scholarly journals A bridging method for global optimization

1999 ◽  
Vol 41 (1) ◽  
pp. 41-57 ◽  
Author(s):  
Y. Liu ◽  
K. L. Teo
Keyword(s):  
Global Optimization ◽  
Differentiable Function ◽  
Numerical Solutions ◽  
Finite Interval ◽  
Optimization Problems ◽  
Optimal Solutions ◽  
Numerical Examples ◽  
One Dimensional ◽  
Differentiable Functions ◽  
Global Optimal

AbstractIn this paper a bridging method is introduced for numerical solutions of one-dimensional global optimization problems where a continuously differentiable function is to be minimized over a finite interval which can be given either explicitly or by constraints involving continuously differentiable functions. The concept of a bridged function is introduced. Some properties of the bridged function are given. On this basis, several bridging algorithm are developed for the computation of global optimal solutions. The algorithms are demonstrated by solving several numerical examples.

Adaptive Grouping Brain Storm Optimization for Multimodal Optimization Problems

2021 ◽  
Vol 12 (4) ◽  
pp. 81-100
Author(s):  
Yao Peng ◽  
Zepeng Shen ◽  
Shiqi Wang
Keyword(s):  
Optimization Problem ◽  
Optimization Problems ◽  
Optimal Algorithm ◽  
Peak Detection ◽  
Multimodal Optimization ◽  
Optimal Solutions ◽  
Grouping Strategy ◽  
Different Dimensions ◽  
Global Optimal ◽  
Localization Ability

Multimodal optimization problem exists in multiple global and many local optimal solutions. The difficulty of solving these problems is finding as many local optimal peaks as possible on the premise of ensuring global optimal precision. This article presents adaptive grouping brainstorm optimization (AGBSO) for solving these problems. In this article, adaptive grouping strategy is proposed for achieving adaptive grouping without providing any prior knowledge by users. For enhancing the diversity and accuracy of the optimal algorithm, elite reservation strategy is proposed to put central particles into an elite pool, and peak detection strategy is proposed to delete particles far from optimal peaks in the elite pool. Finally, this article uses testing functions with different dimensions to compare the convergence, accuracy, and diversity of AGBSO with BSO. Experiments verify that AGBSO has great localization ability for local optimal solutions while ensuring the accuracy of the global optimal solutions.

Genetic Algorithm and Other Meta-Heuristics

2005 ◽  
pp. 144-173 ◽  
Author(s):  
Bernard K.S. Cheung
Keyword(s):  
Genetic Algorithm ◽  
Large Scale ◽  
Optimization Problems ◽  
Optimal Solution ◽  
Schema Theory ◽  
Heuristic Methods ◽  
Optimal Solutions ◽  
Corporate Globalization ◽  
Global Optimal ◽  
Mutation And Selection

Genetic algorithms have been applied in solving various types of large-scale, NP-hard optimization problems. Many researchers have been investigating its global convergence properties using Schema Theory, Markov Chain, etc. A more realistic approach, however, is to estimate the probability of success in finding the global optimal solution within a prescribed number of generations under some function landscapes. Further investigation reveals that its inherent weaknesses that affect its performance can be remedied, while its efficiency can be significantly enhanced through the design of an adaptive scheme that integrates the crossover, mutation and selection operations. The advance of Information Technology and the extensive corporate globalization create great challenges for the solution of modern supply chain models that become more and more complex and size formidable. Meta-heuristic methods have to be employed to obtain near optimal solutions. Recently, a genetic algorithm has been reported to solve these problems satisfactorily and there are reasons for this.

scholarly journals A Note on Double Conformable Laplace Transform Method and Singular One Dimensional Conformable Pseudohyperbolic Equations

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 949 ◽  
Author(s):  
Hassan Eltayeb ◽  
Said Mesloub ◽  
Yahya T. Abdalla ◽  
Adem Kılıçman
Keyword(s):  
Present Method ◽  
Decomposition Method ◽  
Numerical Solutions ◽  
High Accuracy ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Numerical Examples ◽  
One Dimensional ◽  
Linear And Nonlinear ◽  
Laplace Decomposition Method

The purpose of this article is to obtain the exact and approximate numerical solutions of linear and nonlinear singular conformable pseudohyperbolic equations and conformable coupled pseudohyperbolic equations through the conformable double Laplace decomposition method. Further, the numerical examples were provided in order to demonstrate the efficiency, high accuracy, and the simplicity of present method.

On Tolerant Fuzzyc-Means Clustering

2009 ◽  
Vol 13 (4) ◽  
pp. 421-428 ◽  
Author(s):  
Yukihiro Hamasuna ◽  
◽  
Yasunori Endo ◽  
Sadaaki Miyamoto ◽  
Keyword(s):  
Optimization Problems ◽  
Clustering Algorithms ◽  
Fuzzy Classification ◽  
Optimal Solutions ◽  
Numerical Examples ◽  
New Type ◽  
Original Concept ◽  
Classification Function

This paper presents a new type of clustering algorithms by using a tolerance vector called tolerant fuzzyc-means clustering (TFCM). In the proposed algorithms, the new concept of tolerance vector plays very important role. In the original concept of tolerance, a tolerance vector attributes to each data. This concept is developed to handle data flexibly, that is, a tolerance vector attributes not only to each data but also each cluster. Using the new concept, we can consider the influence of clusters to each data by the tolerance. First, the new concept of tolerance is introduced into optimization problems based on conventional fuzzyc-means clustering (FCM). Second, the optimization problems with tolerance are solved by using Karush-Kuhn-Tucker conditions. Third, new clustering algorithms are constructed based on the explicit optimal solutions of the optimization problems. Finally, the effectiveness of the proposed algorithms is verified through numerical examples by fuzzy classification function.

scholarly journals MEEF: A Minimum-Elimination-Escape Function Method for Multimodal Optimization Problems

2015 ◽  
Vol 2015 ◽  
pp. 1-16
Author(s):  
Lei Fan ◽  
Yuping Wang ◽  
Xiyang Liu ◽  
Liping Jia
Keyword(s):  
Function Method ◽  
Optimization Problems ◽  
Auxiliary Function ◽  
Local Optimum ◽  
Multimodal Optimization ◽  
Optimal Solutions ◽  
Mexican Hat ◽  
Global Optimal ◽  
Elimination Function ◽  
Made In

Auxiliary function methods provide us effective and practical ideas to solve multimodal optimization problems. However, improper parameter settings often cause troublesome effects which might lead to the failure of finding global optimal solutions. In this paper, a minimum-elimination-escape function method is proposed for multimodal optimization problems, aiming at avoiding the troublesome “Mexican hat” effect and reducing the influence of local optimal solutions. In the proposed method, the minimum-elimination function is constructed to decrease the number of local optimum first. Then, a minimum-escape function is proposed based on the minimum-elimination function, in which the current minimal solution will be converted to the unique global maximal solution of the minimum-escape function. The minimum-escape function is insensitive to its unique but easy to adopt parameter. At last, an minimum-elimination-escape function method is designed based on these two functions. Experiments on 19 widely used benchmarks are made, in which influences of the parameter and different initial points are analyzed. Comparisons with 11 existing methods indicate that the performance of the proposed algorithm is positive and effective.

On the Sum of Differentiable Functions

1962 ◽  
Vol 58 (2) ◽  
pp. 225-228
Author(s):  
H. T. Croft
Keyword(s):  
Differentiable Function ◽  
Finite Interval ◽  
Differentiable Functions ◽  
Continuous Differentiable Function

Ryll-Nardzewski has proposed the following problem (New Scottish Book, no. 119). If fn(x) are continuous, differentiable* functions in a closed finite interval, do there always exist constants cn (no cn = 0) (depending on the ), such that converges and is also a continuous differentiable function?

scholarly journals Understanding measure-driven algorithms solving irreversibly ill-conditioned problems

2021 ◽  
Author(s):  
Jakub Sawicki ◽  
Marcin Łoś ◽  
Maciej Smołka ◽  
Robert Schaefer
Keyword(s):  
Global Optimization ◽  
Markov Chain ◽  
Dynamic Systems ◽  
Positive Measure ◽  
Numerical Solutions ◽  
Optimization Problems ◽  
Real Life ◽  
Population Based ◽  
Ergodic Markov Chain ◽  
Markov Chain Models

AbstractThe paper helps to understand the essence of stochastic population-based searches that solve ill-conditioned global optimization problems. This condition manifests itself by presence of lowlands, i.e., connected subsets of minimizers of positive measure, and inability to regularize the problem. We show a convenient way to analyze such search strategies as dynamic systems that transform the sampling measure. We can draw informative conclusions for a class of strategies with a focusing heuristic. For this class we can evaluate the amount of information about the problem that can be gathered and suggest ways to verify stopping conditions. Next, we show the Hierarchic Memetic Strategy coupled with Multi-Winner Evolutionary Algorithm (HMS/MWEA) that follow the ideas from the first part of the paper. We introduce a complex, ergodic Markov chain of their dynamics and prove an asymptotic guarantee of success. Finally, we present numerical solutions to ill-conditioned problems: two benchmarks and a real-life engineering one, which show the strategy in action. The paper recalls and synthesizes some results already published by authors, drawing new qualitative conclusions. The totally new parts are Markov chain models of the HMS structure of demes and of the MWEA component, as well as the theorem of their ergodicity.

scholarly journals On the η − (1, 2) approximated optimization problems

2012 ◽  
Vol 28 (1) ◽  
pp. 17-24
Author(s):  
HORATIU VASILE BONCEA ◽  
◽  
DOREL I. DUCA ◽  
Keyword(s):  
Differentiable Function ◽  
Interior Point ◽  
Optimization Problem ◽  
Nonempty Subset ◽  
Optimization Problems ◽  
Saddle Points ◽  
Optimal Solutions

Let X be a nonempty subset of Rn, x 0 be an interior point of X, f : X → R be a differentiable function at x 0 , g : X → Rm be a twice differentiable function at x 0 and η : X × X → Rn be a function. In this paper, we attach to the optimization problem ... the (1, 2)-η- approximated optimization problem ... and we will study the relations between the optimal solutions of Problem (P), the optimal solutions of Problem (AP), the saddle points of Problem (P) and saddle points of Problem (AP).

Generalized Random Tunneling Algorithm for Continuous Design Variables

2003 ◽  
Author(s):  
Satoshi Kitayama ◽  
Koetsu Yamazaki
Keyword(s):  
Global Optimization ◽  
Mathematical Programming ◽  
Structural Optimization ◽  
Global Minimum ◽  
Optimization Problems ◽  
Optimization Method ◽  
Global Optimization Method ◽  
Numerical Examples ◽  
Design Variables ◽  
Side Constraints

A global optimization method for continuous design variables called as Generalized Random Tunneling Algorithm is proposed. Proposed method is called as “Generalized” random tunneling algorithm because this method can treat the behavior constraints as well as the side constraints. This method consists of three phases, that is, the minimization phase, the tunneling phase, and the constraints phase. In the minimization phase, mathematical programming is used, and heuristic approach is introduced in the tunneling and constraint phases. By iterating these phases, global minimum is found. The characteristics of mathematical programming and heuristic approaches are included in the proposed method. Global minimum which may lie on the boundary of constraints is easily found by proposed method. Proposed method is applied to mathematical and structural optimization problems. Through numerical examples, the effectiveness and validity of proposed method have been confirmed.

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