Conjugate Gradient Algorithm Implementation for Unconstrained Optimization

In unconstrained optimization, the conjugate gradient algorithm serves as a highly effective method frequently employed for solving large linear systems and least squares problems. When implementing the conjugate gradient method, a properly structured computational program is essential. Below is a practical implementation example that can be extracted and utilized immediately:

```python

import numpy as np

def conjugate_gradient(A, b, x0, tol=1e-10, maxiter=None):

# Algorithm initialization: Set maximum iterations to vector length if unspecified

if maxiter is None:

maxiter = len(b)

# Initialize solution vector and residual

x = x0.copy()

r = b - A @ x # Compute initial residual vector

p = r.copy() # Set initial search direction

# Core iteration loop implementing conjugate gradient steps

for i in range(maxiter):

Ap = A @ p # Matrix-vector product for step calculation

# Calculate optimal step size using Fletcher-Reeves formula

alpha = np.sum(r ** 2) / np.sum(p * Ap)

x += alpha * p # Update solution vector

r -= alpha * Ap # Update residual vector

# Convergence check using residual norm

if np.linalg.norm(r) < tol:

break

# Compute beta parameter for conjugate direction update

beta = np.sum(r ** 2) / np.sum(p ** 2)

p = r + beta * p # Update search direction maintaining conjugacy

return x # Return optimized solution

```

This Python-based implementation utilizes NumPy for efficient matrix operations and follows the standard conjugate gradient algorithm structure. The function accepts coefficient matrix A, target vector b, initial guess x0, with optional tolerance and iteration limits. Key algorithmic features include automatic iteration management, residual-based convergence checking, and mathematically correct direction updates. For detailed usage instructions and parameter specifications, please refer to the accompanying documentation or contact our technical support team.