Mathematical Optimization
Finding the best solution among all feasible solutions
Optimization Fundamentals
Mathematical optimization is the selection of the best element from a set of available alternatives, playing a crucial role in machine learning, economics, and engineering.
Core Optimization Concepts
Unconstrained Optimization
- Gradient Descent
- Newton's Method
- Quasi-Newton Methods
Constrained Optimization
- Linear Programming
- Lagrange Multipliers
- KKT Conditions
Optimization with Python
import numpy as np
from scipy.optimize import minimize
import matplotlib.pyplot as plt
# Define objective function
def objective(x):
return (x[0] - 1)**2 + (x[1] - 2)**2
# Define gradient
def gradient(x):
return np.array([2*(x[0] - 1), 2*(x[1] - 2)])
# Initial guess
x0 = np.array([0, 0])
# Minimize using gradient descent
result = minimize(objective, x0, method='BFGS', jac=gradient)
# Visualization
x = np.linspace(-1, 3, 100)
y = np.linspace(-1, 4, 100)
X, Y = np.meshgrid(x, y)
Z = (X - 1)**2 + (Y - 2)**2
plt.figure(figsize=(10, 8))
plt.contour(X, Y, Z, levels=20)
plt.colorbar(label='Objective Value')
plt.plot(result.x[0], result.x[1], 'r*', markersize=15)
plt.xlabel('x')
plt.ylabel('y')
plt.title('Optimization Landscape')
plt.grid(True)
plt.show()
print("Optimal solution:", result.x)
print("Minimum value:", result.fun)Advanced Optimization Methods
Convex Optimization
- • Convex Functions
- • Interior Point Methods
- • Barrier Methods
Global Optimization
- • Genetic Algorithms
- • Simulated Annealing
- • Particle Swarm