task2

task2/common/functions.py

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+"""Function definitions with their gradients for optimization."""
+
+import math
+from abc import ABC, abstractmethod
+from typing import Tuple
+
+import numpy as np
+
+
+class Function1D(ABC):
+    """Abstract base class for 1D functions."""
+
+    name: str = "Abstract 1D Function"
+
+    @abstractmethod
+    def __call__(self, x: float) -> float:
+        """Evaluate function at x."""
+        pass
+
+    @abstractmethod
+    def gradient(self, x: float) -> float:
+        """Compute gradient (derivative) at x."""
+        pass
+
+    @property
+    @abstractmethod
+    def domain(self) -> Tuple[float, float]:
+        """Return the domain [a, b] for this function."""
+        pass
+
+
+class Function2D(ABC):
+    """Abstract base class for 2D functions."""
+
+    name: str = "Abstract 2D Function"
+
+    @abstractmethod
+    def __call__(self, x: np.ndarray) -> float:
+        """Evaluate function at point x = [x1, x2]."""
+        pass
+
+    @abstractmethod
+    def gradient(self, x: np.ndarray) -> np.ndarray:
+        """Compute gradient at point x = [x1, x2]."""
+        pass
+
+    @property
+    @abstractmethod
+    def plot_bounds(self) -> Tuple[Tuple[float, float], Tuple[float, float]]:
+        """Return bounds ((x1_min, x1_max), (x2_min, x2_max)) for plotting."""
+        pass
+
+
+class TaskFunction1D(Function1D):
+    """
+    f(x) = sqrt(x^2 + 9) / 4 + (5 - x) / 5
+
+    Derivative: f'(x) = x / (4 * sqrt(x^2 + 9)) - 1/5
+    """
+
+    name = "f(x) = √(x² + 9)/4 + (5 - x)/5"
+
+    def __call__(self, x: float) -> float:
+        return math.sqrt(x**2 + 9) / 4 + (5 - x) / 5
+
+    def gradient(self, x: float) -> float:
+        return x / (4 * math.sqrt(x**2 + 9)) - 1 / 5
+
+    @property
+    def domain(self) -> Tuple[float, float]:
+        return (-3.0, 8.0)
+
+
+class HimmelblauFunction(Function2D):
+    """
+    Himmelblau's function:
+    f(x, y) = (x^2 + y - 11)^2 + (x + y^2 - 7)^2
+
+    Has 4 identical local minima at:
+    - (3.0, 2.0)
+    - (-2.805118, 3.131312)
+    - (-3.779310, -3.283186)
+    - (3.584428, -1.848126)
+
+    Gradient:
+    ∂f/∂x = 4x(x² + y - 11) + 2(x + y² - 7)
+    ∂f/∂y = 2(x² + y - 11) + 4y(x + y² - 7)
+    """
+
+    name = "Himmelblau: (x² + y - 11)² + (x + y² - 7)²"
+
+    def __call__(self, x: np.ndarray) -> float:
+        x1, x2 = x[0], x[1]
+        return (x1**2 + x2 - 11) ** 2 + (x1 + x2**2 - 7) ** 2
+
+    def gradient(self, x: np.ndarray) -> np.ndarray:
+        x1, x2 = x[0], x[1]
+        df_dx1 = 4 * x1 * (x1**2 + x2 - 11) + 2 * (x1 + x2**2 - 7)
+        df_dx2 = 2 * (x1**2 + x2 - 11) + 4 * x2 * (x1 + x2**2 - 7)
+        return np.array([df_dx1, df_dx2])
+
+    @property
+    def plot_bounds(self) -> Tuple[Tuple[float, float], Tuple[float, float]]:
+        return ((-5.0, 5.0), (-5.0, 5.0))
+
+
+class RavineFunction(Function2D):
+    """
+    Овражная функция (эллиптический параболоид):
+    f(x, y) = x² + 20y²
+
+    Минимум в (0, 0), f(0,0) = 0
+
+    Демонстрирует "эффект оврага" - градиент почти перпендикулярен
+    направлению к минимуму, что замедляет сходимость.
+
+    Gradient:
+    ∂f/∂x = 2x
+    ∂f/∂y = 40y
+    """
+
+    name = "Овраг: f(x,y) = x² + 20y²"
+
+    def __call__(self, x: np.ndarray) -> float:
+        x1, x2 = x[0], x[1]
+        return x1**2 + 20 * x2**2
+
+    def gradient(self, x: np.ndarray) -> np.ndarray:
+        x1, x2 = x[0], x[1]
+        df_dx1 = 2 * x1
+        df_dx2 = 40 * x2
+        return np.array([df_dx1, df_dx2])
+
+    @property
+    def plot_bounds(self) -> Tuple[Tuple[float, float], Tuple[float, float]]:
+        return ((-2.0, 2.0), (-0.5, 0.5))
+
+
+# Registry of available functions
+FUNCTIONS_1D = {
+    "task": TaskFunction1D,
+}
+
+FUNCTIONS_2D = {
+    "himmelblau": HimmelblauFunction,
+    "ravine": RavineFunction,
+}