Added implementation for simulated annealing (#1679)

* added hill climbing algorithm

* Shorten long lines, streamline get_neighbors()

* Update hill_climbing.py

* Update and rename optimization/hill_climbing.py to searches/hill_climbing.py

* added hill climbing algorithm

* Shorten long lines, streamline get_neighbors()

* Update hill_climbing.py

* Rebased

* added simulated annealing.py

* added final comments and test

* black formatted

* restricted search domain

Co-authored-by: Christian Clauss <cclauss@me.com>

searches/simulated_annealing.py

@@ -0,0 +1,134 @@
+# https://en.wikipedia.org/wiki/Simulated_annealing
+import math, random
+from hill_climbing import SearchProblem
+
+
+def simulated_annealing(
+    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,
+    start_temperate: float = 100,
+    rate_of_decrease: float = 0.01,
+    threshold_temp: float = 1,
+) -> SearchProblem:
+    """
+        implementation of the simulated annealing algorithm. We start with a given state, find
+            all its neighbors. Pick a random neighbor, if that neighbor improves the solution, we move
+            in that direction, if that neighbor does not improve the solution, we generate a random
+            real number between 0 and 1, if the number is within a certain range (calculated using
+            temperature) we move in that direction, else we pick another neighbor randomly and repeat the process.
+            Args:
+                search_prob: The search state at the start.
+                find_max: If True, the algorithm should find the minimum 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.
+                start_temperate: the initial temperate of the system when the program starts.
+                rate_of_decrease: the rate at which the temperate decreases in each iteration.
+                threshold_temp: the threshold temperature below which we end the search
+            Returns a search state having the maximum (or minimum) score.
+        """
+    search_end = False
+    current_state = search_prob
+    current_temp = start_temperate
+    scores = []
+    iterations = 0
+    best_state = None
+
+    while not search_end:
+        current_score = current_state.score()
+        if best_state is None or current_score > best_state.score():
+            best_state = current_state
+        scores.append(current_score)
+        iterations += 1
+        next_state = None
+        neighbors = current_state.get_neighbors()
+        while (
+            next_state is None and neighbors
+        ):  # till we do not find a neighbor that we can move to
+            index = random.randint(0, len(neighbors) - 1)  # picking a random neighbor
+            picked_neighbor = neighbors.pop(index)
+            change = picked_neighbor.score() - current_score
+
+            if (
+                picked_neighbor.x > max_x
+                or picked_neighbor.x < min_x
+                or picked_neighbor.y > max_y
+                or picked_neighbor.y < min_y
+            ):
+                continue  # neighbor outside our bounds
+
+            if not find_max:
+                change = change * -1  # incase we are finding minimum
+            if change > 0:  # improves the solution
+                next_state = picked_neighbor
+            else:
+                probabililty = (math.e) ** (
+                    change / current_temp
+                )  # probability generation function
+                if random.random() < probabililty:  # random number within probability
+                    next_state = picked_neighbor
+        current_temp = current_temp - (current_temp * rate_of_decrease)
+
+        if (
+            current_temp < threshold_temp or next_state is None
+        ):  # temperature below threshold, or
+            # couldnt find a suitaable neighbor
+            search_end = True
+        else:
+            current_state = next_state
+
+    if visualization:
+        import matplotlib.pyplot as plt
+
+        plt.plot(range(iterations), scores)
+        plt.xlabel("Iterations")
+        plt.ylabel("Function values")
+        plt.show()
+    return best_state
+
+
+if __name__ == "__main__":
+
+    def test_f1(x, y):
+        return (x ** 2) + (y ** 2)
+
+    # starting the problem with initial coordinates (12, 47)
+    prob = SearchProblem(x=12, y=47, step_size=1, function_to_optimize=test_f1)
+    local_min = simulated_annealing(
+        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()}"
+    )
+
+    # starting the problem with initial coordinates (12, 47)
+    prob = SearchProblem(x=12, y=47, step_size=1, function_to_optimize=test_f1)
+    local_min = simulated_annealing(
+        prob, find_max=True, max_x=100, min_x=5, max_y=50, min_y=-5, visualization=True
+    )
+    print(
+        "The maximum 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 = simulated_annealing(prob, find_max=False, visualization=True)
+    print(
+        "The minimum score for f(x, y) = 3*x^2 - 6*y found via hill climbing: "
+        f"{local_min.score()}"
+    )
+
+    prob = SearchProblem(x=3, y=4, step_size=1, function_to_optimize=test_f1)
+    local_min = simulated_annealing(prob, find_max=True, visualization=True)
+    print(
+        "The maximum score for f(x, y) = 3*x^2 - 6*y found via hill climbing: "
+        f"{local_min.score()}"
+    )