        DownhillSimplex<2> dh_fixed(Rosenbrock, starting_point, 1);

        while(dh_fixed.iterate(Rosenbrock))
        {
            cout << dh.get_values()[dh.get_best()] << endl;
        }
        
        cout << dh_fixed.get_simplex()[dh_fixed.get_best()] << endl;
}

    @endcode


    @ingroup gOptimize
    @param   N The dimension of the function to optimize. As usual, the default value of <i>N</i> (-1) indicates
             that the class is sized at run-time.


**/
template<int N=-1, typename Precision=double> class DownhillSimplex
{
    static const int Vertices = (N==-1?-1:N+1);
    typedef Matrix<Vertices, N, Precision> Simplex;
    typedef Vector<Vertices, Precision> Values;

    public:
        /// Initialize the DownhillSimplex class. The simplex is automatically
        /// generated. One point is at <i>c</i>, the remaining points are made by moving
        /// <i>c</i> by <i>spread</i> along each axis aligned unit vector.
        ///
        ///@param func       Functor to minimize.
        ///@param c          Origin of the initial simplex. The dimension of this vector
        ///                  is used to determine the dimension of the run-time sized version.
        ///@param spread     Size of the initial simplex.
        template<class Function> DownhillSimplex(const Function& func, const Vector<N>& c, Precision spread=1)
        :simplex(c.size()+1, c.size()),values(c.size()+1)
        {
            alpha = 1.0;
            rho = 2.0;
            gamma = 0.5;
            sigma = 0.5;

            epsilon = sqrt(numeric_limits<Precision>::epsilon());
            zero_epsilon = 1e-20;

            restart(func, c, spread);
        }
        
        /// This function sets up the simplex around, with one point at \e c and the remaining
        /// points are made by moving by \e spread along each axis aligned unit vector.
        ///
        ///@param func       Functor to minimize.
        ///@param c          \e c corner point of the simplex
        ///@param spread     \e spread simplex size
        template<class Function> void restart(const Function& func, const Vector<N>& c, Precision spread)
        {
            for(int i=0; i < simplex.num_rows(); i++)
                simplex[i] = c;

            for(int i=0; i < simplex.num_cols(); i++)
                simplex[i][i] += spread;

            for(int i=0; i < values.size(); i++)
                values[i] = func(simplex[i]);
        }
        
        ///Check to see it iteration should stop. You probably do not want to use
        ///this function. See iterate() instead. This function updates nothing.
        ///The termination criterion is that the simplex span (distancve between
        ///the best and worst vertices) is small compared to the scale or 
        ///small overall.
        bool finished()
        {
            Precision span =  norm(simplex[get_best()] - simplex[get_worst()]);
            Precision scale = norm(simplex[get_best()]);

            if(span/scale < epsilon || span < zero_epsilon)
                return 1;
            else 
                return 0;
        }
        
        /// This function resets the simplex around the best current point.
        ///
        ///@param func       Functor to minimize.
        ///@param spread     simplex size
        template<class Function> void restart(const Function& func, Precision spread)
        {
            restart(func, simplex[get_best()], spread);
        }

        ///Return the simplex
        const Simplex& get_simplex() const
        {
            return simplex;
        }
        
        ///Return the score at the vertices
        const Values& get_values() const
        {
            return values;
        }
        
        ///Get the index of the best vertex
        int get_best() const 
        {
            return std::min_element(&values[0], &values[0] + values.size()) - &values[0];
        }
        
        ///Get the index of the worst vertex
        int get_worst() const 
        {
            return std::max_element(&values[0], &values[0] + values.size()) - &values[0];
        }

        ///Perform one iteration of the downhill Simplex algorithm
        ///@param func Functor to minimize
        template<class Function> void find_next_point(const Function& func)
        {
            //Find various things:
            // - The worst point
            // - The second worst point
            // - The best point
            // - The centroid of all the points but the worst
            int worst = get_worst();
            Precision second_worst_val=-HUGE_VAL, bestval = HUGE_VAL, worst_val = values[worst];
            int best=0;
            Vector<N> x0 = Zeros(simplex.num_cols());


            for(int i=0; i < simplex.num_rows(); i++)
            {
                if(values[i] < bestval)
                {
                    bestval = values[i];
                    best = i;
                }

                if(i != worst)
                {
                    if(values[i] > second_worst_val)
                        second_worst_val = values[i];

                    //Compute the centroid of the non-worst points;
                    x0 += simplex[i];
                }
            }
            x0 *= 1.0 / simplex.num_cols();


            //Reflect the worst point about the centroid.
            Vector<N> xr = (1 + alpha) * x0 - alpha * simplex[worst];
            Precision fr = func(xr);

            if(fr < bestval)
            {
                //If the new point is better than the smallest, then try expanding the simplex.
                Vector<N> xe = rho * xr + (1-rho) * x0;
                Precision fe = func(xe);

                //Keep whichever is best
                if(fe < fr)
                {
                    simplex[worst] = xe;
                    values[worst] = fe;
                }
                else
                {
                    simplex[worst] = xr;
                    values[worst] = fr;
                }

                return;
            }

            //Otherwise, if the new point lies between the other points
            //then keep it and move on to the next iteration.
            if(fr < second_worst_val)
            {
                simplex[worst] = xr;
                values[worst] = fr;
                return;
            }


            //Otherwise, if the new point is a bit better than the worst point,
            //(ie, it's got just a little bit better) then contract the simplex
            //a bit.
            if(fr < worst_val)
            {
                Vector<N> xc = (1 + gamma) * x0 - gamma * simplex[worst];
                Precision fc = func(xc);

                //If this helped, use it
                if(fc <= fr)
                {
                    simplex[worst] = xc;
                    values[worst] = fc;
                    return;
                }
            }
            
            //Otherwise, fr is worse than the worst point, or the fc was worse
            //than fr. So shrink the whole simplex around the best point.
            for(int i=0; i < simplex.num_rows(); i++)
                if(i != best)
                {
                    simplex[i] = simplex[best] + sigma * (simplex[i] - simplex[best]);
                    values[i] = func(simplex[i]);
                }
        }

        ///Perform one iteration of the downhill Simplex algorithm, and return the result
        ///of not DownhillSimplex::finished.
        ///@param func Functor to minimize
        template<class Function> bool iterate(const Function& func)
        {
            find_next_point(func);
            return !finished();
        }

        Precision alpha; ///< Reflected size. Defaults to 1.
        Precision rho;   ///< Expansion ratio. Defaults to 2.
        Precision gamma; ///< Contraction ratio. Defaults to .5.
        Precision sigma; ///< Shrink ratio. Defaults to .5.
        Precision epsilon;  ///< Tolerance used to determine if the optimization is complete. Defaults to square root of machine precision.
        Precision zero_epsilon; ///< Additive term in tolerance to prevent excessive iterations if \f$x_\mathrm{optimal} = 0\f$. Known as \c ZEPS in numerical recipies. Defaults to 1e-20

    private:

        //Each row is a simplex vertex
        Simplex simplex;

        //Function values for each vertex
        Values values;


};
}
#endif

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