Newton's method is a powerful optimization tool, but it has limitations. Modified Newton methods address these issues, improving stability and efficiency. They tweak the classic approach, making it more practical for real-world problems.
Damped and quasi-Newton methods are key modifications. They introduce step size control and Hessian approximations, respectively. These changes help tackle tricky functions and cut down on computational costs, making optimization more accessible and reliable.
Damped and Quasi-Newton Methods
Damped Newton Method and Its Advantages
Top images from around the web for Damped Newton Method and Its Advantages Top images from around the web for Damped Newton Method and Its Advantages
Damped Newton method modifies traditional Newton's method to improve convergence
Introduces a step size parameter α to control the magnitude of each iteration
Update formula: x k + 1 = x k − α k [ ∇ 2 f ( x k ) ] − 1 ∇ f ( x k ) x_{k+1} = x_k - α_k [∇^2 f(x_k)]^{-1} ∇f(x_k) x k + 1 = x k − α k [ ∇ 2 f ( x k ) ] − 1 ∇ f ( x k )
Helps prevent overshooting in regions where the function is highly nonlinear
Improves stability and convergence rate in challenging optimization problems
Utilizes line search techniques to determine optimal step size α_k at each iteration
Can handle functions with irregular curvature or multiple local minima more effectively
Quasi-Newton Methods: BFGS and DFP
Quasi-Newton methods approximate the Hessian matrix to reduce computational cost
Avoid explicit calculation of second derivatives, making them more efficient for large-scale problems
Update the approximate Hessian or its inverse at each iteration using gradient information
BFGS (Broyden -Fletcher-Goldfarb-Shanno) method updates the inverse Hessian approximation
BFGS update formula: B k + 1 = B k + y k y k T y k T s k − B k s k s k T B k s k T B k s k B_{k+1} = B_k + \frac{y_k y_k^T}{y_k^T s_k} - \frac{B_k s_k s_k^T B_k}{s_k^T B_k s_k} B k + 1 = B k + y k T s k y k y k T − s k T B k s k B k s k s k T B k
DFP (Davidon -Fletcher-Powell) method updates the Hessian approximation directly
DFP update formula: H k + 1 = H k + y k y k T y k T s k − H k s k s k T H k s k T H k s k H_{k+1} = H_k + \frac{y_k y_k^T}{y_k^T s_k} - \frac{H_k s_k s_k^T H_k}{s_k^T H_k s_k} H k + 1 = H k + y k T s k y k y k T − s k T H k s k H k s k s k T H k
Both methods maintain positive definiteness of the approximation, ensuring descent directions
BFGS generally performs better in practice due to its superior numerical stability
Quasi-Newton methods strike a balance between computational efficiency and convergence speed
Trust Region and Gauss-Newton Methods
Trust Region Methods: Concept and Implementation
Trust region methods define a region around the current point where the quadratic model is trusted
Solve a subproblem to find the step within the trust region that minimizes the quadratic model
Trust region subproblem: min p q k ( p ) = f k + g k T p + 1 2 p T B k p subject to ∥ p ∥ ≤ Δ k \min_{p} q_k(p) = f_k + g_k^T p + \frac{1}{2} p^T B_k p \quad \text{subject to} \quad \|p\| \leq \Delta_k min p q k ( p ) = f k + g k T p + 2 1 p T B k p subject to ∥ p ∥ ≤ Δ k
Adjust the trust region radius Δ_k based on the agreement between the model and the actual function
Provide robust convergence properties, especially for ill-conditioned problems
Can handle non-convex optimization problems more effectively than line search methods
Dogleg method offers an efficient approach to solving the trust region subproblem
Gauss-Newton and Levenberg-Marquardt Methods for Nonlinear Least Squares
Gauss-Newton method specializes in solving nonlinear least squares problems
Designed for minimizing sum of squared residuals: f ( x ) = 1 2 ∑ i = 1 m r i ( x ) 2 f(x) = \frac{1}{2} \sum_{i=1}^m r_i(x)^2 f ( x ) = 2 1 ∑ i = 1 m r i ( x ) 2
Approximates the Hessian using the Jacobian of the residuals: ∇ 2 f ( x ) ≈ J ( x ) T J ( x ) \nabla^2 f(x) \approx J(x)^T J(x) ∇ 2 f ( x ) ≈ J ( x ) T J ( x )
Update formula: x k + 1 = x k − [ J ( x k ) T J ( x k ) ] − 1 J ( x k ) T r ( x k ) x_{k+1} = x_k - [J(x_k)^T J(x_k)]^{-1} J(x_k)^T r(x_k) x k + 1 = x k − [ J ( x k ) T J ( x k ) ] − 1 J ( x k ) T r ( x k )
Levenberg-Marquardt method combines Gauss-Newton with trust region ideas
Adds a damping term to the approximated Hessian: [ J ( x k ) T J ( x k ) + λ k I ] [J(x_k)^T J(x_k) + \lambda_k I] [ J ( x k ) T J ( x k ) + λ k I ]
Damping parameter λ_k adjusts between Gauss-Newton and gradient descent behavior
Provides improved stability and convergence for ill-conditioned or highly nonlinear problems
Widely used in curve fitting, parameter estimation, and computer vision applications
Secant Method
Secant Method: Principles and Implementation
Secant method serves as a root-finding algorithm, applicable to optimization by finding zeros of the gradient
Approximates the derivative using two previous points instead of computing it directly
Update formula: x k + 1 = x k − f ( x k ) ( x k − x k − 1 ) f ( x k ) − f ( x k − 1 ) x_{k+1} = x_k - \frac{f(x_k)(x_k - x_{k-1})}{f(x_k) - f(x_{k-1})} x k + 1 = x k − f ( x k ) − f ( x k − 1 ) f ( x k ) ( x k − x k − 1 )
Converges superlinearly with a rate of approximately 1.618 (golden ratio)
Requires two initial points to start the iteration process
Can be extended to multidimensional optimization problems (Broyden's method )
Offers a balance between the simplicity of the bisection method and the fast convergence of Newton's method
Particularly useful when the derivative is difficult or expensive to compute
Can be combined with other methods to create hybrid algorithms with improved performance