# https://en.wikipedia.org/wiki/Hill_climbing import math class SearchProblem: """ An interface to define search problems. The interface will be illustrated using the example of mathematical function. """ def __init__(self, x: int, y: int, step_size: int, function_to_optimize): """ The constructor of the search problem. x: the x coordinate of the current search state. y: the y coordinate of the current search state. step_size: size of the step to take when looking for neighbors. function_to_optimize: a function to optimize having the signature f(x, y). """ self.x = x self.y = y self.step_size = step_size self.function = function_to_optimize def score(self) -> int: """ Returns the output of the function called with current x and y coordinates. >>> def test_function(x, y): ... return x + y >>> SearchProblem(0, 0, 1, test_function).score() # 0 + 0 = 0 0 >>> SearchProblem(5, 7, 1, test_function).score() # 5 + 7 = 12 12 """ return self.function(self.x, self.y) def get_neighbors(self): """ Returns a list of coordinates of neighbors adjacent to the current coordinates. Neighbors: | 0 | 1 | 2 | | 3 | _ | 4 | | 5 | 6 | 7 | """ step_size = self.step_size return [ SearchProblem(x, y, step_size, self.function) for x, y in ( (self.x - step_size, self.y - step_size), (self.x - step_size, self.y), (self.x - step_size, self.y + step_size), (self.x, self.y - step_size), (self.x, self.y + step_size), (self.x + step_size, self.y - step_size), (self.x + step_size, self.y), (self.x + step_size, self.y + step_size), ) ] def __hash__(self): """ hash the string represetation of the current search state. """ return hash(str(self)) def __eq__(self, obj): """ Check if the 2 objects are equal. """ if isinstance(obj, SearchProblem): return hash(str(self)) == hash(str(obj)) return False def __str__(self): """ string representation of the current search state. >>> str(SearchProblem(0, 0, 1, None)) 'x: 0 y: 0' >>> str(SearchProblem(2, 5, 1, None)) 'x: 2 y: 5' """ return f"x: {self.x} y: {self.y}" def hill_climbing( search_prob, find_max: bool = True, max_x: float = math.inf, min_x: float = -math.inf, max_y: float = math.inf, min_y: float = -math.inf, visualization: bool = False, max_iter: int = 10000, ) -> SearchProblem: """ Implementation of the hill climbling algorithm. We start with a given state, find all its neighbors, move towards the neighbor which provides the maximum (or minimum) change. We keep doing this until we are at a state where we do not have any neighbors which can improve the solution. Args: search_prob: The search state at the start. find_max: If True, the algorithm should find the maximum else the minimum. max_x, min_x, max_y, min_y: the maximum and minimum bounds of x and y. visualization: If True, a matplotlib graph is displayed. max_iter: number of times to run the iteration. Returns a search state having the maximum (or minimum) score. """ current_state = search_prob scores = [] # list to store the current score at each iteration iterations = 0 solution_found = False visited = set() while not solution_found and iterations < max_iter: visited.add(current_state) iterations += 1 current_score = current_state.score() scores.append(current_score) neighbors = current_state.get_neighbors() max_change = -math.inf min_change = math.inf next_state = None # to hold the next best neighbor for neighbor in neighbors: if neighbor in visited: continue # do not want to visit the same state again if ( neighbor.x > max_x or neighbor.x < min_x or neighbor.y > max_y or neighbor.y < min_y ): continue # neighbor outside our bounds change = neighbor.score() - current_score if find_max: # finding max # going to direction with greatest ascent if change > max_change and change > 0: max_change = change next_state = neighbor else: # finding min # to direction with greatest descent if change < min_change and change < 0: min_change = change next_state = neighbor if next_state is not None: # we found at least one neighbor which improved the current state current_state = next_state else: # since we have no neighbor that improves the solution we stop the search solution_found = True if visualization: from matplotlib import pyplot as plt plt.plot(range(iterations), scores) plt.xlabel("Iterations") plt.ylabel("Function values") plt.show() return current_state if __name__ == "__main__": import doctest doctest.testmod() def test_f1(x, y): return (x ** 2) + (y ** 2) # starting the problem with initial coordinates (3, 4) prob = SearchProblem(x=3, y=4, step_size=1, function_to_optimize=test_f1) local_min = hill_climbing(prob, find_max=False) print( "The minimum score for f(x, y) = x^2 + y^2 found via hill climbing: " f"{local_min.score()}" ) # starting the problem with initial coordinates (12, 47) prob = SearchProblem(x=12, y=47, step_size=1, function_to_optimize=test_f1) local_min = hill_climbing( prob, find_max=False, max_x=100, min_x=5, max_y=50, min_y=-5, visualization=True ) print( "The minimum score for f(x, y) = x^2 + y^2 with the domain 100 > x > 5 " f"and 50 > y > - 5 found via hill climbing: {local_min.score()}" ) def test_f2(x, y): return (3 * x ** 2) - (6 * y) prob = SearchProblem(x=3, y=4, step_size=1, function_to_optimize=test_f1) local_min = hill_climbing(prob, find_max=True) print( "The maximum score for f(x, y) = x^2 + y^2 found via hill climbing: " f"{local_min.score()}" )