Related Experiment Video
Updated: Nov 12, 2025
13:07
Convergent Polishing: A Simple, Rapid, Full Aperture Polishing Process of High Quality Optical Flats & Spheres
Published on: December 1, 2014
11.4K
New Results on Superlinear Convergence of Classical Quasi-Newton Methods
Anton Rodomanov1, Yurii Nesterov2
1ICTEAM, Catholic University of Louvain, Louvain-la-Neuve, Belgium.
Journal of Optimization Theory and Applications
|March 22, 2021
Summary
We improved convergence rate estimates for quasi-Newton methods. The Broyden-Fletcher-Goldfarb-Shanno method
Area of Science:
- Numerical analysis
- Optimization algorithms
Background:
- Quasi-Newton methods are essential for solving nonlinear equations.
- Classical methods like Broyden's class have known convergence properties.
- Existing estimates for convergence rates can be improved.
Purpose of the Study:
- To provide a new theoretical analysis of local superlinear convergence for convex quasi-Newton methods.
- To improve upon existing convergence rate estimates.
- To characterize the convergence rate of specific methods within Broyden's class.
Main Methods:
- Theoretical analysis of local superlinear convergence.
- Focus on methods within the convex Broyden class.
- Derivation of new convergence rate bounds.
Main Results:
- Significant improvement in known convergence rate estimates.
- Demonstration that the Broyden-Fletcher-Goldfarb-Shanno method's rate depends on problem dimensionality and the logarithm of the condition number.
Conclusions:
- The new analysis offers tighter bounds on convergence rates.
- The findings provide a deeper understanding of the efficiency of quasi-Newton methods.
- This work advances the theoretical foundation for optimization algorithms.
Related Concept Videos
01:29Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving
164
Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a...
In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a...
164
01:29Application of Nonlinear Inequalities
60
A nonlinear inequality describes a comparison involving an expression that curves or behaves more complexly than a straight line. These inequalities often appear in forms that include squares, products, or variables in the denominator.To solve such an inequality, one starts by rewriting it so that zero appears on one side. For example, the inequality: can be factored as: This form makes it easier to identify the values that cause the expression to equal zero. In this case, the key values...
60
01:18Newtonian Fluid: Problem Solving
640
Newtonian fluids exhibit a constant viscosity, meaning their shear stress and shear strain rate are directly proportional. This property ensures a predictable and stable response to applied forces, maintaining a linear relationship between force and flow. Examples include water, air, and light oils, consistently demonstrating this proportional behavior regardless of external conditions.
A velocity gradient forms within the fluid when a Newtonian fluid is placed between two parallel plates, with...
A velocity gradient forms within the fluid when a Newtonian fluid is placed between two parallel plates, with...
640
01:25Introduction to Nonlinear Inequalities
59
Linear and nonlinear inequalities are fundamental for analyzing variable relationships and identifying ranges satisfying specific conditions. A linear inequality involves variables raised only to the first power, resulting in a straight-line graph. This line partitions the coordinate plane into two distinct regions: one that satisfies the inequality and one that does not. Each region represents a set of solutions where the linear relationship holds true under the specified constraint.Nonlinear...
59
01:06Divergence and Stokes' Theorems
2.9K
The divergence and Stokes' theorems are a variation of Green's theorem in a higher dimension. They are also a generalization of the fundamental theorem of calculus. The divergence theorem and Stokes' theorem are in a way similar to each other; The divergence theorem relates to the dot product of a vector, while Stokes' theorem relates to the curl of a vector. Many applications in physics and engineering make use of the divergence and Stokes' theorems, enabling us to write...
2.9K
01:21Convergence of Fourier Series
269
The Fourier series is a powerful mathematical tool for representing periodic signals as an infinite sum of complex exponentials. In practice, this infinite series is truncated to a finite number of terms, yielding a partial sum. This truncation makes the approximation of the signal feasible but introduces certain challenges, particularly near discontinuities, known as the Gibbs phenomenon.
The Gibbs phenomenon refers to the persistent oscillations and overshoots that occur near discontinuities...
The Gibbs phenomenon refers to the persistent oscillations and overshoots that occur near discontinuities...
269