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1670 lines
62 KiB
Python
1670 lines
62 KiB
Python
"""
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shgo: The simplicial homology global optimisation algorithm
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"""
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import numpy as np
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import time
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import logging
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import warnings
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from scipy import spatial
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from scipy.optimize import OptimizeResult, minimize
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from scipy.optimize._shgo_lib import sobol_seq
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from scipy.optimize._shgo_lib.triangulation import Complex
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__all__ = ['shgo']
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def shgo(func, bounds, args=(), constraints=None, n=100, iters=1, callback=None,
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minimizer_kwargs=None, options=None, sampling_method='simplicial'):
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"""
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Finds the global minimum of a function using SHG optimization.
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SHGO stands for "simplicial homology global optimization".
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Parameters
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----------
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func : callable
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The objective function to be minimized. Must be in the form
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``f(x, *args)``, where ``x`` is the argument in the form of a 1-D array
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and ``args`` is a tuple of any additional fixed parameters needed to
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completely specify the function.
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bounds : sequence
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Bounds for variables. ``(min, max)`` pairs for each element in ``x``,
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defining the lower and upper bounds for the optimizing argument of
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`func`. It is required to have ``len(bounds) == len(x)``.
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``len(bounds)`` is used to determine the number of parameters in ``x``.
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Use ``None`` for one of min or max when there is no bound in that
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direction. By default bounds are ``(None, None)``.
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args : tuple, optional
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Any additional fixed parameters needed to completely specify the
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objective function.
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constraints : dict or sequence of dict, optional
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Constraints definition.
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Function(s) ``R**n`` in the form::
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g(x) >= 0 applied as g : R^n -> R^m
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h(x) == 0 applied as h : R^n -> R^p
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Each constraint is defined in a dictionary with fields:
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type : str
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Constraint type: 'eq' for equality, 'ineq' for inequality.
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fun : callable
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The function defining the constraint.
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jac : callable, optional
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The Jacobian of `fun` (only for SLSQP).
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args : sequence, optional
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Extra arguments to be passed to the function and Jacobian.
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Equality constraint means that the constraint function result is to
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be zero whereas inequality means that it is to be non-negative.
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Note that COBYLA only supports inequality constraints.
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.. note::
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Only the COBYLA and SLSQP local minimize methods currently
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support constraint arguments. If the ``constraints`` sequence
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used in the local optimization problem is not defined in
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``minimizer_kwargs`` and a constrained method is used then the
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global ``constraints`` will be used.
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(Defining a ``constraints`` sequence in ``minimizer_kwargs``
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means that ``constraints`` will not be added so if equality
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constraints and so forth need to be added then the inequality
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functions in ``constraints`` need to be added to
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``minimizer_kwargs`` too).
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n : int, optional
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Number of sampling points used in the construction of the simplicial
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complex. Note that this argument is only used for ``sobol`` and other
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arbitrary `sampling_methods`.
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iters : int, optional
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Number of iterations used in the construction of the simplicial complex.
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callback : callable, optional
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Called after each iteration, as ``callback(xk)``, where ``xk`` is the
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current parameter vector.
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minimizer_kwargs : dict, optional
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Extra keyword arguments to be passed to the minimizer
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``scipy.optimize.minimize`` Some important options could be:
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* method : str
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The minimization method (e.g. ``SLSQP``).
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* args : tuple
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Extra arguments passed to the objective function (``func``) and
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its derivatives (Jacobian, Hessian).
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* options : dict, optional
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Note that by default the tolerance is specified as
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``{ftol: 1e-12}``
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options : dict, optional
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A dictionary of solver options. Many of the options specified for the
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global routine are also passed to the scipy.optimize.minimize routine.
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The options that are also passed to the local routine are marked with
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"(L)".
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Stopping criteria, the algorithm will terminate if any of the specified
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criteria are met. However, the default algorithm does not require any to
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be specified:
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* maxfev : int (L)
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Maximum number of function evaluations in the feasible domain.
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(Note only methods that support this option will terminate
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the routine at precisely exact specified value. Otherwise the
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criterion will only terminate during a global iteration)
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* f_min
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Specify the minimum objective function value, if it is known.
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* f_tol : float
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Precision goal for the value of f in the stopping
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criterion. Note that the global routine will also
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terminate if a sampling point in the global routine is
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within this tolerance.
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* maxiter : int
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Maximum number of iterations to perform.
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* maxev : int
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Maximum number of sampling evaluations to perform (includes
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searching in infeasible points).
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* maxtime : float
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Maximum processing runtime allowed
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* minhgrd : int
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Minimum homology group rank differential. The homology group of the
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objective function is calculated (approximately) during every
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iteration. The rank of this group has a one-to-one correspondence
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with the number of locally convex subdomains in the objective
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function (after adequate sampling points each of these subdomains
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contain a unique global minimum). If the difference in the hgr is 0
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between iterations for ``maxhgrd`` specified iterations the
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algorithm will terminate.
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Objective function knowledge:
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* symmetry : bool
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Specify True if the objective function contains symmetric variables.
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The search space (and therefore performance) is decreased by O(n!).
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* jac : bool or callable, optional
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Jacobian (gradient) of objective function. Only for CG, BFGS,
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Newton-CG, L-BFGS-B, TNC, SLSQP, dogleg, trust-ncg. If ``jac`` is a
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boolean and is True, ``fun`` is assumed to return the gradient along
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with the objective function. If False, the gradient will be
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estimated numerically. ``jac`` can also be a callable returning the
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gradient of the objective. In this case, it must accept the same
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arguments as ``fun``. (Passed to `scipy.optimize.minmize` automatically)
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* hess, hessp : callable, optional
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Hessian (matrix of second-order derivatives) of objective function
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or Hessian of objective function times an arbitrary vector p.
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Only for Newton-CG, dogleg, trust-ncg. Only one of ``hessp`` or
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``hess`` needs to be given. If ``hess`` is provided, then
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``hessp`` will be ignored. If neither ``hess`` nor ``hessp`` is
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provided, then the Hessian product will be approximated using
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finite differences on ``jac``. ``hessp`` must compute the Hessian
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times an arbitrary vector. (Passed to `scipy.optimize.minmize`
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automatically)
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Algorithm settings:
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* minimize_every_iter : bool
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If True then promising global sampling points will be passed to a
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local minimization routine every iteration. If False then only the
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final minimizer pool will be run. Defaults to False.
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* local_iter : int
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Only evaluate a few of the best minimizer pool candidates every
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iteration. If False all potential points are passed to the local
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minimization routine.
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* infty_constraints: bool
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If True then any sampling points generated which are outside will
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the feasible domain will be saved and given an objective function
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value of ``inf``. If False then these points will be discarded.
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Using this functionality could lead to higher performance with
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respect to function evaluations before the global minimum is found,
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specifying False will use less memory at the cost of a slight
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decrease in performance. Defaults to True.
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Feedback:
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* disp : bool (L)
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Set to True to print convergence messages.
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sampling_method : str or function, optional
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Current built in sampling method options are ``sobol`` and
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``simplicial``. The default ``simplicial`` uses less memory and provides
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the theoretical guarantee of convergence to the global minimum in finite
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time. The ``sobol`` method is faster in terms of sampling point
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generation at the cost of higher memory resources and the loss of
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guaranteed convergence. It is more appropriate for most "easier"
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problems where the convergence is relatively fast.
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User defined sampling functions must accept two arguments of ``n``
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sampling points of dimension ``dim`` per call and output an array of
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sampling points with shape `n x dim`.
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Returns
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-------
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res : OptimizeResult
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The optimization result represented as a `OptimizeResult` object.
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Important attributes are:
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``x`` the solution array corresponding to the global minimum,
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``fun`` the function output at the global solution,
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``xl`` an ordered list of local minima solutions,
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``funl`` the function output at the corresponding local solutions,
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``success`` a Boolean flag indicating if the optimizer exited
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successfully,
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``message`` which describes the cause of the termination,
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``nfev`` the total number of objective function evaluations including
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the sampling calls,
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``nlfev`` the total number of objective function evaluations
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culminating from all local search optimizations,
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``nit`` number of iterations performed by the global routine.
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Notes
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-----
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Global optimization using simplicial homology global optimization [1]_.
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Appropriate for solving general purpose NLP and blackbox optimization
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problems to global optimality (low-dimensional problems).
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In general, the optimization problems are of the form::
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minimize f(x) subject to
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g_i(x) >= 0, i = 1,...,m
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h_j(x) = 0, j = 1,...,p
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where x is a vector of one or more variables. ``f(x)`` is the objective
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function ``R^n -> R``, ``g_i(x)`` are the inequality constraints, and
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``h_j(x)`` are the equality constraints.
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Optionally, the lower and upper bounds for each element in x can also be
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specified using the `bounds` argument.
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While most of the theoretical advantages of SHGO are only proven for when
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``f(x)`` is a Lipschitz smooth function, the algorithm is also proven to
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converge to the global optimum for the more general case where ``f(x)`` is
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non-continuous, non-convex and non-smooth, if the default sampling method
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is used [1]_.
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The local search method may be specified using the ``minimizer_kwargs``
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parameter which is passed on to ``scipy.optimize.minimize``. By default,
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the ``SLSQP`` method is used. In general, it is recommended to use the
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``SLSQP`` or ``COBYLA`` local minimization if inequality constraints
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are defined for the problem since the other methods do not use constraints.
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The ``sobol`` method points are generated using the Sobol (1967) [2]_
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sequence. The primitive polynomials and various sets of initial direction
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numbers for generating Sobol sequences is provided by [3]_ by Frances Kuo
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and Stephen Joe. The original program sobol.cc (MIT) is available and
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described at https://web.maths.unsw.edu.au/~fkuo/sobol/ translated to
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Python 3 by Carl Sandrock 2016-03-31.
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References
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----------
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.. [1] Endres, SC, Sandrock, C, Focke, WW (2018) "A simplicial homology
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algorithm for lipschitz optimisation", Journal of Global Optimization.
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.. [2] Sobol, IM (1967) "The distribution of points in a cube and the
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approximate evaluation of integrals", USSR Comput. Math. Math. Phys.
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7, 86-112.
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.. [3] Joe, SW and Kuo, FY (2008) "Constructing Sobol sequences with
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better two-dimensional projections", SIAM J. Sci. Comput. 30,
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2635-2654.
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.. [4] Hoch, W and Schittkowski, K (1981) "Test examples for nonlinear
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programming codes", Lecture Notes in Economics and Mathematical
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Systems, 187. Springer-Verlag, New York.
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http://www.ai7.uni-bayreuth.de/test_problem_coll.pdf
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.. [5] Wales, DJ (2015) "Perspective: Insight into reaction coordinates and
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dynamics from the potential energy landscape",
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Journal of Chemical Physics, 142(13), 2015.
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Examples
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--------
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First consider the problem of minimizing the Rosenbrock function, `rosen`:
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>>> from scipy.optimize import rosen, shgo
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>>> bounds = [(0,2), (0, 2), (0, 2), (0, 2), (0, 2)]
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>>> result = shgo(rosen, bounds)
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>>> result.x, result.fun
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(array([ 1., 1., 1., 1., 1.]), 2.9203923741900809e-18)
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Note that bounds determine the dimensionality of the objective
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function and is therefore a required input, however you can specify
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empty bounds using ``None`` or objects like ``np.inf`` which will be
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converted to large float numbers.
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>>> bounds = [(None, None), ]*4
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>>> result = shgo(rosen, bounds)
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>>> result.x
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array([ 0.99999851, 0.99999704, 0.99999411, 0.9999882 ])
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Next, we consider the Eggholder function, a problem with several local
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minima and one global minimum. We will demonstrate the use of arguments and
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the capabilities of `shgo`.
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(https://en.wikipedia.org/wiki/Test_functions_for_optimization)
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>>> def eggholder(x):
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... return (-(x[1] + 47.0)
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... * np.sin(np.sqrt(abs(x[0]/2.0 + (x[1] + 47.0))))
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... - x[0] * np.sin(np.sqrt(abs(x[0] - (x[1] + 47.0))))
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... )
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...
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>>> bounds = [(-512, 512), (-512, 512)]
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`shgo` has two built-in low discrepancy sampling sequences. First, we will
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input 30 initial sampling points of the Sobol sequence:
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>>> result = shgo(eggholder, bounds, n=30, sampling_method='sobol')
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>>> result.x, result.fun
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(array([ 512. , 404.23180542]), -959.64066272085051)
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`shgo` also has a return for any other local minima that was found, these
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can be called using:
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>>> result.xl
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array([[ 512. , 404.23180542],
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[ 283.07593402, -487.12566542],
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[-294.66820039, -462.01964031],
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[-105.87688985, 423.15324143],
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[-242.97923629, 274.38032063],
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[-506.25823477, 6.3131022 ],
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[-408.71981195, -156.10117154],
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[ 150.23210485, 301.31378508],
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[ 91.00922754, -391.28375925],
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[ 202.8966344 , -269.38042147],
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[ 361.66625957, -106.96490692],
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[-219.40615102, -244.06022436],
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[ 151.59603137, -100.61082677]])
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>>> result.funl
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array([-959.64066272, -718.16745962, -704.80659592, -565.99778097,
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-559.78685655, -557.36868733, -507.87385942, -493.9605115 ,
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-426.48799655, -421.15571437, -419.31194957, -410.98477763,
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-202.53912972])
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These results are useful in applications where there are many global minima
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and the values of other global minima are desired or where the local minima
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can provide insight into the system (for example morphologies
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in physical chemistry [5]_).
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If we want to find a larger number of local minima, we can increase the
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number of sampling points or the number of iterations. We'll increase the
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number of sampling points to 60 and the number of iterations from the
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default of 1 to 5. This gives us 60 x 5 = 300 initial sampling points.
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>>> result_2 = shgo(eggholder, bounds, n=60, iters=5, sampling_method='sobol')
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>>> len(result.xl), len(result_2.xl)
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(13, 39)
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Note the difference between, e.g., ``n=180, iters=1`` and ``n=60, iters=3``.
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In the first case the promising points contained in the minimiser pool
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is processed only once. In the latter case it is processed every 60 sampling
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points for a total of 3 times.
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To demonstrate solving problems with non-linear constraints consider the
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following example from Hock and Schittkowski problem 73 (cattle-feed) [4]_::
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minimize: f = 24.55 * x_1 + 26.75 * x_2 + 39 * x_3 + 40.50 * x_4
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subject to: 2.3 * x_1 + 5.6 * x_2 + 11.1 * x_3 + 1.3 * x_4 - 5 >= 0,
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12 * x_1 + 11.9 * x_2 + 41.8 * x_3 + 52.1 * x_4 - 21
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-1.645 * sqrt(0.28 * x_1**2 + 0.19 * x_2**2 +
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20.5 * x_3**2 + 0.62 * x_4**2) >= 0,
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x_1 + x_2 + x_3 + x_4 - 1 == 0,
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1 >= x_i >= 0 for all i
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The approximate answer given in [4]_ is::
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f([0.6355216, -0.12e-11, 0.3127019, 0.05177655]) = 29.894378
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>>> def f(x): # (cattle-feed)
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... return 24.55*x[0] + 26.75*x[1] + 39*x[2] + 40.50*x[3]
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...
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>>> def g1(x):
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... return 2.3*x[0] + 5.6*x[1] + 11.1*x[2] + 1.3*x[3] - 5 # >=0
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...
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>>> def g2(x):
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... return (12*x[0] + 11.9*x[1] +41.8*x[2] + 52.1*x[3] - 21
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... - 1.645 * np.sqrt(0.28*x[0]**2 + 0.19*x[1]**2
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... + 20.5*x[2]**2 + 0.62*x[3]**2)
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... ) # >=0
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...
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>>> def h1(x):
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... return x[0] + x[1] + x[2] + x[3] - 1 # == 0
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...
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>>> cons = ({'type': 'ineq', 'fun': g1},
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... {'type': 'ineq', 'fun': g2},
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... {'type': 'eq', 'fun': h1})
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>>> bounds = [(0, 1.0),]*4
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>>> res = shgo(f, bounds, iters=3, constraints=cons)
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>>> res
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fun: 29.894378159142136
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funl: array([29.89437816])
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message: 'Optimization terminated successfully.'
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nfev: 114
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nit: 3
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nlfev: 35
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nlhev: 0
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nljev: 5
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success: True
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x: array([6.35521569e-01, 1.13700270e-13, 3.12701881e-01, 5.17765506e-02])
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xl: array([[6.35521569e-01, 1.13700270e-13, 3.12701881e-01, 5.17765506e-02]])
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>>> g1(res.x), g2(res.x), h1(res.x)
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(-5.0626169922907138e-14, -2.9594104944408173e-12, 0.0)
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"""
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# Initiate SHGO class
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shc = SHGO(func, bounds, args=args, constraints=constraints, n=n,
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iters=iters, callback=callback,
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minimizer_kwargs=minimizer_kwargs,
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options=options, sampling_method=sampling_method)
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# Run the algorithm, process results and test success
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shc.construct_complex()
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if not shc.break_routine:
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if shc.disp:
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print("Successfully completed construction of complex.")
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# Test post iterations success
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if len(shc.LMC.xl_maps) == 0:
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# If sampling failed to find pool, return lowest sampled point
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# with a warning
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shc.find_lowest_vertex()
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shc.break_routine = True
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shc.fail_routine(mes="Failed to find a feasible minimizer point. "
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"Lowest sampling point = {}".format(shc.f_lowest))
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shc.res.fun = shc.f_lowest
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shc.res.x = shc.x_lowest
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shc.res.nfev = shc.fn
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# Confirm the routine ran successfully
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if not shc.break_routine:
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shc.res.message = 'Optimization terminated successfully.'
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shc.res.success = True
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# Return the final results
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return shc.res
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class SHGO(object):
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def __init__(self, func, bounds, args=(), constraints=None, n=None,
|
|
iters=None, callback=None, minimizer_kwargs=None,
|
|
options=None, sampling_method='sobol'):
|
|
|
|
# Input checks
|
|
methods = ['sobol', 'simplicial']
|
|
if isinstance(sampling_method, str) and sampling_method not in methods:
|
|
raise ValueError(("Unknown sampling_method specified."
|
|
" Valid methods: {}").format(', '.join(methods)))
|
|
|
|
# Initiate class
|
|
self.func = func
|
|
self.bounds = bounds
|
|
self.args = args
|
|
self.callback = callback
|
|
|
|
# Bounds
|
|
abound = np.array(bounds, float)
|
|
self.dim = np.shape(abound)[0] # Dimensionality of problem
|
|
|
|
# Set none finite values to large floats
|
|
infind = ~np.isfinite(abound)
|
|
abound[infind[:, 0], 0] = -1e50
|
|
abound[infind[:, 1], 1] = 1e50
|
|
|
|
# Check if bounds are correctly specified
|
|
bnderr = abound[:, 0] > abound[:, 1]
|
|
if bnderr.any():
|
|
raise ValueError('Error: lb > ub in bounds {}.'
|
|
.format(', '.join(str(b) for b in bnderr)))
|
|
|
|
self.bounds = abound
|
|
|
|
# Constraints
|
|
# Process constraint dict sequence:
|
|
if constraints is not None:
|
|
self.min_cons = constraints
|
|
self.g_cons = []
|
|
self.g_args = []
|
|
if (type(constraints) is not tuple) and (type(constraints)
|
|
is not list):
|
|
constraints = (constraints,)
|
|
|
|
for cons in constraints:
|
|
if cons['type'] == 'ineq':
|
|
self.g_cons.append(cons['fun'])
|
|
try:
|
|
self.g_args.append(cons['args'])
|
|
except KeyError:
|
|
self.g_args.append(())
|
|
self.g_cons = tuple(self.g_cons)
|
|
self.g_args = tuple(self.g_args)
|
|
else:
|
|
self.g_cons = None
|
|
self.g_args = None
|
|
|
|
# Define local minimization keyword arguments
|
|
# Start with defaults
|
|
self.minimizer_kwargs = {'args': self.args,
|
|
'method': 'SLSQP',
|
|
'bounds': self.bounds,
|
|
'options': {},
|
|
'callback': self.callback
|
|
}
|
|
if minimizer_kwargs is not None:
|
|
# Overwrite with supplied values
|
|
self.minimizer_kwargs.update(minimizer_kwargs)
|
|
|
|
else:
|
|
self.minimizer_kwargs['options'] = {'ftol': 1e-12}
|
|
|
|
if (self.minimizer_kwargs['method'] in ('SLSQP', 'COBYLA') and
|
|
(minimizer_kwargs is not None and
|
|
'constraints' not in minimizer_kwargs and
|
|
constraints is not None) or
|
|
(self.g_cons is not None)):
|
|
self.minimizer_kwargs['constraints'] = self.min_cons
|
|
|
|
# Process options dict
|
|
if options is not None:
|
|
self.init_options(options)
|
|
else: # Default settings:
|
|
self.f_min_true = None
|
|
self.minimize_every_iter = False
|
|
|
|
# Algorithm limits
|
|
self.maxiter = None
|
|
self.maxfev = None
|
|
self.maxev = None
|
|
self.maxtime = None
|
|
self.f_min_true = None
|
|
self.minhgrd = None
|
|
|
|
# Objective function knowledge
|
|
self.symmetry = False
|
|
|
|
# Algorithm functionality
|
|
self.local_iter = False
|
|
self.infty_cons_sampl = True
|
|
|
|
# Feedback
|
|
self.disp = False
|
|
|
|
# Remove unknown arguments in self.minimizer_kwargs
|
|
# Start with arguments all the solvers have in common
|
|
self.min_solver_args = ['fun', 'x0', 'args',
|
|
'callback', 'options', 'method']
|
|
# then add the ones unique to specific solvers
|
|
solver_args = {
|
|
'_custom': ['jac', 'hess', 'hessp', 'bounds', 'constraints'],
|
|
'nelder-mead': [],
|
|
'powell': [],
|
|
'cg': ['jac'],
|
|
'bfgs': ['jac'],
|
|
'newton-cg': ['jac', 'hess', 'hessp'],
|
|
'l-bfgs-b': ['jac', 'bounds'],
|
|
'tnc': ['jac', 'bounds'],
|
|
'cobyla': ['constraints'],
|
|
'slsqp': ['jac', 'bounds', 'constraints'],
|
|
'dogleg': ['jac', 'hess'],
|
|
'trust-ncg': ['jac', 'hess', 'hessp'],
|
|
'trust-krylov': ['jac', 'hess', 'hessp'],
|
|
'trust-exact': ['jac', 'hess'],
|
|
}
|
|
method = self.minimizer_kwargs['method']
|
|
self.min_solver_args += solver_args[method.lower()]
|
|
|
|
# Only retain the known arguments
|
|
def _restrict_to_keys(dictionary, goodkeys):
|
|
"""Remove keys from dictionary if not in goodkeys - inplace"""
|
|
existingkeys = set(dictionary)
|
|
for key in existingkeys - set(goodkeys):
|
|
dictionary.pop(key, None)
|
|
|
|
_restrict_to_keys(self.minimizer_kwargs, self.min_solver_args)
|
|
_restrict_to_keys(self.minimizer_kwargs['options'],
|
|
self.min_solver_args + ['ftol'])
|
|
|
|
# Algorithm controls
|
|
# Global controls
|
|
self.stop_global = False # Used in the stopping_criteria method
|
|
self.break_routine = False # Break the algorithm globally
|
|
self.iters = iters # Iterations to be ran
|
|
self.iters_done = 0 # Iterations to be ran
|
|
self.n = n # Sampling points per iteration
|
|
self.nc = n # Sampling points to sample in current iteration
|
|
self.n_prc = 0 # Processed points (used to track Delaunay iters)
|
|
self.n_sampled = 0 # To track number of sampling points already generated
|
|
self.fn = 0 # Number of feasible sampling points evaluations performed
|
|
self.hgr = 0 # Homology group rank
|
|
|
|
# Default settings if no sampling criteria.
|
|
if self.iters is None:
|
|
self.iters = 1
|
|
if self.n is None:
|
|
self.n = 100
|
|
self.nc = self.n
|
|
|
|
if not ((self.maxiter is None) and (self.maxfev is None) and (
|
|
self.maxev is None)
|
|
and (self.minhgrd is None) and (self.f_min_true is None)):
|
|
self.iters = None
|
|
|
|
# Set complex construction mode based on a provided stopping criteria:
|
|
# Choose complex constructor
|
|
if sampling_method == 'simplicial':
|
|
self.iterate_complex = self.iterate_hypercube
|
|
self.minimizers = self.simplex_minimizers
|
|
self.sampling_method = sampling_method
|
|
|
|
elif sampling_method == 'sobol' or not isinstance(sampling_method, str):
|
|
self.iterate_complex = self.iterate_delaunay
|
|
self.minimizers = self.delaunay_complex_minimisers
|
|
# Sampling method used
|
|
if sampling_method == 'sobol':
|
|
self.sampling_method = sampling_method
|
|
self.sampling = self.sampling_sobol
|
|
self.Sobol = sobol_seq.Sobol() # Init Sobol class
|
|
if self.dim < 40:
|
|
self.sobol_points = self.sobol_points_40
|
|
else:
|
|
self.sobol_points = self.sobol_points_10k
|
|
else:
|
|
# A user defined sampling method:
|
|
# self.sampling_points = sampling_method
|
|
self.sampling = self.sampling_custom
|
|
self.sampling_function = sampling_method # F(n, d)
|
|
self.sampling_method = 'custom'
|
|
|
|
# Local controls
|
|
self.stop_l_iter = False # Local minimisation iterations
|
|
self.stop_complex_iter = False # Sampling iterations
|
|
|
|
# Initiate storage objects used in algorithm classes
|
|
self.minimizer_pool = []
|
|
|
|
# Cache of local minimizers mapped
|
|
self.LMC = LMapCache()
|
|
|
|
# Initialize return object
|
|
self.res = OptimizeResult() # scipy.optimize.OptimizeResult object
|
|
self.res.nfev = 0 # Includes each sampling point as func evaluation
|
|
self.res.nlfev = 0 # Local function evals for all minimisers
|
|
self.res.nljev = 0 # Local Jacobian evals for all minimisers
|
|
self.res.nlhev = 0 # Local Hessian evals for all minimisers
|
|
|
|
# Initiation aids
|
|
def init_options(self, options):
|
|
"""
|
|
Initiates the options.
|
|
|
|
Can also be useful to change parameters after class initiation.
|
|
|
|
Parameters
|
|
----------
|
|
options : dict
|
|
|
|
Returns
|
|
-------
|
|
None
|
|
|
|
"""
|
|
self.minimizer_kwargs['options'].update(options)
|
|
# Default settings:
|
|
self.minimize_every_iter = options.get('minimize_every_iter', False)
|
|
|
|
# Algorithm limits
|
|
# Maximum number of iterations to perform.
|
|
self.maxiter = options.get('maxiter', None)
|
|
# Maximum number of function evaluations in the feasible domain
|
|
self.maxfev = options.get('maxfev', None)
|
|
# Maximum number of sampling evaluations (includes searching in
|
|
# infeasible points
|
|
self.maxev = options.get('maxev', None)
|
|
# Maximum processing runtime allowed
|
|
self.init = time.time()
|
|
self.maxtime = options.get('maxtime', None)
|
|
if 'f_min' in options:
|
|
# Specify the minimum objective function value, if it is known.
|
|
self.f_min_true = options['f_min']
|
|
self.f_tol = options.get('f_tol', 1e-4)
|
|
else:
|
|
self.f_min_true = None
|
|
|
|
self.minhgrd = options.get('minhgrd', None)
|
|
|
|
# Objective function knowledge
|
|
self.symmetry = 'symmetry' in options
|
|
|
|
# Algorithm functionality
|
|
# Only evaluate a few of the best candiates
|
|
self.local_iter = options.get('local_iter', False)
|
|
|
|
self.infty_cons_sampl = options.get('infty_constraints', True)
|
|
|
|
# Feedback
|
|
self.disp = options.get('disp', False)
|
|
|
|
# Iteration properties
|
|
# Main construction loop:
|
|
def construct_complex(self):
|
|
"""
|
|
Construct for `iters` iterations.
|
|
|
|
If uniform sampling is used, every iteration adds 'n' sampling points.
|
|
|
|
Iterations if a stopping criteria (e.g., sampling points or
|
|
processing time) has been met.
|
|
|
|
"""
|
|
if self.disp:
|
|
print('Splitting first generation')
|
|
|
|
while not self.stop_global:
|
|
if self.break_routine:
|
|
break
|
|
# Iterate complex, process minimisers
|
|
self.iterate()
|
|
self.stopping_criteria()
|
|
|
|
# Build minimiser pool
|
|
# Final iteration only needed if pools weren't minimised every iteration
|
|
if not self.minimize_every_iter:
|
|
if not self.break_routine:
|
|
self.find_minima()
|
|
|
|
self.res.nit = self.iters_done + 1
|
|
|
|
def find_minima(self):
|
|
"""
|
|
Construct the minimizer pool, map the minimizers to local minima
|
|
and sort the results into a global return object.
|
|
"""
|
|
self.minimizers()
|
|
if len(self.X_min) != 0:
|
|
# Minimize the pool of minimizers with local minimization methods
|
|
# Note that if Options['local_iter'] is an `int` instead of default
|
|
# value False then only that number of candidates will be minimized
|
|
self.minimise_pool(self.local_iter)
|
|
# Sort results and build the global return object
|
|
self.sort_result()
|
|
|
|
# Lowest values used to report in case of failures
|
|
self.f_lowest = self.res.fun
|
|
self.x_lowest = self.res.x
|
|
else:
|
|
self.find_lowest_vertex()
|
|
|
|
def find_lowest_vertex(self):
|
|
# Find the lowest objective function value on one of
|
|
# the vertices of the simplicial complex
|
|
if self.sampling_method == 'simplicial':
|
|
self.f_lowest = np.inf
|
|
for x in self.HC.V.cache:
|
|
if self.HC.V[x].f < self.f_lowest:
|
|
self.f_lowest = self.HC.V[x].f
|
|
self.x_lowest = self.HC.V[x].x_a
|
|
if self.f_lowest == np.inf: # no feasible point
|
|
self.f_lowest = None
|
|
self.x_lowest = None
|
|
else:
|
|
if self.fn == 0:
|
|
self.f_lowest = None
|
|
self.x_lowest = None
|
|
else:
|
|
self.f_I = np.argsort(self.F, axis=-1)
|
|
self.f_lowest = self.F[self.f_I[0]]
|
|
self.x_lowest = self.C[self.f_I[0]]
|
|
|
|
# Stopping criteria functions:
|
|
def finite_iterations(self):
|
|
if self.iters is not None:
|
|
if self.iters_done >= (self.iters - 1):
|
|
self.stop_global = True
|
|
|
|
if self.maxiter is not None: # Stop for infeasible sampling
|
|
if self.iters_done >= (self.maxiter - 1):
|
|
self.stop_global = True
|
|
return self.stop_global
|
|
|
|
def finite_fev(self):
|
|
# Finite function evals in the feasible domain
|
|
if self.fn >= self.maxfev:
|
|
self.stop_global = True
|
|
return self.stop_global
|
|
|
|
def finite_ev(self):
|
|
# Finite evaluations including infeasible sampling points
|
|
if self.n_sampled >= self.maxev:
|
|
self.stop_global = True
|
|
|
|
def finite_time(self):
|
|
if (time.time() - self.init) >= self.maxtime:
|
|
self.stop_global = True
|
|
|
|
def finite_precision(self):
|
|
"""
|
|
Stop the algorithm if the final function value is known
|
|
|
|
Specify in options (with ``self.f_min_true = options['f_min']``)
|
|
and the tolerance with ``f_tol = options['f_tol']``
|
|
"""
|
|
# If no minimizer has been found use the lowest sampling value
|
|
if len(self.LMC.xl_maps) == 0:
|
|
self.find_lowest_vertex()
|
|
|
|
# Function to stop algorithm at specified percentage error:
|
|
if self.f_lowest == 0.0:
|
|
if self.f_min_true == 0.0:
|
|
if self.f_lowest <= self.f_tol:
|
|
self.stop_global = True
|
|
else:
|
|
pe = (self.f_lowest - self.f_min_true) / abs(self.f_min_true)
|
|
if self.f_lowest <= self.f_min_true:
|
|
self.stop_global = True
|
|
# 2if (pe - self.f_tol) <= abs(1.0 / abs(self.f_min_true)):
|
|
if abs(pe) >= 2 * self.f_tol:
|
|
warnings.warn("A much lower value than expected f* =" +
|
|
" {} than".format(self.f_min_true) +
|
|
" the was found f_lowest =" +
|
|
"{} ".format(self.f_lowest))
|
|
if pe <= self.f_tol:
|
|
self.stop_global = True
|
|
|
|
return self.stop_global
|
|
|
|
def finite_homology_growth(self):
|
|
if self.LMC.size == 0:
|
|
return # pass on no reason to stop yet.
|
|
self.hgrd = self.LMC.size - self.hgr
|
|
|
|
self.hgr = self.LMC.size
|
|
if self.hgrd <= self.minhgrd:
|
|
self.stop_global = True
|
|
return self.stop_global
|
|
|
|
def stopping_criteria(self):
|
|
"""
|
|
Various stopping criteria ran every iteration
|
|
|
|
Returns
|
|
-------
|
|
stop : bool
|
|
"""
|
|
if self.maxiter is not None:
|
|
self.finite_iterations()
|
|
if self.iters is not None:
|
|
self.finite_iterations()
|
|
if self.maxfev is not None:
|
|
self.finite_fev()
|
|
if self.maxev is not None:
|
|
self.finite_ev()
|
|
if self.maxtime is not None:
|
|
self.finite_time()
|
|
if self.f_min_true is not None:
|
|
self.finite_precision()
|
|
if self.minhgrd is not None:
|
|
self.finite_homology_growth()
|
|
|
|
def iterate(self):
|
|
self.iterate_complex()
|
|
|
|
# Build minimizer pool
|
|
if self.minimize_every_iter:
|
|
if not self.break_routine:
|
|
self.find_minima() # Process minimizer pool
|
|
|
|
# Algorithm updates
|
|
self.iters_done += 1
|
|
|
|
def iterate_hypercube(self):
|
|
"""
|
|
Iterate a subdivision of the complex
|
|
|
|
Note: called with ``self.iterate_complex()`` after class initiation
|
|
"""
|
|
# Iterate the complex
|
|
if self.n_sampled == 0:
|
|
# Initial triangulation of the hyper-rectangle
|
|
self.HC = Complex(self.dim, self.func, self.args,
|
|
self.symmetry, self.bounds, self.g_cons,
|
|
self.g_args)
|
|
else:
|
|
self.HC.split_generation()
|
|
|
|
# feasible sampling points counted by the triangulation.py routines
|
|
self.fn = self.HC.V.nfev
|
|
self.n_sampled = self.HC.V.size # nevs counted in triangulation.py
|
|
return
|
|
|
|
def iterate_delaunay(self):
|
|
"""
|
|
Build a complex of Delaunay triangulated points
|
|
|
|
Note: called with ``self.iterate_complex()`` after class initiation
|
|
"""
|
|
self.nc += self.n
|
|
self.sampled_surface(infty_cons_sampl=self.infty_cons_sampl)
|
|
self.n_sampled = self.nc
|
|
return
|
|
|
|
# Hypercube minimizers
|
|
def simplex_minimizers(self):
|
|
"""
|
|
Returns the indexes of all minimizers
|
|
"""
|
|
self.minimizer_pool = []
|
|
# Note: Can implement parallelization here
|
|
for x in self.HC.V.cache:
|
|
if self.HC.V[x].minimiser():
|
|
if self.disp:
|
|
logging.info('=' * 60)
|
|
logging.info(
|
|
'v.x = {} is minimizer'.format(self.HC.V[x].x_a))
|
|
logging.info('v.f = {} is minimizer'.format(self.HC.V[x].f))
|
|
logging.info('=' * 30)
|
|
|
|
if self.HC.V[x] not in self.minimizer_pool:
|
|
self.minimizer_pool.append(self.HC.V[x])
|
|
|
|
if self.disp:
|
|
logging.info('Neighbors:')
|
|
logging.info('=' * 30)
|
|
for vn in self.HC.V[x].nn:
|
|
logging.info('x = {} || f = {}'.format(vn.x, vn.f))
|
|
|
|
logging.info('=' * 60)
|
|
|
|
self.minimizer_pool_F = []
|
|
self.X_min = []
|
|
# normalized tuple in the Vertex cache
|
|
self.X_min_cache = {} # Cache used in hypercube sampling
|
|
|
|
for v in self.minimizer_pool:
|
|
self.X_min.append(v.x_a)
|
|
self.minimizer_pool_F.append(v.f)
|
|
self.X_min_cache[tuple(v.x_a)] = v.x
|
|
|
|
self.minimizer_pool_F = np.array(self.minimizer_pool_F)
|
|
self.X_min = np.array(self.X_min)
|
|
|
|
# TODO: Only do this if global mode
|
|
self.sort_min_pool()
|
|
|
|
return self.X_min
|
|
|
|
# Local minimization
|
|
# Minimizer pool processing
|
|
def minimise_pool(self, force_iter=False):
|
|
"""
|
|
This processing method can optionally minimise only the best candidate
|
|
solutions in the minimizer pool
|
|
|
|
Parameters
|
|
----------
|
|
force_iter : int
|
|
Number of starting minimizers to process (can be sepcified
|
|
globally or locally)
|
|
|
|
"""
|
|
# Find first local minimum
|
|
# NOTE: Since we always minimize this value regardless it is a waste to
|
|
# build the topograph first before minimizing
|
|
lres_f_min = self.minimize(self.X_min[0], ind=self.minimizer_pool[0])
|
|
|
|
# Trim minimized point from current minimizer set
|
|
self.trim_min_pool(0)
|
|
|
|
# Force processing to only
|
|
if force_iter:
|
|
self.local_iter = force_iter
|
|
|
|
while not self.stop_l_iter:
|
|
# Global stopping criteria:
|
|
if self.f_min_true is not None:
|
|
if (lres_f_min.fun - self.f_min_true) / abs(
|
|
self.f_min_true) <= self.f_tol:
|
|
self.stop_l_iter = True
|
|
break
|
|
# Note first iteration is outside loop:
|
|
if self.local_iter is not None:
|
|
if self.disp:
|
|
logging.info(
|
|
'SHGO.iters in function minimise_pool = {}'.format(
|
|
self.local_iter))
|
|
self.local_iter -= 1
|
|
if self.local_iter == 0:
|
|
self.stop_l_iter = True
|
|
break
|
|
|
|
if np.shape(self.X_min)[0] == 0:
|
|
self.stop_l_iter = True
|
|
break
|
|
|
|
# Construct topograph from current minimizer set
|
|
# (NOTE: This is a very small topograph using only the minizer pool
|
|
# , it might be worth using some graph theory tools instead.
|
|
self.g_topograph(lres_f_min.x, self.X_min)
|
|
|
|
# Find local minimum at the miniser with the greatest Euclidean
|
|
# distance from the current solution
|
|
ind_xmin_l = self.Z[:, -1]
|
|
lres_f_min = self.minimize(self.Ss[-1, :], self.minimizer_pool[-1])
|
|
|
|
# Trim minimised point from current minimizer set
|
|
self.trim_min_pool(ind_xmin_l)
|
|
|
|
# Reset controls
|
|
self.stop_l_iter = False
|
|
return
|
|
|
|
def sort_min_pool(self):
|
|
# Sort to find minimum func value in min_pool
|
|
self.ind_f_min = np.argsort(self.minimizer_pool_F)
|
|
self.minimizer_pool = np.array(self.minimizer_pool)[self.ind_f_min]
|
|
self.minimizer_pool_F = np.array(self.minimizer_pool_F)[
|
|
self.ind_f_min]
|
|
return
|
|
|
|
def trim_min_pool(self, trim_ind):
|
|
self.X_min = np.delete(self.X_min, trim_ind, axis=0)
|
|
self.minimizer_pool_F = np.delete(self.minimizer_pool_F, trim_ind)
|
|
self.minimizer_pool = np.delete(self.minimizer_pool, trim_ind)
|
|
return
|
|
|
|
def g_topograph(self, x_min, X_min):
|
|
"""
|
|
Returns the topographical vector stemming from the specified value
|
|
``x_min`` for the current feasible set ``X_min`` with True boolean
|
|
values indicating positive entries and False values indicating
|
|
negative entries.
|
|
|
|
"""
|
|
x_min = np.array([x_min])
|
|
self.Y = spatial.distance.cdist(x_min, X_min, 'euclidean')
|
|
# Find sorted indexes of spatial distances:
|
|
self.Z = np.argsort(self.Y, axis=-1)
|
|
|
|
self.Ss = X_min[self.Z][0]
|
|
self.minimizer_pool = self.minimizer_pool[self.Z]
|
|
self.minimizer_pool = self.minimizer_pool[0]
|
|
return self.Ss
|
|
|
|
# Local bound functions
|
|
def construct_lcb_simplicial(self, v_min):
|
|
"""
|
|
Construct locally (approximately) convex bounds
|
|
|
|
Parameters
|
|
----------
|
|
v_min : Vertex object
|
|
The minimizer vertex
|
|
|
|
Returns
|
|
-------
|
|
cbounds : list of lists
|
|
List of size dimension with length-2 list of bounds for each dimension
|
|
|
|
"""
|
|
cbounds = [[x_b_i[0], x_b_i[1]] for x_b_i in self.bounds]
|
|
# Loop over all bounds
|
|
for vn in v_min.nn:
|
|
for i, x_i in enumerate(vn.x_a):
|
|
# Lower bound
|
|
if (x_i < v_min.x_a[i]) and (x_i > cbounds[i][0]):
|
|
cbounds[i][0] = x_i
|
|
|
|
# Upper bound
|
|
if (x_i > v_min.x_a[i]) and (x_i < cbounds[i][1]):
|
|
cbounds[i][1] = x_i
|
|
|
|
if self.disp:
|
|
logging.info('cbounds found for v_min.x_a = {}'.format(v_min.x_a))
|
|
logging.info('cbounds = {}'.format(cbounds))
|
|
|
|
return cbounds
|
|
|
|
def construct_lcb_delaunay(self, v_min, ind=None):
|
|
"""
|
|
Construct locally (approximately) convex bounds
|
|
|
|
Parameters
|
|
----------
|
|
v_min : Vertex object
|
|
The minimizer vertex
|
|
|
|
Returns
|
|
-------
|
|
cbounds : list of lists
|
|
List of size dimension with length-2 list of bounds for each dimension
|
|
"""
|
|
cbounds = [[x_b_i[0], x_b_i[1]] for x_b_i in self.bounds]
|
|
|
|
return cbounds
|
|
|
|
# Minimize a starting point locally
|
|
def minimize(self, x_min, ind=None):
|
|
"""
|
|
This function is used to calculate the local minima using the specified
|
|
sampling point as a starting value.
|
|
|
|
Parameters
|
|
----------
|
|
x_min : vector of floats
|
|
Current starting point to minimize.
|
|
|
|
Returns
|
|
-------
|
|
lres : OptimizeResult
|
|
The local optimization result represented as a `OptimizeResult`
|
|
object.
|
|
"""
|
|
# Use minima maps if vertex was already run
|
|
if self.disp:
|
|
logging.info('Vertex minimiser maps = {}'.format(self.LMC.v_maps))
|
|
|
|
if self.LMC[x_min].lres is not None:
|
|
return self.LMC[x_min].lres
|
|
|
|
# TODO: Check discarded bound rules
|
|
|
|
if self.callback is not None:
|
|
print('Callback for '
|
|
'minimizer starting at {}:'.format(x_min))
|
|
|
|
if self.disp:
|
|
print('Starting '
|
|
'minimization at {}...'.format(x_min))
|
|
|
|
if self.sampling_method == 'simplicial':
|
|
x_min_t = tuple(x_min)
|
|
# Find the normalized tuple in the Vertex cache:
|
|
x_min_t_norm = self.X_min_cache[tuple(x_min_t)]
|
|
|
|
x_min_t_norm = tuple(x_min_t_norm)
|
|
|
|
g_bounds = self.construct_lcb_simplicial(self.HC.V[x_min_t_norm])
|
|
if 'bounds' in self.min_solver_args:
|
|
self.minimizer_kwargs['bounds'] = g_bounds
|
|
|
|
else:
|
|
g_bounds = self.construct_lcb_delaunay(x_min, ind=ind)
|
|
if 'bounds' in self.min_solver_args:
|
|
self.minimizer_kwargs['bounds'] = g_bounds
|
|
|
|
if self.disp and 'bounds' in self.minimizer_kwargs:
|
|
print('bounds in kwarg:')
|
|
print(self.minimizer_kwargs['bounds'])
|
|
|
|
# Local minimization using scipy.optimize.minimize:
|
|
lres = minimize(self.func, x_min, **self.minimizer_kwargs)
|
|
|
|
if self.disp:
|
|
print('lres = {}'.format(lres))
|
|
|
|
# Local function evals for all minimizers
|
|
self.res.nlfev += lres.nfev
|
|
if 'njev' in lres:
|
|
self.res.nljev += lres.njev
|
|
if 'nhev' in lres:
|
|
self.res.nlhev += lres.nhev
|
|
|
|
try: # Needed because of the brain dead 1x1 NumPy arrays
|
|
lres.fun = lres.fun[0]
|
|
except (IndexError, TypeError):
|
|
lres.fun
|
|
|
|
# Append minima maps
|
|
self.LMC[x_min]
|
|
self.LMC.add_res(x_min, lres, bounds=g_bounds)
|
|
|
|
return lres
|
|
|
|
# Post local minimization processing
|
|
def sort_result(self):
|
|
"""
|
|
Sort results and build the global return object
|
|
"""
|
|
# Sort results in local minima cache
|
|
results = self.LMC.sort_cache_result()
|
|
self.res.xl = results['xl']
|
|
self.res.funl = results['funl']
|
|
self.res.x = results['x']
|
|
self.res.fun = results['fun']
|
|
|
|
# Add local func evals to sampling func evals
|
|
# Count the number of feasible vertices and add to local func evals:
|
|
self.res.nfev = self.fn + self.res.nlfev
|
|
return self.res
|
|
|
|
# Algorithm controls
|
|
def fail_routine(self, mes=("Failed to converge")):
|
|
self.break_routine = True
|
|
self.res.success = False
|
|
self.X_min = [None]
|
|
self.res.message = mes
|
|
|
|
def sampled_surface(self, infty_cons_sampl=False):
|
|
"""
|
|
Sample the function surface.
|
|
|
|
There are 2 modes, if ``infty_cons_sampl`` is True then the sampled
|
|
points that are generated outside the feasible domain will be
|
|
assigned an ``inf`` value in accordance with SHGO rules.
|
|
This guarantees convergence and usually requires less objective function
|
|
evaluations at the computational costs of more Delaunay triangulation
|
|
points.
|
|
|
|
If ``infty_cons_sampl`` is False, then the infeasible points are discarded
|
|
and only a subspace of the sampled points are used. This comes at the
|
|
cost of the loss of guaranteed convergence and usually requires more
|
|
objective function evaluations.
|
|
"""
|
|
# Generate sampling points
|
|
if self.disp:
|
|
print('Generating sampling points')
|
|
self.sampling(self.nc, self.dim)
|
|
|
|
if not infty_cons_sampl:
|
|
# Find subspace of feasible points
|
|
if self.g_cons is not None:
|
|
self.sampling_subspace()
|
|
|
|
# Sort remaining samples
|
|
self.sorted_samples()
|
|
|
|
# Find objective function references
|
|
self.fun_ref()
|
|
|
|
self.n_sampled = self.nc
|
|
|
|
def delaunay_complex_minimisers(self):
|
|
# Construct complex minimizers on the current sampling set.
|
|
# if self.fn >= (self.dim + 1):
|
|
if self.fn >= (self.dim + 2):
|
|
# TODO: Check on strange Qhull error where the number of vertices
|
|
# required for an initial simplex is higher than n + 1?
|
|
if self.dim < 2: # Scalar objective functions
|
|
if self.disp:
|
|
print('Constructing 1-D minimizer pool')
|
|
|
|
self.ax_subspace()
|
|
self.surface_topo_ref()
|
|
self.minimizers_1D()
|
|
|
|
else: # Multivariate functions.
|
|
if self.disp:
|
|
print('Constructing Gabrial graph and minimizer pool')
|
|
|
|
if self.iters == 1:
|
|
self.delaunay_triangulation(grow=False)
|
|
else:
|
|
self.delaunay_triangulation(grow=True, n_prc=self.n_prc)
|
|
self.n_prc = self.C.shape[0]
|
|
|
|
if self.disp:
|
|
print('Triangulation completed, building minimizer pool')
|
|
|
|
self.delaunay_minimizers()
|
|
|
|
if self.disp:
|
|
logging.info(
|
|
"Minimizer pool = SHGO.X_min = {}".format(self.X_min))
|
|
else:
|
|
if self.disp:
|
|
print(
|
|
'Not enough sampling points found in the feasible domain.')
|
|
self.minimizer_pool = [None]
|
|
try:
|
|
self.X_min
|
|
except AttributeError:
|
|
self.X_min = []
|
|
|
|
def sobol_points_40(self, n, d, skip=0):
|
|
"""
|
|
Wrapper for ``sobol_seq.i4_sobol_generate``
|
|
|
|
Generate N sampling points in D dimensions
|
|
"""
|
|
points = self.Sobol.i4_sobol_generate(d, n, skip=0)
|
|
|
|
return points
|
|
|
|
def sobol_points_10k(self, N, D):
|
|
"""
|
|
sobol.cc by Frances Kuo and Stephen Joe translated to Python 3 by
|
|
Carl Sandrock 2016-03-31
|
|
|
|
The original program is available and described at
|
|
https://web.maths.unsw.edu.au/~fkuo/sobol/
|
|
"""
|
|
import gzip
|
|
import os
|
|
path = os.path.join(os.path.dirname(__file__), '_shgo_lib',
|
|
'sobol_vec.gz')
|
|
f = gzip.open(path, 'rb')
|
|
unsigned = "uint64"
|
|
# swallow header
|
|
next(f)
|
|
|
|
L = int(np.log(N) // np.log(2.0)) + 1
|
|
|
|
C = np.ones(N, dtype=unsigned)
|
|
for i in range(1, N):
|
|
value = i
|
|
while value & 1:
|
|
value >>= 1
|
|
C[i] += 1
|
|
|
|
points = np.zeros((N, D), dtype='double')
|
|
|
|
# XXX: This appears not to set the first element of V
|
|
V = np.empty(L + 1, dtype=unsigned)
|
|
for i in range(1, L + 1):
|
|
V[i] = 1 << (32 - i)
|
|
|
|
X = np.empty(N, dtype=unsigned)
|
|
X[0] = 0
|
|
for i in range(1, N):
|
|
X[i] = X[i - 1] ^ V[C[i - 1]]
|
|
points[i, 0] = X[i] / 2 ** 32
|
|
|
|
for j in range(1, D):
|
|
F_int = [int(item) for item in next(f).strip().split()]
|
|
(_, s, a), m = F_int[:3], [0] + F_int[3:]
|
|
|
|
if L <= s:
|
|
for i in range(1, L + 1):
|
|
V[i] = m[i] << (32 - i)
|
|
else:
|
|
for i in range(1, s + 1):
|
|
V[i] = m[i] << (32 - i)
|
|
for i in range(s + 1, L + 1):
|
|
V[i] = V[i - s] ^ (
|
|
V[i - s] >> np.array(s, dtype=unsigned))
|
|
for k in range(1, s):
|
|
V[i] ^= np.array(
|
|
(((a >> (s - 1 - k)) & 1) * V[i - k]),
|
|
dtype=unsigned)
|
|
|
|
X[0] = 0
|
|
for i in range(1, N):
|
|
X[i] = X[i - 1] ^ V[C[i - 1]]
|
|
points[i, j] = X[i] / 2 ** 32 # *** the actual points
|
|
|
|
f.close()
|
|
return points
|
|
|
|
def sampling_sobol(self, n, dim):
|
|
"""
|
|
Generates uniform sampling points in a hypercube and scales the points
|
|
to the bound limits.
|
|
"""
|
|
# Generate sampling points.
|
|
# Generate uniform sample points in [0, 1]^m \subset R^m
|
|
if self.n_sampled == 0:
|
|
self.C = self.sobol_points(n, dim)
|
|
else:
|
|
self.C = self.sobol_points(n, dim, skip=self.n_sampled)
|
|
# Distribute over bounds
|
|
for i in range(len(self.bounds)):
|
|
self.C[:, i] = (self.C[:, i] *
|
|
(self.bounds[i][1] - self.bounds[i][0])
|
|
+ self.bounds[i][0])
|
|
return self.C
|
|
|
|
def sampling_custom(self, n, dim):
|
|
"""
|
|
Generates uniform sampling points in a hypercube and scales the points
|
|
to the bound limits.
|
|
"""
|
|
# Generate sampling points.
|
|
# Generate uniform sample points in [0, 1]^m \subset R^m
|
|
self.C = self.sampling_function(n, dim)
|
|
# Distribute over bounds
|
|
for i in range(len(self.bounds)):
|
|
self.C[:, i] = (self.C[:, i] *
|
|
(self.bounds[i][1] - self.bounds[i][0])
|
|
+ self.bounds[i][0])
|
|
return self.C
|
|
|
|
def sampling_subspace(self):
|
|
"""Find subspace of feasible points from g_func definition"""
|
|
# Subspace of feasible points.
|
|
for ind, g in enumerate(self.g_cons):
|
|
self.C = self.C[g(self.C.T, *self.g_args[ind]) >= 0.0]
|
|
if self.C.size == 0:
|
|
self.res.message = ('No sampling point found within the '
|
|
+ 'feasible set. Increasing sampling '
|
|
+ 'size.')
|
|
# sampling correctly for both 1-D and >1-D cases
|
|
if self.disp:
|
|
print(self.res.message)
|
|
|
|
def sorted_samples(self): # Validated
|
|
"""Find indexes of the sorted sampling points"""
|
|
self.Ind_sorted = np.argsort(self.C, axis=0)
|
|
self.Xs = self.C[self.Ind_sorted]
|
|
return self.Ind_sorted, self.Xs
|
|
|
|
def ax_subspace(self): # Validated
|
|
"""
|
|
Finds the subspace vectors along each component axis.
|
|
"""
|
|
self.Ci = []
|
|
self.Xs_i = []
|
|
self.Ii = []
|
|
for i in range(self.dim):
|
|
self.Ci.append(self.C[:, i])
|
|
self.Ii.append(self.Ind_sorted[:, i])
|
|
self.Xs_i.append(self.Xs[:, i])
|
|
|
|
def fun_ref(self):
|
|
"""
|
|
Find the objective function output reference table
|
|
"""
|
|
# TODO: Replace with cached wrapper
|
|
|
|
# Note: This process can be pooled easily
|
|
# Obj. function returns to be used as reference table.:
|
|
f_cache_bool = False
|
|
if self.fn > 0: # Store old function evaluations
|
|
Ftemp = self.F
|
|
fn_old = self.fn
|
|
f_cache_bool = True
|
|
|
|
self.F = np.zeros(np.shape(self.C)[0])
|
|
# NOTE: It might be easier to replace this with a cached
|
|
# objective function
|
|
for i in range(self.fn, np.shape(self.C)[0]):
|
|
eval_f = True
|
|
if self.g_cons is not None:
|
|
for g in self.g_cons:
|
|
if g(self.C[i, :], *self.args) < 0.0:
|
|
eval_f = False
|
|
break # Breaks the g loop
|
|
|
|
if eval_f:
|
|
self.F[i] = self.func(self.C[i, :], *self.args)
|
|
self.fn += 1
|
|
elif self.infty_cons_sampl:
|
|
self.F[i] = np.inf
|
|
self.fn += 1
|
|
if f_cache_bool:
|
|
if fn_old > 0: # Restore saved function evaluations
|
|
self.F[0:fn_old] = Ftemp
|
|
|
|
return self.F
|
|
|
|
def surface_topo_ref(self): # Validated
|
|
"""
|
|
Find the BD and FD finite differences along each component vector.
|
|
"""
|
|
# Replace numpy inf, -inf and nan objects with floating point numbers
|
|
# nan --> float
|
|
self.F[np.isnan(self.F)] = np.inf
|
|
# inf, -inf --> floats
|
|
self.F = np.nan_to_num(self.F)
|
|
|
|
self.Ft = self.F[self.Ind_sorted]
|
|
self.Ftp = np.diff(self.Ft, axis=0) # FD
|
|
self.Ftm = np.diff(self.Ft[::-1], axis=0)[::-1] # BD
|
|
|
|
def sample_topo(self, ind):
|
|
# Find the position of the sample in the component axial directions
|
|
self.Xi_ind_pos = []
|
|
self.Xi_ind_topo_i = []
|
|
|
|
for i in range(self.dim):
|
|
for x, I_ind in zip(self.Ii[i], range(len(self.Ii[i]))):
|
|
if x == ind:
|
|
self.Xi_ind_pos.append(I_ind)
|
|
|
|
# Use the topo reference tables to find if point is a minimizer on
|
|
# the current axis
|
|
|
|
# First check if index is on the boundary of the sampling points:
|
|
if self.Xi_ind_pos[i] == 0:
|
|
# if boundary is in basin
|
|
self.Xi_ind_topo_i.append(self.Ftp[:, i][0] > 0)
|
|
|
|
elif self.Xi_ind_pos[i] == self.fn - 1:
|
|
# Largest value at sample size
|
|
self.Xi_ind_topo_i.append(self.Ftp[:, i][self.fn - 2] < 0)
|
|
|
|
# Find axial reference for other points
|
|
else:
|
|
Xi_ind_top_p = self.Ftp[:, i][self.Xi_ind_pos[i]] > 0
|
|
Xi_ind_top_m = self.Ftm[:, i][self.Xi_ind_pos[i] - 1] > 0
|
|
self.Xi_ind_topo_i.append(Xi_ind_top_p and Xi_ind_top_m)
|
|
|
|
if np.array(self.Xi_ind_topo_i).all():
|
|
self.Xi_ind_topo = True
|
|
else:
|
|
self.Xi_ind_topo = False
|
|
self.Xi_ind_topo = np.array(self.Xi_ind_topo_i).all()
|
|
|
|
return self.Xi_ind_topo
|
|
|
|
def minimizers_1D(self):
|
|
"""
|
|
Returns the indices of all minimizers
|
|
"""
|
|
self.minimizer_pool = []
|
|
# Note: Can implement parallelization here
|
|
for ind in range(self.fn):
|
|
min_bool = self.sample_topo(ind)
|
|
if min_bool:
|
|
self.minimizer_pool.append(ind)
|
|
|
|
self.minimizer_pool_F = self.F[self.minimizer_pool]
|
|
|
|
# Sort to find minimum func value in min_pool
|
|
self.sort_min_pool()
|
|
if not len(self.minimizer_pool) == 0:
|
|
self.X_min = self.C[self.minimizer_pool]
|
|
# If function is called again and pool is found unbreak:
|
|
else:
|
|
self.X_min = []
|
|
|
|
return self.X_min
|
|
|
|
def delaunay_triangulation(self, grow=False, n_prc=0):
|
|
if not grow:
|
|
self.Tri = spatial.Delaunay(self.C)
|
|
else:
|
|
if hasattr(self, 'Tri'):
|
|
self.Tri.add_points(self.C[n_prc:, :])
|
|
else:
|
|
self.Tri = spatial.Delaunay(self.C, incremental=True)
|
|
|
|
return self.Tri
|
|
|
|
@staticmethod
|
|
def find_neighbors_delaunay(pindex, triang):
|
|
"""
|
|
Returns the indices of points connected to ``pindex`` on the Gabriel
|
|
chain subgraph of the Delaunay triangulation.
|
|
"""
|
|
return triang.vertex_neighbor_vertices[1][
|
|
triang.vertex_neighbor_vertices[0][pindex]:
|
|
triang.vertex_neighbor_vertices[0][pindex + 1]]
|
|
|
|
def sample_delaunay_topo(self, ind):
|
|
self.Xi_ind_topo_i = []
|
|
|
|
# Find the position of the sample in the component Gabrial chain
|
|
G_ind = self.find_neighbors_delaunay(ind, self.Tri)
|
|
|
|
# Find finite deference between each point
|
|
for g_i in G_ind:
|
|
rel_topo_bool = self.F[ind] < self.F[g_i]
|
|
self.Xi_ind_topo_i.append(rel_topo_bool)
|
|
|
|
# Check if minimizer
|
|
self.Xi_ind_topo = np.array(self.Xi_ind_topo_i).all()
|
|
|
|
return self.Xi_ind_topo
|
|
|
|
def delaunay_minimizers(self):
|
|
"""
|
|
Returns the indices of all minimizers
|
|
"""
|
|
self.minimizer_pool = []
|
|
# Note: Can easily be parralized
|
|
if self.disp:
|
|
logging.info('self.fn = {}'.format(self.fn))
|
|
logging.info('self.nc = {}'.format(self.nc))
|
|
logging.info('np.shape(self.C)'
|
|
' = {}'.format(np.shape(self.C)))
|
|
for ind in range(self.fn):
|
|
min_bool = self.sample_delaunay_topo(ind)
|
|
if min_bool:
|
|
self.minimizer_pool.append(ind)
|
|
|
|
self.minimizer_pool_F = self.F[self.minimizer_pool]
|
|
|
|
# Sort to find minimum func value in min_pool
|
|
self.sort_min_pool()
|
|
if self.disp:
|
|
logging.info('self.minimizer_pool = {}'.format(self.minimizer_pool))
|
|
if not len(self.minimizer_pool) == 0:
|
|
self.X_min = self.C[self.minimizer_pool]
|
|
else:
|
|
self.X_min = [] # Empty pool breaks main routine
|
|
return self.X_min
|
|
|
|
|
|
class LMap:
|
|
def __init__(self, v):
|
|
self.v = v
|
|
self.x_l = None
|
|
self.lres = None
|
|
self.f_min = None
|
|
self.lbounds = []
|
|
|
|
|
|
class LMapCache:
|
|
def __init__(self):
|
|
self.cache = {}
|
|
|
|
# Lists for search queries
|
|
self.v_maps = []
|
|
self.xl_maps = []
|
|
self.f_maps = []
|
|
self.lbound_maps = []
|
|
self.size = 0
|
|
|
|
def __getitem__(self, v):
|
|
v = np.ndarray.tolist(v)
|
|
v = tuple(v)
|
|
try:
|
|
return self.cache[v]
|
|
except KeyError:
|
|
xval = LMap(v)
|
|
self.cache[v] = xval
|
|
|
|
return self.cache[v]
|
|
|
|
def add_res(self, v, lres, bounds=None):
|
|
v = np.ndarray.tolist(v)
|
|
v = tuple(v)
|
|
self.cache[v].x_l = lres.x
|
|
self.cache[v].lres = lres
|
|
self.cache[v].f_min = lres.fun
|
|
self.cache[v].lbounds = bounds
|
|
|
|
# Update cache size
|
|
self.size += 1
|
|
|
|
# Cache lists for search queries
|
|
self.v_maps.append(v)
|
|
self.xl_maps.append(lres.x)
|
|
self.f_maps.append(lres.fun)
|
|
self.lbound_maps.append(bounds)
|
|
|
|
def sort_cache_result(self):
|
|
"""
|
|
Sort results and build the global return object
|
|
"""
|
|
results = {}
|
|
# Sort results and save
|
|
self.xl_maps = np.array(self.xl_maps)
|
|
self.f_maps = np.array(self.f_maps)
|
|
|
|
# Sorted indexes in Func_min
|
|
ind_sorted = np.argsort(self.f_maps)
|
|
|
|
# Save ordered list of minima
|
|
results['xl'] = self.xl_maps[ind_sorted] # Ordered x vals
|
|
self.f_maps = np.array(self.f_maps)
|
|
results['funl'] = self.f_maps[ind_sorted]
|
|
results['funl'] = results['funl'].T
|
|
|
|
# Find global of all minimizers
|
|
results['x'] = self.xl_maps[ind_sorted[0]] # Save global minima
|
|
results['fun'] = self.f_maps[ind_sorted[0]] # Save global fun value
|
|
|
|
self.xl_maps = np.ndarray.tolist(self.xl_maps)
|
|
self.f_maps = np.ndarray.tolist(self.f_maps)
|
|
return results
|