Module « scipy.optimize »

Fonction shgo - module scipy.optimize

Signature de la fonction shgo

def shgo(func, bounds, args=(), constraints=None, n=None, iters=1, callback=None, minimizer_kwargs=None, options=None, sampling_method='simplicial') 

Description

shgo.__doc__

    Finds the global minimum of a function using SHG optimization.

    SHGO stands for "simplicial homology global optimization".

    Parameters
    ----------
    func : callable
        The objective function to be minimized.  Must be in the form
        ``f(x, *args)``, where ``x`` is the argument in the form of a 1-D array
        and ``args`` is a tuple of any additional fixed parameters needed to
        completely specify the function.
    bounds : sequence
        Bounds for variables.  ``(min, max)`` pairs for each element in ``x``,
        defining the lower and upper bounds for the optimizing argument of
        `func`. It is required to have ``len(bounds) == len(x)``.
        ``len(bounds)`` is used to determine the number of parameters in ``x``.
        Use ``None`` for one of min or max when there is no bound in that
        direction. By default bounds are ``(None, None)``.
    args : tuple, optional
        Any additional fixed parameters needed to completely specify the
        objective function.
    constraints : dict or sequence of dict, optional
        Constraints definition.
        Function(s) ``R**n`` in the form::

            g(x) >= 0 applied as g : R^n -> R^m
            h(x) == 0 applied as h : R^n -> R^p

        Each constraint is defined in a dictionary with fields:

            type : str
                Constraint type: 'eq' for equality, 'ineq' for inequality.
            fun : callable
                The function defining the constraint.
            jac : callable, optional
                The Jacobian of `fun` (only for SLSQP).
            args : sequence, optional
                Extra arguments to be passed to the function and Jacobian.

        Equality constraint means that the constraint function result is to
        be zero whereas inequality means that it is to be non-negative.
        Note that COBYLA only supports inequality constraints.

        .. note::

           Only the COBYLA and SLSQP local minimize methods currently
           support constraint arguments. If the ``constraints`` sequence
           used in the local optimization problem is not defined in
           ``minimizer_kwargs`` and a constrained method is used then the
           global ``constraints`` will be used.
           (Defining a ``constraints`` sequence in ``minimizer_kwargs``
           means that ``constraints`` will not be added so if equality
           constraints and so forth need to be added then the inequality
           functions in ``constraints`` need to be added to
           ``minimizer_kwargs`` too).

    n : int, optional
        Number of sampling points used in the construction of the simplicial
        complex. Note that this argument is only used for ``sobol`` and other
        arbitrary `sampling_methods`. In case of ``sobol``, it must be a
        power of 2: ``n=2**m``, and the argument will automatically be
        converted to the next higher power of 2. Default is 100 for
        ``sampling_method='simplicial'`` and 128 for
        ``sampling_method='sobol'``.
    iters : int, optional
        Number of iterations used in the construction of the simplicial
        complex. Default is 1.
    callback : callable, optional
        Called after each iteration, as ``callback(xk)``, where ``xk`` is the
        current parameter vector.
    minimizer_kwargs : dict, optional
        Extra keyword arguments to be passed to the minimizer
        ``scipy.optimize.minimize`` Some important options could be:

            * method : str
                The minimization method, the default is ``SLSQP``.
            * args : tuple
                Extra arguments passed to the objective function (``func``) and
                its derivatives (Jacobian, Hessian).
            * options : dict, optional
                Note that by default the tolerance is specified as
                ``{ftol: 1e-12}``

    options : dict, optional
        A dictionary of solver options. Many of the options specified for the
        global routine are also passed to the scipy.optimize.minimize routine.
        The options that are also passed to the local routine are marked with
        "(L)".

        Stopping criteria, the algorithm will terminate if any of the specified
        criteria are met. However, the default algorithm does not require any to
        be specified:

        * maxfev : int (L)
            Maximum number of function evaluations in the feasible domain.
            (Note only methods that support this option will terminate
            the routine at precisely exact specified value. Otherwise the
            criterion will only terminate during a global iteration)
        * f_min
            Specify the minimum objective function value, if it is known.
        * f_tol : float
            Precision goal for the value of f in the stopping
            criterion. Note that the global routine will also
            terminate if a sampling point in the global routine is
            within this tolerance.
        * maxiter : int
            Maximum number of iterations to perform.
        * maxev : int
            Maximum number of sampling evaluations to perform (includes
            searching in infeasible points).
        * maxtime : float
            Maximum processing runtime allowed
        * minhgrd : int
            Minimum homology group rank differential. The homology group of the
            objective function is calculated (approximately) during every
            iteration. The rank of this group has a one-to-one correspondence
            with the number of locally convex subdomains in the objective
            function (after adequate sampling points each of these subdomains
            contain a unique global minimum). If the difference in the hgr is 0
            between iterations for ``maxhgrd`` specified iterations the
            algorithm will terminate.

        Objective function knowledge:

        * symmetry : bool
            Specify True if the objective function contains symmetric variables.
            The search space (and therefore performance) is decreased by O(n!).

        * jac : bool or callable, optional
            Jacobian (gradient) of objective function. Only for CG, BFGS,
            Newton-CG, L-BFGS-B, TNC, SLSQP, dogleg, trust-ncg. If ``jac`` is a
            boolean and is True, ``fun`` is assumed to return the gradient along
            with the objective function. If False, the gradient will be
            estimated numerically. ``jac`` can also be a callable returning the
            gradient of the objective. In this case, it must accept the same
            arguments as ``fun``. (Passed to `scipy.optimize.minmize` automatically)

        * hess, hessp : callable, optional
            Hessian (matrix of second-order derivatives) of objective function
            or Hessian of objective function times an arbitrary vector p.
            Only for Newton-CG, dogleg, trust-ncg. Only one of ``hessp`` or
            ``hess`` needs to be given. If ``hess`` is provided, then
            ``hessp`` will be ignored. If neither ``hess`` nor ``hessp`` is
            provided, then the Hessian product will be approximated using
            finite differences on ``jac``. ``hessp`` must compute the Hessian
            times an arbitrary vector. (Passed to `scipy.optimize.minmize`
            automatically)

        Algorithm settings:

        * minimize_every_iter : bool
            If True then promising global sampling points will be passed to a
            local minimization routine every iteration. If False then only the
            final minimizer pool will be run. Defaults to False.
        * local_iter : int
            Only evaluate a few of the best minimizer pool candidates every
            iteration. If False all potential points are passed to the local
            minimization routine.
        * infty_constraints: bool
            If True then any sampling points generated which are outside will
            the feasible domain will be saved and given an objective function
            value of ``inf``. If False then these points will be discarded.
            Using this functionality could lead to higher performance with
            respect to function evaluations before the global minimum is found,
            specifying False will use less memory at the cost of a slight
            decrease in performance. Defaults to True.

        Feedback:

        * disp : bool (L)
            Set to True to print convergence messages.

    sampling_method : str or function, optional
        Current built in sampling method options are ``halton``, ``sobol`` and
        ``simplicial``. The default ``simplicial`` provides
        the theoretical guarantee of convergence to the global minimum in finite
        time. ``halton`` and ``sobol`` method are faster in terms of sampling
        point generation at the cost of the loss of
        guaranteed convergence. It is more appropriate for most "easier"
        problems where the convergence is relatively fast.
        User defined sampling functions must accept two arguments of ``n``
        sampling points of dimension ``dim`` per call and output an array of
        sampling points with shape `n x dim`.

    Returns
    -------
    res : OptimizeResult
        The optimization result represented as a `OptimizeResult` object.
        Important attributes are:
        ``x`` the solution array corresponding to the global minimum,
        ``fun`` the function output at the global solution,
        ``xl`` an ordered list of local minima solutions,
        ``funl`` the function output at the corresponding local solutions,
        ``success`` a Boolean flag indicating if the optimizer exited
        successfully,
        ``message`` which describes the cause of the termination,
        ``nfev`` the total number of objective function evaluations including
        the sampling calls,
        ``nlfev`` the total number of objective function evaluations
        culminating from all local search optimizations,
        ``nit`` number of iterations performed by the global routine.

    Notes
    -----
    Global optimization using simplicial homology global optimization [1]_.
    Appropriate for solving general purpose NLP and blackbox optimization
    problems to global optimality (low-dimensional problems).

    In general, the optimization problems are of the form::

        minimize f(x) subject to

        g_i(x) >= 0,  i = 1,...,m
        h_j(x)  = 0,  j = 1,...,p

    where x is a vector of one or more variables. ``f(x)`` is the objective
    function ``R^n -> R``, ``g_i(x)`` are the inequality constraints, and
    ``h_j(x)`` are the equality constraints.

    Optionally, the lower and upper bounds for each element in x can also be
    specified using the `bounds` argument.

    While most of the theoretical advantages of SHGO are only proven for when
    ``f(x)`` is a Lipschitz smooth function, the algorithm is also proven to
    converge to the global optimum for the more general case where ``f(x)`` is
    non-continuous, non-convex and non-smooth, if the default sampling method
    is used [1]_.

    The local search method may be specified using the ``minimizer_kwargs``
    parameter which is passed on to ``scipy.optimize.minimize``. By default,
    the ``SLSQP`` method is used. In general, it is recommended to use the
    ``SLSQP`` or ``COBYLA`` local minimization if inequality constraints
    are defined for the problem since the other methods do not use constraints.

    The ``halton`` and ``sobol`` method points are generated using
    `scipy.stats.qmc`. Any other QMC method could be used.

    References
    ----------
    .. [1] Endres, SC, Sandrock, C, Focke, WW (2018) "A simplicial homology
           algorithm for lipschitz optimisation", Journal of Global Optimization.
    .. [2] Joe, SW and Kuo, FY (2008) "Constructing Sobol' sequences with
           better  two-dimensional projections", SIAM J. Sci. Comput. 30,
           2635-2654.
    .. [3] Hoch, W and Schittkowski, K (1981) "Test examples for nonlinear
           programming codes", Lecture Notes in Economics and Mathematical
           Systems, 187. Springer-Verlag, New York.
           http://www.ai7.uni-bayreuth.de/test_problem_coll.pdf
    .. [4] Wales, DJ (2015) "Perspective: Insight into reaction coordinates and
           dynamics from the potential energy landscape",
           Journal of Chemical Physics, 142(13), 2015.

    Examples
    --------
    First consider the problem of minimizing the Rosenbrock function, `rosen`:

    >>> from scipy.optimize import rosen, shgo
    >>> bounds = [(0,2), (0, 2), (0, 2), (0, 2), (0, 2)]
    >>> result = shgo(rosen, bounds)
    >>> result.x, result.fun
    (array([1., 1., 1., 1., 1.]), 2.920392374190081e-18)

    Note that bounds determine the dimensionality of the objective
    function and is therefore a required input, however you can specify
    empty bounds using ``None`` or objects like ``np.inf`` which will be
    converted to large float numbers.

    >>> bounds = [(None, None), ]*4
    >>> result = shgo(rosen, bounds)
    >>> result.x
    array([0.99999851, 0.99999704, 0.99999411, 0.9999882 ])

    Next, we consider the Eggholder function, a problem with several local
    minima and one global minimum. We will demonstrate the use of arguments and
    the capabilities of `shgo`.
    (https://en.wikipedia.org/wiki/Test_functions_for_optimization)

    >>> def eggholder(x):
    ...     return (-(x[1] + 47.0)
    ...             * np.sin(np.sqrt(abs(x[0]/2.0 + (x[1] + 47.0))))
    ...             - x[0] * np.sin(np.sqrt(abs(x[0] - (x[1] + 47.0))))
    ...             )
    ...
    >>> bounds = [(-512, 512), (-512, 512)]

    `shgo` has built-in low discrepancy sampling sequences. First, we will
    input 64 initial sampling points of the *Sobol'* sequence:

    >>> result = shgo(eggholder, bounds, n=64, sampling_method='sobol')
    >>> result.x, result.fun
    (array([512.        , 404.23180824]), -959.6406627208397)

    `shgo` also has a return for any other local minima that was found, these
    can be called using:

    >>> result.xl
    array([[ 512.        ,  404.23180824],
           [ 283.0759062 , -487.12565635],
           [-294.66820039, -462.01964031],
           [-105.87688911,  423.15323845],
           [-242.97926   ,  274.38030925],
           [-506.25823477,    6.3131022 ],
           [-408.71980731, -156.10116949],
           [ 150.23207937,  301.31376595],
           [  91.00920901, -391.283763  ],
           [ 202.89662724, -269.38043241],
           [ 361.66623976, -106.96493868],
           [-219.40612786, -244.06020508]])

    >>> result.funl
    array([-959.64066272, -718.16745962, -704.80659592, -565.99778097,
           -559.78685655, -557.36868733, -507.87385942, -493.9605115 ,
           -426.48799655, -421.15571437, -419.31194957, -410.98477763])

    These results are useful in applications where there are many global minima
    and the values of other global minima are desired or where the local minima
    can provide insight into the system (for example morphologies
    in physical chemistry [4]_).

    If we want to find a larger number of local minima, we can increase the
    number of sampling points or the number of iterations. We'll increase the
    number of sampling points to 64 and the number of iterations from the
    default of 1 to 3. Using ``simplicial`` this would have given us
    64 x 3 = 192 initial sampling points.

    >>> result_2 = shgo(eggholder, bounds, n=64, iters=3, sampling_method='sobol')
    >>> len(result.xl), len(result_2.xl)
    (12, 20)

    Note the difference between, e.g., ``n=192, iters=1`` and ``n=64,
    iters=3``.
    In the first case the promising points contained in the minimiser pool
    are processed only once. In the latter case it is processed every 64
    sampling points for a total of 3 times.

    To demonstrate solving problems with non-linear constraints consider the
    following example from Hock and Schittkowski problem 73 (cattle-feed) [3]_::

        minimize: f = 24.55 * x_1 + 26.75 * x_2 + 39 * x_3 + 40.50 * x_4

        subject to: 2.3 * x_1 + 5.6 * x_2 + 11.1 * x_3 + 1.3 * x_4 - 5     >= 0,

                    12 * x_1 + 11.9 * x_2 + 41.8 * x_3 + 52.1 * x_4 - 21
                        -1.645 * sqrt(0.28 * x_1**2 + 0.19 * x_2**2 +
                                      20.5 * x_3**2 + 0.62 * x_4**2)       >= 0,

                    x_1 + x_2 + x_3 + x_4 - 1                              == 0,

                    1 >= x_i >= 0 for all i

    The approximate answer given in [3]_ is::

        f([0.6355216, -0.12e-11, 0.3127019, 0.05177655]) = 29.894378

    >>> def f(x):  # (cattle-feed)
    ...     return 24.55*x[0] + 26.75*x[1] + 39*x[2] + 40.50*x[3]
    ...
    >>> def g1(x):
    ...     return 2.3*x[0] + 5.6*x[1] + 11.1*x[2] + 1.3*x[3] - 5  # >=0
    ...
    >>> def g2(x):
    ...     return (12*x[0] + 11.9*x[1] +41.8*x[2] + 52.1*x[3] - 21
    ...             - 1.645 * np.sqrt(0.28*x[0]**2 + 0.19*x[1]**2
    ...                             + 20.5*x[2]**2 + 0.62*x[3]**2)
    ...             ) # >=0
    ...
    >>> def h1(x):
    ...     return x[0] + x[1] + x[2] + x[3] - 1  # == 0
    ...
    >>> cons = ({'type': 'ineq', 'fun': g1},
    ...         {'type': 'ineq', 'fun': g2},
    ...         {'type': 'eq', 'fun': h1})
    >>> bounds = [(0, 1.0),]*4
    >>> res = shgo(f, bounds, iters=3, constraints=cons)
    >>> res
         fun: 29.894378159142136
        funl: array([29.89437816])
     message: 'Optimization terminated successfully.'
        nfev: 114
         nit: 3
       nlfev: 35
       nlhev: 0
       nljev: 5
     success: True
           x: array([6.35521569e-01, 1.13700270e-13, 3.12701881e-01, 5.17765506e-02])
          xl: array([[6.35521569e-01, 1.13700270e-13, 3.12701881e-01, 5.17765506e-02]])

    >>> g1(res.x), g2(res.x), h1(res.x)
    (-5.062616992290714e-14, -2.9594104944408173e-12, 0.0)