scholarly journals Convergence of the Nelder-Mead method

2021 ◽  
Author(s):  
Aurél Galántai
Keyword(s):  
Convergence Theorem ◽  
Simplex Method ◽  
Failure Modes ◽  
Matrix Form ◽  
Infinite Matrix ◽  
Matrix Products ◽  
Low Dimensional ◽  
General Convergence

AbstractWe develop a matrix form of the Nelder-Mead simplex method and show that its convergence is related to the convergence of infinite matrix products. We then characterize the spectra of the involved matrices necessary for the study of convergence. Using these results, we discuss several examples of possible convergence or failure modes. Then, we prove a general convergence theorem for the simplex sequences generated by the method. The key assumption of the convergence theorem is proved in low-dimensional spaces up to 8 dimensions.

scholarly journals Convergence theorems for the Nelder-Mead method

2020 ◽  
Vol 15 (2) ◽  
pp. 115-133
Author(s):  
Aurél Galántai
Keyword(s):  
Convergence Theorem ◽  
Matrix Form ◽  
Convergence Theorems ◽  
Low Dimensional ◽  
General Convergence

We develop a matrix form of the Nelder-Mead method and after discussing the concept of convergence we prove a general convergence theorem. The new theorem is demonstrated in low dimensional spaces.

scholarly journals Infinite Matrix Products and the Representation of the Matrix Gamma Function

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
J.-C. Cortés ◽  
L. Jódar ◽  
Francisco J. Solís ◽  
Roberto Ku-Carrillo
Keyword(s):  
Gamma Function ◽  
Infinite Product ◽  
Infinite Matrix ◽  
Convergence Results ◽  
Matrix Products ◽  
The Matrix ◽  
Definition Of

We introduce infinite matrix products including some of their main properties and convergence results. We apply them in order to extend to the matrix scenario the definition of the scalar gamma function given by an infinite product due to Weierstrass. A limit representation of the matrix gamma function is also provided.

Quadratic transformations: a model for population growth. II

1970 ◽  
Vol 2 (02) ◽  
pp. 179-228 ◽  
Author(s):  
Harry Kesten
Keyword(s):  
Population Growth ◽  
Convergence Theorem ◽  
Conditional Expectation ◽  
High Probability ◽  
Random Variables ◽  
Convergence Results ◽  
General Convergence

In this last part theFn(i) andMn(i) are considered as random variables whose distributions are described by the model and various mating rules of Section 2. Several convergence results will be proved for those specific mating rules, but we begin with the more general convergence theorem 6.1. The proof of this theorem brings out the basic idea of this section, namely that whenFnandMnare large,Fn + 1(i) andMn + 1(i) will, with high probability, be close to a certain function ofFn(·) andMn(·) (roughly the conditional expectation ofFn+1(i) andMn + 1(i) givenFn(·) andMn(·)).

scholarly journals A general convergence theorem for multiple-set split feasibility problem in Hilbert spaces

2015 ◽  
Vol 31 (3) ◽  
pp. 349-357
Author(s):  
ABDUL RAHIM KHAN ◽  
◽  
MUJAHID ABBAS ◽  
YEKINI SHEHU ◽  
◽  
...  
Keyword(s):  
Strong Convergence ◽  
Hilbert Spaces ◽  
Convergence Theorem ◽  
Split Feasibility Problem ◽  
Contractive Mapping ◽  
Convergence Result ◽  
Feasibility Problem ◽  
Infinite Dimensional ◽  
Split Feasibility ◽  
General Convergence

We establish strong convergence result of split feasibility problem for a family of quasi-nonexpansive multi-valued mappings and a total asymptotically strict pseudo-contractive mapping in infinite dimensional Hilbert spaces.

An Adaptive Population-based Simplex Method for Continuous Optimization

2016 ◽  
Vol 7 (4) ◽  
pp. 23-51 ◽  
Author(s):  
Mahamed G.H. Omran ◽  
Maurice Clerc
Keyword(s):  
Simplex Method ◽  
Continuous Optimization ◽  
Optimization Method ◽  
Population Based ◽  
Function Optimization ◽  
Test Functions ◽  
Dimensional Simplex ◽  
Low Dimensional ◽  
Better Than ◽  
Real World Problems

This paper proposes a new population-based simplex method for continuous function optimization. The proposed method, called Adaptive Population-based Simplex (APS), is inspired by the Low-Dimensional Simplex Evolution (LDSE) method. LDSE is a recent optimization method, which uses the reflection and contraction steps of the Nelder-Mead Simplex method. Like LDSE, APS uses a population from which different simplexes are selected. In addition, a local search is performed using a hyper-sphere generated around the best individual in a simplex. APS is a tuning-free approach, it is easy to code and easy to understand. APS is compared with five state-of-the-art approaches on 23 functions where five of them are quasi-real-world problems. The experimental results show that APS generally performs better than the other methods on the test functions. In addition, a scalability study has been conducted and the results show that APS can work well with relatively high-dimensional problems.

scholarly journals XI. On the series of Strum and Liouville , as derived from a pair of fundamental integral equations instead of a differential equation

1912 ◽  
Vol 211 (471-483) ◽  
pp. 411-432 ◽  
Keyword(s):  
Differential Equation ◽  
Integral Equations ◽  
Convergence Theorem ◽  
Approximate Solutions ◽  
General Series ◽  
General Convergence

The series of Liouville and Strum are generally treated by means of approximate solutions of the fundamental differential equation, these approximations being valid when certain functions involved in the differential equation have differential co­efficients. The object of the present paper is to relax this restriction, and for this purpose integral equations are used in place of a differential equation, and an approximation is investigated (§§ 4—11) depending on a function which is constant throughout each of a system of sub-intervals. In §§ 15-18 the results are applied, by help of Hobson's general convergence theorem, to that one of the Liouville series which is usually valid at the two ends of the fundamental interval, and in §§ 19-22 to the more general series discussed by me in ‘Proc. L. M. S.,’ ser. 2, vol. 3, pp. 83-103.

Sign in / Sign up

Export Citation Format

Share Document