A Note on the Extended Rosenbrock Function

2006 ◽  
Vol 14 (1) ◽  
pp. 119-126 ◽  
Author(s):  
Yun-Wei Shang ◽  
Yu-Huang Qiu
Keyword(s):  
Theoretical Analysis ◽  
Evolutionary Algorithms ◽  
Numerical Optimization ◽  
Local Minimum ◽  
Optimization Problems ◽  
Higher Dimensions ◽  
High Dimensional ◽  
Two Dimensional ◽  
Local Minima ◽  
Unimodal Function

The Rosenbrock function is a well-known benchmark for numerical optimization problems, which is frequently used to assess the performance of Evolutionary Algorithms. The classical Rosenbrock function, which is a two-dimensional unimodal function, has been extended to higher dimensions in recent years. Many researchers take the high-dimensional Rosenbrock function as a unimodal function by instinct. In 2001 and 2002, Hansen and Deb found that the Rosenbrock function is not a unimodal function for higher dimensions although no theoretical analysis was provided. This paper shows that the n-dimensional (n = 4 ∼ 30) Rosenbrock function has 2 minima, and analysis is proposed to verify this. The local minima in some cases are presented. In addition, this paper demonstrates that one of the “local minima” for the 20-variable Rosenbrock function found by Deb might not in fact be a local minimum.

scholarly journals Investigation of Improved Cooperative Coevolution for Large-Scale Global Optimization Problems

Algorithms ◽  
2021 ◽  
Vol 14 (5) ◽  
pp. 146
Author(s):  
Aleksei Vakhnin ◽  
Evgenii Sopov
Keyword(s):  
Global Optimization ◽  
Evolutionary Algorithms ◽  
Numerical Experiments ◽  
Large Scale ◽  
Optimization Problems ◽  
State Of The Art ◽  
Fixed Number ◽  
High Dimensional ◽  
Cooperative Coevolution ◽  
Large Scale Problems

Modern real-valued optimization problems are complex and high-dimensional, and they are known as “large-scale global optimization (LSGO)” problems. Classic evolutionary algorithms (EAs) perform poorly on this class of problems because of the curse of dimensionality. Cooperative Coevolution (CC) is a high-performed framework for performing the decomposition of large-scale problems into smaller and easier subproblems by grouping objective variables. The efficiency of CC strongly depends on the size of groups and the grouping approach. In this study, an improved CC (iCC) approach for solving LSGO problems has been proposed and investigated. iCC changes the number of variables in subcomponents dynamically during the optimization process. The SHADE algorithm is used as a subcomponent optimizer. We have investigated the performance of iCC-SHADE and CC-SHADE on fifteen problems from the LSGO CEC’13 benchmark set provided by the IEEE Congress of Evolutionary Computation. The results of numerical experiments have shown that iCC-SHADE outperforms, on average, CC-SHADE with a fixed number of subcomponents. Also, we have compared iCC-SHADE with some state-of-the-art LSGO metaheuristics. The experimental results have shown that the proposed algorithm is competitive with other efficient metaheuristics.

scholarly journals Evolutionary Algorithms for Constrained Parameter Optimization Problems

1996 ◽  
Vol 4 (1) ◽  
pp. 1-32 ◽  
Author(s):  
Zbigniew Michalewicz ◽  
Marc Schoenauer
Keyword(s):  
Nonlinear Programming ◽  
Evolutionary Algorithms ◽  
Evolutionary Computation ◽  
Numerical Optimization ◽  
Optimization Problems ◽  
Optimization Techniques ◽  
Test Cases ◽  
Nonlinear Constraints ◽  
Nonlinear Programming Problem ◽  
General Nonlinear Programming

Evolutionary computation techniques have received a great deal of attention regarding their potential as optimization techniques for complex numerical functions. However, they have not produced a significant breakthrough in the area of nonlinear programming due to the fact that they have not addressed the issue of constraints in a systematic way. Only recently have several methods been proposed for handling nonlinear constraints by evolutionary algorithms for numerical optimization problems; however, these methods have several drawbacks, and the experimental results on many test cases have been disappointing. In this paper we (1) discuss difficulties connected with solving the general nonlinear programming problem; (2) survey several approaches that have emerged in the evolutionary computation community; and (3) provide a set of 11 interesting test cases that may serve as a handy reference for future methods.

Fuzzy C-Means in High Dimensional Spaces

2013 ◽  
pp. 1-16
Author(s):  
Roland Winkler ◽  
Frank Klawonn ◽  
Rudolf Kruse
Keyword(s):  
Objective Function ◽  
Local Minimum ◽  
High Dimensional ◽  
Controlled Environment ◽  
Data Sets ◽  
Local Minima ◽  
High Dimensions ◽  
Data Set ◽  
Fcm Algorithm ◽  
Centre Of Gravity

High dimensions have a devastating effect on the FCM algorithm and similar algorithms. One effect is that the prototypes run into the centre of gravity of the entire data set. The objective function must have a local minimum in the centre of gravity that causes FCM’s behaviour. In this paper, examine this problem. This paper answers the following questions: How many dimensions are necessary to cause an ill behaviour of FCM? How does the number of prototypes influence the behaviour? Why has the objective function a local minimum in the centre of gravity? How must FCM be initialised to avoid the local minima in the centre of gravity? To understand the behaviour of the FCM algorithm and answer the above questions, the authors examine the values of the objective function and develop three test environments that consist of artificially generated data sets to provide a controlled environment. The paper concludes that FCM can only be applied successfully in high dimensions if the prototypes are initialized very close to the cluster centres.

Fuzzy C-Means in High Dimensional Spaces

2011 ◽  
Vol 1 (1) ◽  
pp. 1-16 ◽  
Author(s):  
Roland Winkler ◽  
Frank Klawonn ◽  
Rudolf Kruse
Keyword(s):  
Objective Function ◽  
Local Minimum ◽  
High Dimensional ◽  
Controlled Environment ◽  
Data Sets ◽  
Local Minima ◽  
High Dimensions ◽  
Data Set ◽  
Fcm Algorithm ◽  
Centre Of Gravity

High dimensions have a devastating effect on the FCM algorithm and similar algorithms. One effect is that the prototypes run into the centre of gravity of the entire data set. The objective function must have a local minimum in the centre of gravity that causes FCM’s behaviour. In this paper, examine this problem. This paper answers the following questions: How many dimensions are necessary to cause an ill behaviour of FCM? How does the number of prototypes influence the behaviour? Why has the objective function a local minimum in the centre of gravity? How must FCM be initialised to avoid the local minima in the centre of gravity? To understand the behaviour of the FCM algorithm and answer the above questions, the authors examine the values of the objective function and develop three test environments that consist of artificially generated data sets to provide a controlled environment. The paper concludes that FCM can only be applied successfully in high dimensions if the prototypes are initialized very close to the cluster centres.

scholarly journals Evolutionary Algorithms, Homomorphous Mappings, and Constrained Parameter Optimization

1999 ◽  
Vol 7 (1) ◽  
pp. 19-44 ◽  
Author(s):  
Slawomir Koziel ◽  
Zbigniew Michalewicz
Keyword(s):  
Evolutionary Algorithms ◽  
Parameter Optimization ◽  
Numerical Optimization ◽  
Optimization Problems ◽  
Search Space ◽  
Test Cases ◽  
Nonlinear Constraints ◽  
New Approach ◽  
Survey Papers ◽  
Dimensional Cube

During the last five years, several methods have been proposed for handling nonlinear constraints using evolutionary algorithms (EAs) for numerical optimization problems. Recent survey papers classify these methods into four categories: preservation of feasibility, penalty functions, searching for feasibility, and other hybrids. In this paper we investigate a new approach for solving constrained numerical optimization problems which incorporates a homomorphous mapping between n-dimensional cube and a feasible search space. This approach constitutes an example of the fifth decoder-based category of constraint handling techniques. We demonstrate the power of this new approach on several test cases and discuss its further potential.

scholarly journals Optimization of High-Dimensional Functions through Hypercube Evaluation

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Rahib H. Abiyev ◽  
Mustafa Tunay
Keyword(s):  
Numerical Optimization ◽  
Optimization Problems ◽  
Learning Algorithm ◽  
Evaluation Process ◽  
Stochastic Search ◽  
High Dimensional ◽  
Potential Candidate ◽  
Objective Functions ◽  
Global Numerical Optimization ◽  
Search Area

A novel learning algorithm for solving global numerical optimization problems is proposed. The proposed learning algorithm is intense stochastic search method which is based on evaluation and optimization of a hypercube and is called the hypercube optimization (HO) algorithm. The HO algorithm comprises the initialization and evaluation process, displacement-shrink process, and searching space process. The initialization and evaluation process initializes initial solution and evaluates the solutions in given hypercube. The displacement-shrink process determines displacement and evaluates objective functions using new points, and the search area process determines next hypercube using certain rules and evaluates the new solutions. The algorithms for these processes have been designed and presented in the paper. The designed HO algorithm is tested on specific benchmark functions. The simulations of HO algorithm have been performed for optimization of functions of 1000-, 5000-, or even 10000 dimensions. The comparative simulation results with other approaches demonstrate that the proposed algorithm is a potential candidate for optimization of both low and high dimensional functions.

Nelder-Mead Evolutionary Hybrid Algorithms

2011 ◽  
pp. 1191-1196 ◽  
Author(s):  
Sanjoy Das
Keyword(s):  
Evolutionary Algorithms ◽  
Local Search ◽  
Optimization Problems ◽  
Search Algorithm ◽  
Search Space ◽  
Simplex Algorithm ◽  
Stochastic Methods ◽  
Great Success ◽  
Local Minima ◽  
Swarm Optimization

Real world optimization problems are often too complex to be solved through analytic means. Evolutionary algorithms are a class of algorithms that borrow paradigms from nature to address them. These are stochastic methods of optimization that maintain a population of individual solutions, which correspond to points in the search space of the problem. These algorithms have been immensely popular as they are derivativefree techniques, are not as prone to getting trapped in local minima, and can be tailored specifically to suit any given problem. The performance of evolutionary algorithms can be improved further by adding a local search component to them. The Nelder-Mead simplex algorithm (Nelder & Mead, 1965) is a simple local search algorithm that has been routinely applied to improve the search process in evolutionary algorithms, and such a strategy has met with great success. In this article, we provide an overview of the various strategies that have been adopted to hybridize two wellknown evolutionary algorithms - genetic algorithms (GA) and particle swarm optimization (PSO).

Multi-Objective Evolutionary Algorithms

2011 ◽  
pp. 1145-1151 ◽  
Author(s):  
Sanjoy Das ◽  
Bijaya K. Panigrahi
Keyword(s):  
Evolutionary Algorithms ◽  
Optimization Problems ◽  
Population Based ◽  
Stochastic Methods ◽  
Local Minima ◽  
Multi Objective ◽  
Derivative Free ◽  
Derivative Free Methods ◽  
Complex Optimization ◽  
Single Objective

Real world optimization problems are often too complex to be solved through analytical means. Evolutionary algorithms, a class of algorithms that borrow paradigms from nature, are particularly well suited to address such problems. These algorithms are stochastic methods of optimization that have become immensely popular recently, because they are derivative-free methods, are not as prone to getting trapped in local minima (as they are population based), and are shown to work well for many complex optimization problems. Although evolutionary algorithms have conventionally focussed on optimizing single objective functions, most practical problems in engineering are inherently multi-objective in nature. Multi-objective evolutionary optimization is a relatively new, and rapidly expanding area of research in evolutionary computation that looks at ways to address these problems. In this chapter, we provide an overview of some of the most significant issues in multi-objective optimization (Deb, 2001).

Introduction of a Tradeoff Index for Efficient Trade Space Exploration

2015 ◽  
Author(s):  
Mehmet Unal ◽  
Gordon Warn ◽  
Timothy W. Simpson
Keyword(s):  
Approximate Solution ◽  
Design Optimization ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Space Exploration ◽  
High Dimensional ◽  
Two Dimensional ◽  
Complex Design ◽  
Trade Space Exploration ◽  
Tradeoff Index

The development of many-objective evolutionary algorithms has facilitated solving complex design optimization problems, that is, optimization problems with four or more competing objectives. The outcome of many-objective optimization is often a rich set of solutions, including the non-dominated solutions, with varying degrees of tradeoff amongst the objectives, herein referred to as the trade space. As the number of objectives increases, exploring the trade space and identifying acceptable solutions becomes less straightforward. Visual analytic techniques that transform a high-dimensional trade space into two-dimensional (2D) presentations have been developed to overcome the cognitive challenges associated with exploring high-dimensional trade spaces. Existing visual analytic techniques either identify acceptable solutions using algorithms that do not allow preferences to be formed and applied iteratively, or they rely on exhaustive sets of 2D representations to identify tradeoffs from which acceptable solutions are selected. In this paper, an index is introduced to quantify tradeoffs between any two objectives and integrated into a visual analytic technique. The tradeoff index enables efficient trade space exploration by quickly pinpointing those objectives that have tradeoffs for further exploration, thus reducing the number of 2D representations that must be generated and interpreted while allowing preferences to be formed and applied when selecting a solution. Furthermore, the proposed index is scalable to any number of objectives. Finally, to illustrate the utility of the proposed tradeoff index, a visual analytic technique that is based on this index is applied to a Pareto approximate solution set from a design optimization problem with ten objectives.

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