Source code for

from math import inf
import numpy as np
from typing import List, Dict

from ase import Atoms
from import chemical_symbols as periodic_table
from .. import ClusterExpansion
from ..core.local_orbit_list_generator import LocalOrbitListGenerator
from ..core.structure import Structure
from .variable_transformation import transform_parameters
from ..input_output.logging_tools import logger

    import mip
    from mip.constants import BINARY, INTEGER
except ImportError:
    raise ImportError('Python-MIP ( is required in '
                      'order to use the ground state finder.')

[docs]class GroundStateFinder: """ This class provides functionality for determining the ground states using a binary cluster expansion. This is efficiently achieved through the use of mixed integer programming (MIP) as developed by Larsen *et al.* in `Phys. Rev. Lett. 120, 256101 (2018) <>`_. This class relies on the `Python-MIP package <>`_. Python-MIP can be used together with `Gurobi <>`_, which is not open source but issues academic licenses free of charge. Pleaase note that Gurobi needs to be installed separately. The `GroundStateFinder` works also without Gurobi, but if performance is critical, Gurobi is highly recommended. Warning ------- In order to be able to use Gurobi with python-mip one must ensure that `GUROBI_HOME` should point to the installation directory (``<installdir>``):: export GUROBI_HOME=<installdir> Note ---- The current implementation only works for binary systems. Parameters ---------- cluster_expansion : ClusterExpansion cluster expansion for which to find ground states structure : Atoms atomic configuration solver_name : str, optional 'gurobi', alternatively 'grb', or 'cbc', searches for available solvers if not informed verbose : bool, optional whether to display solver messages on the screen (default: True) Example ------- The following snippet illustrates how to determine the ground state for a Au-Ag alloy. Here, the parameters of the cluster expansion are set to emulate a simple Ising model in order to obtain an example that can be run without modification. In practice, one should of course use a proper cluster expansion:: >>> from import bulk >>> from icet import ClusterExpansion, ClusterSpace >>> # prepare cluster expansion >>> # the setup emulates a second nearest-neighbor (NN) Ising model >>> # (zerolet and singlet parameters are zero; only first and second neighbor >>> # pairs are included) >>> prim = bulk('Au') >>> chemical_symbols = ['Ag', 'Au'] >>> cs = ClusterSpace(prim, cutoffs=[4.3], chemical_symbols=chemical_symbols) >>> ce = ClusterExpansion(cs, [0, 0, 0.1, -0.02]) >>> # prepare initial configuration >>> structure = prim.repeat(3) >>> # set up the ground state finder and calculate the ground state energy >>> gsf = GroundStateFinder(ce, structure) >>> ground_state = gsf.get_ground_state({'Ag': 5}) >>> print('Ground state energy:', ce.predict(ground_state)) """ def __init__(self, cluster_expansion: ClusterExpansion, structure: Atoms, solver_name: str = None, verbose: bool = True) -> None: # Check that there is only one active sublattice self._cluster_expansion = cluster_expansion self._fractional_position_tolerance = cluster_expansion.fractional_position_tolerance self.structure = structure cluster_space = self._cluster_expansion.get_cluster_space_copy() primitive_structure = cluster_space.primitive_structure self._active_sublattices = cluster_space.get_sublattices(structure).active_sublattices # Check that there are no more than two allowed species active_species = [set(subl.chemical_symbols) for subl in self._active_sublattices] if any(len(species) > 2 for species in active_species): raise NotImplementedError('Currently, systems with more than two allowed species on ' 'any sublattice are not supported.') # Check that there are no merged orbits if any(['merge' in elem for elem in cluster_space._pruning_history]): raise NotImplementedError('Currently, systems with merged orbits are not supported.') self._active_species = active_species # Define cluster functions for elements self._reverse_id_maps = [] for species in active_species: for species_map in cluster_space.species_maps: symbols = [periodic_table[n] for n in species_map] if set(symbols) == species: reverse_id_map = {1 - species_map[n]: periodic_table[n] for n in species_map} self._reverse_id_maps.append(reverse_id_map) break self._count_symbols = [reverse_id_map[1] for reverse_id_map in self._reverse_id_maps] # Generate full orbit list self._orbit_list = cluster_space.orbit_list lolg = LocalOrbitListGenerator( orbit_list=self._orbit_list, structure=Structure.from_atoms(primitive_structure), fractional_position_tolerance=self._fractional_position_tolerance) self._full_orbit_list = lolg.generate_full_orbit_list() # Transform the parameters binary_parameters = transform_parameters(primitive_structure, self._full_orbit_list, self._cluster_expansion.parameters) self._transformed_parameters = binary_parameters # Build model if solver_name is None: solver_name = '' self._model = self._build_model(structure, solver_name, verbose) # Properties that are defined when searching for a ground state self._optimization_status = None def _build_model(self, structure: Atoms, solver_name: str, verbose: bool) -> mip.Model: """ Build a Python-MIP model based on the provided structure Parameters ---------- structure atomic configuration solver_name 'gurobi', alternatively 'grb', or 'cbc', searches for available solvers if not informed verbose whether to display solver messages on the screen """ # Create cluster maps self._create_cluster_maps(structure) # Initiate MIP model model = mip.Model('CE', solver_name=solver_name) model.solver.set_mip_gap(0) # avoid stopping prematurely model.solver.set_emphasis(2) # focus on finding optimal solution model.preprocess = 2 # maximum preprocessing # Set verbosity model.verbose = int(verbose) # Spin variables (remapped) for all atoms in the structure xs = {i: model.add_var(name='atom_{}'.format(i), var_type=BINARY) for subl in self._active_sublattices for i in subl.indices} ys = [model.add_var(name='cluster_{}'.format(i), var_type=BINARY) for i in range(len(self._cluster_to_orbit_map))] # The objective function is added to 'model' first model.objective = mip.minimize(mip.xsum(self._get_total_energy(ys))) # Connect cluster variables to spin variables with cluster constraints # TODO: don't create cluster constraints for singlets constraint_count = 0 for i, cluster in enumerate(self._cluster_to_sites_map): orbit = self._cluster_to_orbit_map[i] parameter = self._transformed_parameters[orbit + 1] assert parameter != 0 if len(cluster) < 2 or parameter < 0: # no "downwards" pressure for atom in cluster: model.add_constr(ys[i] <= xs[atom], 'Decoration -> cluster {}'.format(constraint_count)) constraint_count = constraint_count + 1 if len(cluster) < 2 or parameter > 0: # no "upwards" pressure model.add_constr(ys[i] >= 1 - len(cluster) + mip.xsum(xs[atom] for atom in cluster), 'Decoration -> cluster {}'.format(constraint_count)) constraint_count = constraint_count + 1 for sym, subl in zip(self._count_symbols, self._active_sublattices): # Create slack variable slack = model.add_var(name='slackvar_{}'.format(sym), var_type=INTEGER, lb=0, ub=len(subl.indices)) # Add slack constraint model.add_constr(slack <= -1, name='{} slack'.format(sym)) # Set species constraint model.add_constr(mip.xsum([xs[i] for i in subl.indices]) + slack == -1, name='{} count'.format(sym)) # Update the model so that variables and constraints can be queried if model.solver_name.upper() in ['GRB', 'GUROBI']: model.solver.update() return model def _create_cluster_maps(self, structure: Atoms) -> None: """ Create maps that include information regarding which sites and orbits are associated with each cluster as well as the number of clusters per orbit Parameters ---------- structure atomic configuration """ # Generate full orbit list lolg = LocalOrbitListGenerator( orbit_list=self._orbit_list, structure=Structure.from_atoms(structure), fractional_position_tolerance=self._fractional_position_tolerance) full_orbit_list = lolg.generate_full_orbit_list() # Create maps of site indices and orbits for all clusters cluster_to_sites_map = [] cluster_to_orbit_map = [] for orb_index in range(len(full_orbit_list)): clusters = full_orbit_list.get_orbit(orb_index).clusters # Determine the sites and the orbit associated with each cluster for cluster in clusters: # Do not include clusters for which the parameter is 0 parameter = self._transformed_parameters[orb_index + 1] if parameter == 0: continue # Add the the list of sites and the orbit to the respective cluster maps cluster_sites = [site.index for site in cluster.lattice_sites] cluster_to_sites_map.append(cluster_sites) cluster_to_orbit_map.append(orb_index) # calculate the number of clusters per orbit nclusters_per_orbit = [cluster_to_orbit_map.count(i) for i in range(cluster_to_orbit_map[-1] + 1)] nclusters_per_orbit = [1] + nclusters_per_orbit self._cluster_to_sites_map = cluster_to_sites_map self._cluster_to_orbit_map = cluster_to_orbit_map self._nclusters_per_orbit = nclusters_per_orbit def _get_total_energy(self, cluster_instance_activities: List[int]) -> List[float]: """ Calculates the total energy using the expression based on binary variables .. math:: H({\\boldsymbol x}, {\\boldsymbol E})=E_0+ \\sum\\limits_j\\sum\\limits_{{\\boldsymbol c} \\in{\\boldsymbol C}_j}E_jy_{{\\boldsymbol c}}, where (:math:`y_{{\\boldsymbol c}}= \\prod\\limits_{i\\in{\\boldsymbol c}}x_i`). Parameters ---------- cluster_instance_activities list of cluster instance activities, (:math:`y_{{\\boldsymbol c}}`) """ E = [0.0 for _ in self._transformed_parameters] for i in range(len(cluster_instance_activities)): orbit = self._cluster_to_orbit_map[i] E[orbit + 1] = E[orbit + 1] + cluster_instance_activities[i] E[0] = 1 E = [0.0 if np.isclose(self._transformed_parameters[orbit], 0.0) else E[orbit] * self._transformed_parameters[orbit] / self._nclusters_per_orbit[orbit] for orbit in range(len(self._transformed_parameters))] return E
[docs] def get_ground_state(self, species_count: Dict[str, int] = None, max_seconds: float = inf, threads: int = 0) -> Atoms: """ Finds the ground state for a given structure and species count, which refers to the `count_species`, if provided when initializing the instance of this class, or the first species in the list of chemical symbols for the active sublattice. Parameters ---------- species_count dictionary with count for one of the species on each active sublattice. If no count is provided for a sublattice, the concentration is allowed to vary. max_seconds maximum runtime in seconds (default: inf) threads number of threads to be used when solving the problem, given that a positive integer has been provided. If set to 0 the solver default configuration is used while -1 corresponds to all available processing cores. """ if species_count is None: species_count = {} # Check that the species_count is consistent with the cluster space all_active_species = set.union(*self._active_species) for symbol in species_count: if symbol not in all_active_species: raise ValueError('The species {} is not present on any of the active sublattices' ' ({})'.format(symbol, self._active_species)) # The model is solved using python-MIPs choice of solver, which is # Gurobi, if available, and COIN-OR Branch-and-Cut, otherwise. model = self._model # Update the species counts for i, species in enumerate(self._active_species): count_symbol = self._count_symbols[i] max_count = len(self._active_sublattices[i].indices) symbols_to_add = set.intersection(set(species_count), set(species)) if len(symbols_to_add) > 1: raise ValueError('Provide counts for at most one of the species on each active ' 'sublattice ({}), not {}!'.format(self._active_species, list(species_count))) elif len(symbols_to_add) == 1: sym = symbols_to_add.pop() count = species_count[sym] if count < 0 or count > max_count: raise ValueError('The count for species {} ({}) must be a positive integer and' ' cannot exceed the number of sites on the active sublattice ' '({})'.format(sym, count, max_count)) if sym == count_symbol: xcount = count else: xcount = max_count - count max_slack = 0 else: xcount = max_slack = max_count model.constr_by_name('{} count'.format(count_symbol)).rhs = xcount model.constr_by_name('{} slack'.format(count_symbol)).rhs = max_slack # Set the number of threads model.threads = threads # Optimize the model self._optimization_status = model.optimize(max_seconds=max_seconds) # The status of the solution is printed to the screen if str(self._optimization_status) != 'OptimizationStatus.OPTIMAL': if str(self._optimization_status) == 'OptimizationStatus.FEASIBLE': logger.warning('Solution optimality not proven.') else: raise Exception('Optimization failed ({0})'.format(str(self._optimization_status))) # Each of the variables is printed with it's resolved optimum value gs = self.structure.copy() active_index_to_sublattice_map = {i: j for j, subl in enumerate(self._active_sublattices) for i in subl.indices} for v in model.vars: if 'atom' in index = int('_')[-1]) sublattice_index = active_index_to_sublattice_map[index] gs[index].symbol = self._reverse_id_maps[sublattice_index][int(v.x)] # Assert that the solution agrees with the prediction prediction = self._cluster_expansion.predict(gs) assert abs(model.objective_value - prediction) < 1e-6 return gs
@property def optimization_status(self) -> mip.OptimizationStatus: """Optimization status""" return self._optimization_status @property def model(self) -> mip.Model: """Python-MIP model""" return self._model.copy()