Improved convergence results for the Nelder-Mead simplex method in low dimensional spaces

2021 ◽  
Author(s):  
Aurél
Keyword(s):  
Limit Point ◽  
Simplex Method ◽  
Convergence Theorems ◽  
Convergence Results ◽  
Common Limit ◽  
Low Dimensional

Abstract We prove two results on the convergence of the Nelder-Mead simplex method. Both theorems prove the convergence of the simplex vertices to a common limit point. The first result is an improved variant of the convergence theorems of [7] and [8], while the second one proves the convergence with probability 1.

scholarly journals Convergence theorems for Banach space valued integrable multifunctions

1987 ◽  
Vol 10 (3) ◽  
pp. 433-442 ◽  
Author(s):  
Nikolaos S. Papageorgiou
Keyword(s):  
Banach Space ◽  
Convergent Sequence ◽  
Mosco Convergence ◽  
Convergence Theorems ◽  
Aumann Integral ◽  
Convergence Results ◽  
Measurable Multifunctions ◽  
Dominated Convergence ◽  
Convergence Of Sets

In this work we generalize a result of Kato on the pointwise behavior of a weakly convergent sequence in the Lebesgue-Bochner spacesLXP(Ω) (1≤p≤∞). Then we use that result to prove Fatou's type lemmata and dominated convergence theorems for the Aumann integral of Banach space valued measurable multifunctions. Analogous convergence results are also proved for the sets of integrable selectors of those multifunctions. In the process of proving those convergence theorems we make some useful observations concerning the Kuratowski-Mosco convergence of sets.

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.

Monotone convergence theorems for variational triples with applications to intermediate problems

1991 ◽  
Vol 117 (1-2) ◽  
pp. 39-58
Author(s):  
R. D. Brown
Keyword(s):  
Eigenvalue Problem ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Monotone Convergence ◽  
Approximation Methods ◽  
Convergence Theorems ◽  
Convergence Results ◽  
Increasing Sequences ◽  
Real Quadratic ◽  
General Convergence

SynopsisThe variational eigenvalue problem for a real quadratic form b with respect to a positive definite form a on a vector space V may be represented by the triple (b, a, V). Methods of intermediate problems provide approximations from below to the lower eigenvalues of (b, a, V) using monotone increasing sequences of such triples. It is shown that every such approximation method is canonically equivalent to Weinstein's method. General convergence theorems are proved for such methods. These results generalise known convergence results for increasing sequences of quadratic forms. The results are applied to some specific approximation methods and are illustrated using a differential eigenvalue problem.

scholarly journals Local and Semilocal Convergence of Nourein’s Iterative Method for Finding All Zeros of a Polynomial Simultaneously

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1801 ◽  
Author(s):  
Petko D. Proinov ◽  
Maria T. Vasileva
Keyword(s):  
Iterative Method ◽  
Local Convergence ◽  
Semilocal Convergence ◽  
Posteriori Error ◽  
A Posteriori ◽  
Convergence Theorems ◽  
A Posteriori Error ◽  
Convergence Results ◽  
Local Convergence Analysis ◽  
Appl Math

In 1977, Nourein (Intern. J. Comput. Math. 6:3, 1977) constructed a fourth-order iterative method for finding all zeros of a polynomial simultaneously. This method is also known as Ehrlich’s method with Newton’s correction because it is obtained by combining Ehrlich’s method (Commun. ACM 10:2, 1967) and the classical Newton’s method. The paper provides a detailed local convergence analysis of a well-known but not well-studied generalization of Nourein’s method for simultaneous finding of multiple polynomial zeros. As a consequence, we obtain two types of local convergence theorems as well as semilocal convergence theorems (with verifiable initial condition and a posteriori error bound) for the classical Nourein’s method. Each of the new semilocal convergence results improves the result of Petković, Petković and Rančić (J. Comput. Appl. Math. 205:1, 2007) in several directions. The paper ends with several examples that show the applicability of our semilocal convergence theorems.

Some convergence results for nonexpansive mappings in uniformly convex hyperbolic spaces

2017 ◽  
Vol 26 (3) ◽  
pp. 331-338
Author(s):  
AYNUR SAHIN ◽  
◽  
METIN BASARIR ◽  
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Hyperbolic Space ◽  
Nonexpansive Mappings ◽  
Iteration Process ◽  
Hyperbolic Spaces ◽  
Uniformly Convex ◽  
Convergence Theorems ◽  
Convergence Results ◽  
Uniformly Convex Hyperbolic Space

In this paper, we establish some strong and 4-convergence theorems of an iteration process for approximating a common fixed point of three nonexpansive mappings in a uniformly convex hyperbolic space. The results presented here extend and improve various results in the existing literature.

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 Mean convergence theorems and weak laws of large numbers for double arrays of random variables

2006 ◽  
Vol 2006 ◽  
pp. 1-15 ◽  
Author(s):  
Le Van Thanh
Keyword(s):  
Positive Integer ◽  
Random Variables ◽  
Convergence Theorems ◽  
Laws Of Large Numbers ◽  
Mean Convergence ◽  
Convergence Results ◽  
Large Numbers ◽  
The Mean ◽  
Weak Laws ◽  
Positive Integers

For a double array of random variables {Xmn, m ≥ 1, n ≥ 1}, mean convergence theorems and weak laws of large numbers are established. For the mean convergence results, conditions are provided under which ∑i=1km∑j=1lnamnij(Xij−EXij)→Lr0(0<r≤2) where {amnij;m,n,i,j≥1} are constants, and {kn,n≥1} and {ln,n≥1} are sequences of positive integers. The weak law results provide conditions for ∑i=1Tm∑j=1τnamnij(Xij−EXij)→p0 to hold where {Tm,m≥1} and {τn,n≥1} are sequences of positive integer-valued random variables. The sharpness of the results is illustrated by examples.

scholarly journals Learning in Bayesian Games with Binary Actions

2009 ◽  
Vol 9 (1) ◽  
Author(s):  
Alan Beggs
Keyword(s):  
Global Convergence ◽  
Learning Process ◽  
Adaptive Learning ◽  
Limit Point ◽  
Learning Rule ◽  
Potential Games ◽  
Unique Equilibrium ◽  
Supermodular Games ◽  
Bayesian Games ◽  
Convergence Results

This paper considers a simple adaptive learning rule in Bayesian games with binary actions where players employ threshold strategies. Global convergence results are given for supermodular games and potential games. If there is a unique equilibrium, players' strategies converge almost surely to it. Even if there is not, in potential games and in the two-player case in supermodular games, any limit point of the learning process must be an equilibrium. In particular, if equilibria are isolated, the learning process converges to one of them almost surely.

scholarly journals Convergence Results for Proximal Point Algorithm in Complete Cat(0) Space for Multivalued Mappings

2021 ◽  
Vol 27 (1) ◽  
pp. 29-47
Author(s):  
Samir Dashputre ◽  
C. Padmavati ◽  
Kavita Sakure
Keyword(s):  
Proximal Point Algorithm ◽  
Nonexpansive Mappings ◽  
Multivalued Mappings ◽  
Proximal Point ◽  
Convergence Theorems ◽  
Numerical Example ◽  
Asymptotically Nonexpansive Mappings ◽  
Convergence Results ◽  
Asymptotically Nonexpansive

In this paper, we propose the modified proximal point algorithm with the process for three nearly Lipschitzian asymptotically nonexpansive mappings and multivalued mappings in CAT(0) space under certain conditions. We prove some convergence theorems for the algorithm which was introduced by Shamshad Hussain et al. [18]. A numerical example is given to illustrate the efficiency of proximal point algorithm for supporting our result.

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