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### 1.18 A differential evolution (stochastic) optimizer

#### Helptext:

    de_min: global optimisation using differential evolution

    Usage: [x, obj_value, nfeval, convergence] = de_min(fcn, control)

    minimization of a user-supplied function with respect to x(1:D),
    using the differential evolution (DE) method based on an algorithm
    by  Rainer Storn (http://www.icsi.berkeley.edu/~storn/code.html)
    See: http://www.softcomputing.net/tevc2009_1.pdf

    Arguments:
    ---------------
    fcn        string : Name of function. Must return a real value
    control    vector : (Optional) Control variables, described below
            or struct

    Returned values:
    ----------------
    x          vector : parameter vector of best solution
    obj_value  scalar : objective function value of best solution
    nfeval     scalar : number of function evaluations
    convergence       : 1 = best below value to reach (VTR)
                        0 = population has reached defined quality (tol)
                       -1 = some values are close to constraints/boundaries
                       -2 = max number of iterations reached (maxiter)
                       -3 = max number of functions evaluations reached (maxnfe)

    Control variable:   (optional) may be named arguments (i.e. "name",value
    ----------------    pairs), a struct, or a vector, where
                        NaN's are ignored.

    XVmin        : vector of lower bounds of initial population
                   *** note: by default these are no constraints ***
    XVmax        : vector of upper bounds of initial population
    constr       : 1 -> enforce the bounds not just for the initial population
    const        : data vector (remains fixed during the minimization)
    NP           : number of population members
    F            : difference factor from interval [0, 2]
    CR           : crossover probability constant from interval [0, 1]
    strategy     : 1 --> DE/best/1/exp           7 --> DE/best/1/bin
                   2 --> DE/rand/1/exp           8 --> DE/rand/1/bin
                   3 --> DE/target-to-best/1/exp 9 --> DE/target-to-best/1/bin
                   4 --> DE/best/2/exp           10--> DE/best/2/bin
                   5 --> DE/rand/2/exp           11--> DE/rand/2/bin
                   6 --> DEGL/SAW/exp            else  DEGL/SAW/bin
    refresh      : intermediate output will be produced after "refresh"
                   iterations. No intermediate output will be produced
                   if refresh is < 1
    VTR          : Stopping criterion: "Value To Reach"
                   de_min will stop when obj_value <= VTR.
                   Use this if you know which value you expect.
    tol          : Stopping criterion: "tolerance"
                   stops if (best-worst)/max(1,worst) < tol
                   This stops basically if the whole population is "good".
    maxnfe       : maximum number of function evaluations
    maxiter      : maximum number of iterations (generations)

          The algorithm seems to work well only if [XVmin,XVmax] covers the
          region where the global minimum is expected.
          DE is also somewhat sensitive to the choice of the
          difference factor F. A good initial guess is to choose F from
          interval [0.5, 1], e.g. 0.8.
          CR, the crossover probability constant from interval [0, 1]
          helps to maintain the diversity of the population and is
          rather uncritical but affects strongly the convergence speed.
          If the parameters are correlated, high values of CR work better.
          The reverse is true for no correlation.
          Experiments suggest that /bin likes to have a slightly
          larger CR than /exp.
          The number of population members NP is also not very critical. A
          good initial guess is 10*D. Depending on the difficulty of the
          problem NP can be lower than 10*D or must be higher than 10*D
          to achieve convergence.

    Default Values:
    ---------------
    XVmin = [-2];
    XVmax = [ 2];
    constr= 0;
    const = [];
    NP    = 10 *D
    F     = 0.8;
    CR    = 0.9;
    strategy = 12;
    refresh  = 0;
    VTR   = -Inf;
    tol   = 1.e-3;
    maxnfe  = 1e6;
    maxiter = 1000;

    Example to find the minimum of the Rosenbrock saddle:
    ----------------------------------------------------
    Define f as:
                       function result = f(x);
                         result = 100 * (x(2) - x(1)^2)^2 + (1 - x(1))^2;
                       end
    Then type:

    	ctl.XVmin = [-2 -2];
    	ctl.XVmax = [ 2  2];
    	[x, obj_value, nfeval, convergence] = de_min (@f, ctl);

    Keywords: global-optimisation optimisation minimisation

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