"""shgo: The simplicial homology global optimisation algorithm.""" from collections import namedtuple import time import logging import warnings import sys import numpy as np from scipy import spatial from scipy.optimize import OptimizeResult, minimize, Bounds from scipy.optimize._optimize import MemoizeJac from scipy.optimize._constraints import new_bounds_to_old from scipy.optimize._minimize import standardize_constraints from scipy._lib._util import _FunctionWrapper from scipy.optimize._shgo_lib._complex import Complex __all__ = ['shgo'] def shgo( func, bounds, args=(), constraints=None, n=100, iters=1, callback=None, minimizer_kwargs=None, options=None, sampling_method='simplicial', *, workers=1 ): """ 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 or `Bounds` Bounds for variables. There are two ways to specify the bounds: 1. Instance of `Bounds` class. 2. Sequence of ``(min, max)`` pairs for each element in `x`. args : tuple, optional Any additional fixed parameters needed to completely specify the objective function. constraints : {Constraint, dict} or List of {Constraint, dict}, optional Constraints definition. Only for COBYLA, COBYQA, SLSQP and trust-constr. See the tutorial [5]_ for further details on specifying constraints. .. note:: Only COBYLA, COBYQA, SLSQP, and trust-constr 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). COBYLA only supports inequality constraints. .. versionchanged:: 1.11.0 ``constraints`` accepts `NonlinearConstraint`, `LinearConstraint`. n : int, optional Number of sampling points used in the construction of the simplicial complex. For the default ``simplicial`` sampling method 2**dim + 1 sampling points are generated instead of the default ``n=100``. For all other specified values `n` sampling points are generated. For ``sobol``, ``halton`` and other arbitrary `sampling_methods` ``n=100`` or another specified number of sampling points are generated. 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. If not given, chosen to be one of BFGS, L-BFGS-B, SLSQP, depending on whether or not the problem has constraints or bounds. 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 : float 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 : list or bool Specify if the objective function contains symmetric variables. The search space (and therefore performance) is decreased by up to O(n!) times in the fully symmetric case. If `True` is specified then all variables will be set symmetric to the first variable. Default is set to False. E.g. f(x) = (x_1 + x_2 + x_3) + (x_4)**2 + (x_5)**2 + (x_6)**2 In this equation x_2 and x_3 are symmetric to x_1, while x_5 and x_6 are symmetric to x_4, this can be specified to the solver as:: symmetry = [0, # Variable 1 0, # symmetric to variable 1 0, # symmetric to variable 1 3, # Variable 4 3, # symmetric to variable 4 3, # symmetric to variable 4 ] 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.minimize` 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.minimize` 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 True then only the final minimizer pool will be run. Defaults to True. 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`. workers : int or map-like callable, optional Sample and run the local serial minimizations in parallel. Supply -1 to use all available CPU cores, or an int to use that many Processes (uses `multiprocessing.Pool `). Alternatively supply a map-like callable, such as `multiprocessing.Pool.map` for parallel evaluation. This evaluation is carried out as ``workers(func, iterable)``. Requires that `func` be pickleable. .. versionadded:: 1.11.0 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``, ``COBYLA``, or ``COBYQA`` 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] Hock, 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. .. [5] https://docs.scipy.org/doc/scipy/tutorial/optimize.html#constrained-minimization-of-multivariate-scalar-functions-minimize 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) >>> import numpy as np >>> 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, 23) 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, n=150, constraints=cons) >>> res message: Optimization terminated successfully. success: True fun: 29.894378159142136 funl: [ 2.989e+01] x: [ 6.355e-01 1.137e-13 3.127e-01 5.178e-02] # may vary xl: [[ 6.355e-01 1.137e-13 3.127e-01 5.178e-02]] # may vary nit: 1 nfev: 142 # may vary nlfev: 35 # may vary nljev: 5 nlhev: 0 >>> g1(res.x), g2(res.x), h1(res.x) (-5.062616992290714e-14, -2.9594104944408173e-12, 0.0) """ # if necessary, convert bounds class to old bounds if isinstance(bounds, Bounds): bounds = new_bounds_to_old(bounds.lb, bounds.ub, len(bounds.lb)) # Initiate SHGO class # use in context manager to make sure that any parallelization # resources are freed. with SHGO(func, bounds, args=args, constraints=constraints, n=n, iters=iters, callback=callback, minimizer_kwargs=minimizer_kwargs, options=options, sampling_method=sampling_method, workers=workers) as shc: # Run the algorithm, process results and test success shc.iterate_all() if not shc.break_routine: if shc.disp: logging.info("Successfully completed construction of complex.") # Test post iterations success if len(shc.LMC.xl_maps) == 0: # If sampling failed to find pool, return lowest sampled point # with a warning shc.find_lowest_vertex() shc.break_routine = True shc.fail_routine(mes="Failed to find a feasible minimizer point. " f"Lowest sampling point = {shc.f_lowest}") shc.res.fun = shc.f_lowest shc.res.x = shc.x_lowest shc.res.nfev = shc.fn shc.res.tnev = shc.n_sampled else: # Test that the optimal solutions do not violate any constraints pass # TODO # Confirm the routine ran successfully if not shc.break_routine: shc.res.message = 'Optimization terminated successfully.' shc.res.success = True # Return the final results return shc.res class SHGO: def __init__(self, func, bounds, args=(), constraints=None, n=None, iters=None, callback=None, minimizer_kwargs=None, options=None, sampling_method='simplicial', workers=1): from scipy.stats import qmc # Input checks methods = ['halton', 'sobol', 'simplicial'] if isinstance(sampling_method, str) and sampling_method not in methods: raise ValueError(("Unknown sampling_method specified." " Valid methods: {}").format(', '.join(methods))) # Split obj func if given with Jac try: if ((minimizer_kwargs['jac'] is True) and (not callable(minimizer_kwargs['jac']))): self.func = MemoizeJac(func) jac = self.func.derivative minimizer_kwargs['jac'] = jac func = self.func # .fun else: self.func = func # Normal definition of objective function except (TypeError, KeyError): self.func = func # Normal definition of objective function # Initiate class self.func = _FunctionWrapper(func, args) 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 " f"{', '.join(str(b) for b in bnderr)}.") self.bounds = abound # Constraints # Process constraint dict sequence: self.constraints = constraints if constraints is not None: self.min_cons = constraints self.g_cons = [] self.g_args = [] # shgo internals deals with old-style constraints # self.constraints is used to create Complex, so need # to be stored internally in old-style. # `minimize` takes care of normalising these constraints # for slsqp/cobyla/cobyqa/trust-constr. self.constraints = standardize_constraints( constraints, np.empty(self.dim, float), 'old' ) for cons in self.constraints: if cons['type'] in ('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 = {'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'].lower() in ('slsqp', 'cobyla', 'cobyqa', 'trust-constr') 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 = True # 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 = None # Algorithm functionality self.infty_cons_sampl = True self.local_iter = False # 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', 'catol'], 'cobyqa': ['bounds', 'constraints', 'feasibility_tol'], 'slsqp': ['jac', 'bounds', 'constraints'], 'dogleg': ['jac', 'hess'], 'trust-ncg': ['jac', 'hess', 'hessp'], 'trust-krylov': ['jac', 'hess', 'hessp'], 'trust-exact': ['jac', 'hess'], 'trust-constr': ['jac', 'hess', 'hessp', 'constraints'], } 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 completed self.n = n # Sampling points per iteration self.nc = 0 # 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 no. of sampling points already generated self.fn = 0 # Number of feasible sampling points evaluations performed self.hgr = 0 # Homology group rank # Initially attempt to build the triangulation incrementally: self.qhull_incremental = True # Default settings if no sampling criteria. if (self.n is None) and (self.iters is None) \ and (sampling_method == 'simplicial'): self.n = 2 ** self.dim + 1 self.nc = 0 # self.n if self.iters is None: self.iters = 1 if (self.n is None) and not (sampling_method == 'simplicial'): self.n = self.n = 100 self.nc = 0 # self.n if (self.n == 100) and (sampling_method == 'simplicial'): self.n = 2 ** self.dim + 1 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: # Initialise sampling Complex and function cache # Note that sfield_args=() since args are already wrapped in self.func # using the_FunctionWrapper class. self.HC = Complex(dim=self.dim, domain=self.bounds, sfield=self.func, sfield_args=(), symmetry=self.symmetry, constraints=self.constraints, workers=workers) # Choose complex constructor if sampling_method == 'simplicial': self.iterate_complex = self.iterate_hypercube self.sampling_method = sampling_method elif sampling_method in ['halton', 'sobol'] or \ not isinstance(sampling_method, str): self.iterate_complex = self.iterate_delaunay # Sampling method used if sampling_method in ['halton', 'sobol']: if sampling_method == 'sobol': self.n = int(2 ** np.ceil(np.log2(self.n))) # self.n #TODO: Should always be self.n, this is # unacceptable for shgo, check that nfev behaves as # expected. self.nc = 0 self.sampling_method = 'sobol' self.qmc_engine = qmc.Sobol(d=self.dim, scramble=False, seed=0) else: self.sampling_method = 'halton' self.qmc_engine = qmc.Halton(d=self.dim, scramble=True, seed=0) def sampling_method(n, d): return self.qmc_engine.random(n) else: # A user defined sampling method: self.sampling_method = 'custom' self.sampling = self.sampling_custom self.sampling_function = sampling_method # F(n, d) # 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 """ # Update 'options' dict passed to optimize.minimize # Do this first so we don't mutate `options` below. self.minimizer_kwargs['options'].update(options) # Ensure that 'jac', 'hess', and 'hessp' are passed directly to # `minimize` as keywords, not as part of its 'options' dictionary. for opt in ['jac', 'hess', 'hessp']: if opt in self.minimizer_kwargs['options']: self.minimizer_kwargs[opt] = ( self.minimizer_kwargs['options'].pop(opt)) # Default settings: self.minimize_every_iter = options.get('minimize_every_iter', True) # 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 = options.get('symmetry', False) if self.symmetry: self.symmetry = [0, ]*len(self.bounds) else: self.symmetry = None # Algorithm functionality # Only evaluate a few of the best candidates self.local_iter = options.get('local_iter', False) self.infty_cons_sampl = options.get('infty_constraints', True) # Feedback self.disp = options.get('disp', False) def __enter__(self): return self def __exit__(self, *args): return self.HC.V._mapwrapper.__exit__(*args) # Iteration properties # Main construction loop: def iterate_all(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: logging.info('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 self.fn = self.HC.V.nfev def find_minima(self): """ Construct the minimizer pool, map the minimizers to local minima and sort the results into a global return object. """ if self.disp: logging.info('Searching for minimizer pool...') 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() if self.disp: logging.info(f"Minimiser pool = SHGO.X_min = {self.X_min}") def find_lowest_vertex(self): # Find the lowest objective function value on one of # the vertices of the simplicial complex self.f_lowest = np.inf for x in self.HC.V.cache: if self.HC.V[x].f < self.f_lowest: if self.disp: logging.info(f'self.HC.V[x].f = {self.HC.V[x].f}') self.f_lowest = self.HC.V[x].f self.x_lowest = self.HC.V[x].x_a for lmc in self.LMC.cache: if self.LMC[lmc].f_min < self.f_lowest: self.f_lowest = self.LMC[lmc].f_min self.x_lowest = self.LMC[lmc].x_l if self.f_lowest == np.inf: # no feasible point self.f_lowest = None self.x_lowest = None # Stopping criteria functions: def finite_iterations(self): mi = min(x for x in [self.iters, self.maxiter] if x is not None) if self.disp: logging.info(f'Iterations done = {self.iters_done} / {mi}') if self.iters is not None: if self.iters_done >= (self.iters): self.stop_global = True if self.maxiter is not None: # Stop for infeasible sampling if self.iters_done >= (self.maxiter): self.stop_global = True return self.stop_global def finite_fev(self): # Finite function evals in the feasible domain if self.disp: logging.info(f'Function evaluations done = {self.fn} / {self.maxfev}') 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.disp: logging.info(f'Sampling evaluations done = {self.n_sampled} ' f'/ {self.maxev}') if self.n_sampled >= self.maxev: self.stop_global = True def finite_time(self): if self.disp: logging.info(f'Time elapsed = {time.time() - self.init} ' f'/ {self.maxtime}') 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 self.find_lowest_vertex() if self.disp: logging.info(f'Lowest function evaluation = {self.f_lowest}') logging.info(f'Specified minimum = {self.f_min_true}') # If no feasible point was return from test if self.f_lowest is None: return self.stop_global # Function to stop algorithm at specified percentage error: 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( f"A much lower value than expected f* = {self.f_min_true} " f"was found f_lowest = {self.f_lowest}", stacklevel=3 ) if pe <= self.f_tol: self.stop_global = True return self.stop_global def finite_homology_growth(self): """ Stop the algorithm if homology group rank did not grow in iteration. """ 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 if self.disp: logging.info(f'Current homology growth = {self.hgrd} ' f' (minimum growth = {self.minhgrd})') 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() return self.stop_global 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.disp: logging.info('Constructing and refining simplicial complex graph ' 'structure') if self.n is None: self.HC.refine_all() self.n_sampled = self.HC.V.size() # nevs counted else: self.HC.refine(self.n) self.n_sampled += self.n if self.disp: logging.info('Triangulation completed, evaluating all constraints ' 'and objective function values.') # Re-add minimisers to complex if len(self.LMC.xl_maps) > 0: for xl in self.LMC.cache: v = self.HC.V[xl] v_near = v.star() for v in v.nn: v_near = v_near.union(v.nn) # Reconnect vertices to complex # if self.HC.connect_vertex_non_symm(tuple(self.LMC[xl].x_l), # near=v_near): # continue # else: # If failure to find in v_near, then search all vertices # (very expensive operation: # self.HC.connect_vertex_non_symm(tuple(self.LMC[xl].x_l) # ) # Evaluate all constraints and functions self.HC.V.process_pools() if self.disp: logging.info('Evaluations completed.') # feasible sampling points counted by the triangulation.py routines self.fn = self.HC.V.nfev 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) # Add sampled points to a triangulation, construct self.Tri if self.disp: logging.info(f'self.n = {self.n}') logging.info(f'self.nc = {self.nc}') logging.info('Constructing and refining simplicial complex graph ' 'structure from sampling points.') if self.dim < 2: self.Ind_sorted = np.argsort(self.C, axis=0) self.Ind_sorted = self.Ind_sorted.flatten() tris = [] for ind, ind_s in enumerate(self.Ind_sorted): if ind > 0: tris.append(self.Ind_sorted[ind - 1:ind + 1]) tris = np.array(tris) # Store 1D triangulation: self.Tri = namedtuple('Tri', ['points', 'simplices'])(self.C, tris) self.points = {} else: if self.C.shape[0] > self.dim + 1: # Ensure a simplex can be built self.delaunay_triangulation(n_prc=self.n_prc) self.n_prc = self.C.shape[0] if self.disp: logging.info('Triangulation completed, evaluating all ' 'constraints and objective function values.') if hasattr(self, 'Tri'): self.HC.vf_to_vv(self.Tri.points, self.Tri.simplices) # Process all pools # Evaluate all constraints and functions if self.disp: logging.info('Triangulation completed, evaluating all constraints ' 'and objective function values.') # Evaluate all constraints and functions self.HC.V.process_pools() if self.disp: logging.info('Evaluations completed.') # feasible sampling points counted by the triangulation.py routines self.fn = self.HC.V.nfev self.n_sampled = self.nc # nevs counted in triangulation return # Hypercube minimizers def minimizers(self): """ Returns the indexes of all minimizers """ self.minimizer_pool = [] # Note: Can implement parallelization here for x in self.HC.V.cache: in_LMC = False if len(self.LMC.xl_maps) > 0: for xlmi in self.LMC.xl_maps: if np.all(np.array(x) == np.array(xlmi)): in_LMC = True if in_LMC: continue if self.HC.V[x].minimiser(): if self.disp: logging.info('=' * 60) logging.info(f'v.x = {self.HC.V[x].x_a} is minimizer') logging.info(f'v.f = {self.HC.V[x].f} is minimizer') 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(f'x = {vn.x} || f = {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 minimisation # Minimiser pool processing def minimise_pool(self, force_iter=False): """ This processing method can optionally minimise only the best candidate solutions in the minimiser pool Parameters ---------- force_iter : int Number of starting minimizers to process (can be specified 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) while not self.stop_l_iter: # Global stopping criteria: self.stopping_criteria() # Note first iteration is outside loop: if force_iter: force_iter -= 1 if force_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(f'cbounds found for v_min.x_a = {v_min.x_a}') logging.info(f'cbounds = {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(f'Vertex minimiser maps = {self.LMC.v_maps}') if self.LMC[x_min].lres is not None: logging.info(f'Found self.LMC[x_min].lres = ' f'{self.LMC[x_min].lres}') return self.LMC[x_min].lres if self.callback is not None: logging.info(f'Callback for minimizer starting at {x_min}:') if self.disp: logging.info(f'Starting minimization at {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 logging.info(self.minimizer_kwargs['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 logging.info(self.minimizer_kwargs['bounds']) if self.disp and 'bounds' in self.minimizer_kwargs: logging.info('bounds in kwarg:') logging.info(self.minimizer_kwargs['bounds']) # Local minimization using scipy.optimize.minimize: lres = minimize(self.func, x_min, **self.minimizer_kwargs) if self.disp: logging.info(f'lres = {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: logging.info('Generating sampling points') self.sampling(self.nc, self.dim) if len(self.LMC.xl_maps) > 0: self.C = np.vstack((self.C, np.array(self.LMC.xl_maps))) 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.n_sampled = self.nc 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 if self.n_sampled == 0: self.C = self.sampling_function(n, dim) else: 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): # C.shape = (Z, dim) where Z is the number of sampling points to # evaluate and dim is the dimensionality of the problem. # the constraint function may not be vectorised so have to step # through each sampling point sequentially. feasible = np.array( [np.all(g(x_C, *self.g_args[ind]) >= 0.0) for x_C in self.C], dtype=bool ) self.C = self.C[feasible] 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: logging.info(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 delaunay_triangulation(self, n_prc=0): if hasattr(self, 'Tri') and self.qhull_incremental: # TODO: Uncertain if n_prc needs to add len(self.LMC.xl_maps) # in self.sampled_surface self.Tri.add_points(self.C[n_prc:, :]) else: try: self.Tri = spatial.Delaunay(self.C, incremental=self.qhull_incremental, ) except spatial.QhullError: if str(sys.exc_info()[1])[:6] == 'QH6239': logging.warning('QH6239 Qhull precision error detected, ' 'this usually occurs when no bounds are ' 'specified, Qhull can only run with ' 'handling cocircular/cospherical points' ' and in this case incremental mode is ' 'switched off. The performance of shgo ' 'will be reduced in this mode.') self.qhull_incremental = False self.Tri = spatial.Delaunay(self.C, incremental= self.qhull_incremental) else: raise return self.Tri 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.xl_maps_set = set() self.f_maps = [] self.lbound_maps = [] self.size = 0 def __getitem__(self, v): try: v = np.ndarray.tolist(v) except TypeError: pass 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.xl_maps_set.add(tuple(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