Line search methods are crucial for unconstrained optimization. They find the best direction and to move towards the minimum. These methods balance between computational efficiency and convergence speed, making them practical for many real-world problems.
Steepest descent and are two key line search approaches. , like BFGS, offer a middle ground. They use clever approximations to avoid costly calculations while still converging quickly in most cases.
Search Direction and Step Size
Fundamentals of Line Search Methods
Search direction determines the path of optimization in multidimensional space
Step size controls the magnitude of movement along the search direction
Line search methods iterate between choosing a direction and determining an appropriate step size
Optimization process continues until convergence criteria are met (gradient norm below threshold)
Steepest Descent and Newton's Method
Steepest descent method uses negative gradient as search direction
Computationally inexpensive but can be slow to converge
Formula: dk=−∇f(xk)
Performs well for functions with circular contours
Newton's method utilizes both gradient and Hessian information
Faster convergence rate, especially near the optimum
Search direction given by: dk=−Hk−1∇f(xk)
Requires computation and inversion of
Both methods update the current point using: xk+1=xk+αkdk
αk represents the step size
Determined through line search procedures (exact or inexact)
Quasi-Newton Methods
Principles and Advantages
Quasi-Newton methods approximate the Hessian matrix or its inverse
Balance between computational efficiency and convergence speed
Avoid explicit calculation and inversion of the Hessian matrix
Update formula maintains positive definiteness of approximation
rate in many practical applications
BFGS Method Implementation
Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm widely used in practice