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When ordering of weights, the local search heuristic takes the first uncolored vertex and determines a set of adjacent vertices including it. This set of vertices is then ordered descending with regard to the values of weights. This local search heuristic determines the [MATH] neighborhood of current solution, where [M... |
The swap local search heuristic finds the first uncolored vertex and descendingly orders the set of all predecessors in the solution according to the saturation degree. Then, the uncolored vertex is swapped with the vertex from the set of predecessors with the highest saturation degree. When more vertices with the same... |
IV.1.2 Analysis of the Hybrid Self-adaptive Evolutionary Algorithm for Graph 3-coloring The goal of this subsection is twofold. At the first, an influence of the local search heuristics on results of the HSA-EA is analyzed in details. Further, a comparison of the HSA-EA hybridized with the neutral survivor selection an... |
Characteristics of the HSA-EA used in experiments were as follows. The normal distributed mutation was employed and applied with mutation probability of 1.0. The crossover was not used. The tournament selection with size 3 selects the parents for mutation. The population model [MATH] was suitable for the self-adaptatio... |
The Culberson ( 2008 random graph generator was employed for generation of random graphs that constituted the test suite. It is capable to generate the graphs of various types, number of vertices, edge densities and seeds of random generator. In this study we concentrated on the equi-partite type of graphs. This type o... |
All generated graphs consisted of [MATH] vertices. An edge density is controlled by parameter [MATH] of the random graph generator that determines probability that two vertices [MATH] and [MATH] in the graph [MATH] are connected with an edge [MATH] |
inproc:chiarandini2010 . However, if [MATH] is small the graph is not connected because the edges are sparse. When [MATH] is increased the number of edges raised and the graph becomes interconnected. As a result, the number of constraints that needs to be satisfied by the coloring algorithm increases until suddenly the... |
To capture a phenomenon of the threshold, the parameter [MATH] by generation of the equi-partite graphs was varied from [MATH] to [MATH] in a step of [MATH] . In this way, the test suite consisted of 15 instances of graphs, in which the hardest graph with [MATH] was presented as well. In fact, the evolutionary algorith... |
The impact of the local search heuristics In this experiments, the impact of four implemented local search heuristics on results of the HSA-EA was taken into consideration. Results of the experiments are illustrated in the Fig. that is divided into six graphs and arranged according to the particular measures [MATH] [MA... |
A seen by the Fig. .a, no one of the HSA-EA versions was succeed to solve the hardest instance of graph with [MATH] . The best results in the vicinity of the threshold is observed by the HSA-EA hybridizing with the ordering by saturation local search heuristic ( [MATH] by [MATH] ). The overall best performance is shown... |
In average, results according to the [MATH] (Fig. .c) show that the HSA-EA hybridized with the swap local search heuristic finds the solutions with the smallest number of the fitness evaluations. However, troubles are arisen in the vicinity of the threshold, where the HSA-EA with other local search heuristics are faced... |
The HSA-EA hybridizing with the ordering by saturation local search heuristic demonstrates the worst results according to the [MATH] , as presented in the Fig. .e. The graph instance by [MATH] was exposed as the most critical by this algorithm ( [MATH] ) although this is not the closest to the threshold. In average, wh... |
In the right side of the Fig. , results of different versions of the HSA-EA are collected. The first version that is designated as [MATH] operates with the same parameters as the original HSA-EA but without the local search heuristics. The label [MATH] in this figure indicates the original version of the HSA-EA. Finall... |
In Fig. .b results of different versions of the HSA-EA according to the [MATH] are presented. The best results by the instances the nearest to the threshold ( [MATH] ) are observed by the original HSA-EA. Conversely, the HSA-EA with the random initialization procedure ( [MATH] ) gained the worst results by the instance... |
In contrary, the best results by the instances the nearest to the threshold according to the [MATH] was observed by the HSA-EA without local search heuristics ( [MATH] ). Here, the turning point regarding the performance of the HSA-EA ( [MATH] ) was observed as well. After this point results of the HSA-EA without local... |
As illustrated by Fig. .f, all versions of the HSA-EA leaved in average less than [MATH] uncolored vertices by the 3-coloring. The bad result by the original HSA-EA coloring the graph with [MATH] was caused because of the ordering by saturation local search heuristic that got stuck in the local optima. Nevertheless, no... |
The impact of the neutral survivor selection In this experiments the impact of the neutral survivor selection on results of the HSA-EA was analyzed. In this context, the HSA-EA with deterministic survivor selection was developed with the following characteristic: |
The Equation 12 that prevents the generation of neutral solutions was used instead of the Equation 10 The deterministic survivor selection was employed instead of the neutral survivor solution. This selection orders the solutions according to the increasing values of the fitness function. In the next generation the fir... |
Before starting with the analysis, we need to prove the existence of neutral solution and to establish they characteristics. Therefore, a typical run of the HSA-EA with neutral survivor selection is compared with the typical run of the HSA-EA with the deterministic survivor selection. As example, the 3-coloring of the ... |
In the Fig. .a the best and the average number of uncolored nodes that were achieved by the HSA-EA with neutral and the HSA-EA with deterministic survivor selection are presented. The figure shows that the HSA-EA with the neutral survivor selection converge to the optimal solution very fast. To improve the number of un... |
More interestingly, the neutral solution occurs when the average fitness values comes near to the best (Fig. .b). As illustrated by this figure, the most neutral solutions arise in the later generations when the population becomes matured. In example from Fig. .b, the most neutral solutions occurred after 20,000 and 30... |
In contrary, the HSA-EA with deterministic survivor selection starts with the lower number of uncolored vertices (Fig. .c) than the HSA-EA with neutral selection. However, the convergence of this algorithm is slower than by its counterpart with the neutral selection. A closer look at the average fitness value uncovers ... |
In the Fig. .d a diversity of population as produced by the HSA-EA with different survivor selections is presented. The diversity of population is calculated as a standard deviation of the vector consisting of the mean weight values in the population. Both HSA-EA from this figure lose diversity of the initial populatio... |
To determine what impact the neutral survivor selection has on results of the HSA-EA, a comparison between results of the HSA-EA with neutral survivor selection ( [MATH] ) and the HSA-EA with deterministic survivor selection ( [MATH] ) was done. However, both versions of the HSA-EA run without local search heuristics. ... |
As shown by the Fig. 10 .a the HSA-EA with neutral survivor selection ( [MATH] ) exposes better results by the instances near to the threshold ( [MATH] ) while the HSA-EA with deterministic survivor selection ( [MATH] ) was slightly better by the instance of graph with [MATH] . Interestingly, while the curve of the for... |
In summary, the original HSA-EA with swap local search heuristic used as reference outperforms all observed versions of the HSA-EA. |
IV.1.3 Summary In this subsection the characteristics of the HSA-EA were studied on the collection of equi-partite graphs, where we focused on the behavior of the algorithm in the vicinity of the threshold. Therefore, an impact of the hybridizing elements, like the initialization procedure, the local search heuristics,... |
As shown by the table , results of the HSA-EA with deterministic survivor selection without local search heuristics and without random initialization procedure ( [MATH] , denoted as [MATH] ) were worse than results or its counterpart with neutral survivor selection ( [MATH] , denoted as [MATH] ) in average for more tha... |
In summary, the construction heuristics has the most impact on results of the HSA-EA. That is, the basis of the graph 3-coloring represents the self-adaptive evolutionary algorithm with corresponding construction heuristic. However, to improve results of this base algorithm additional hybrid elements were developed. As... |
Conclusion Evolutionary algorithms are a good general problem solver but suffer from a lack of domain specific knowledge. However, the problem specific knowledge can be added to evolutionary algorithms by hybridizing different parts of evolutionary algorithms. In this chapter, the hybridization of search and selection ... |
In continuation of work the graph [MATH] -coloring will be investigated. On the other hand, the neutral selection operator needs to be improved with tabu search that will prevent that the reference solution will be selected repeatedly. Updated 1 December 2012. |
# Source: arxiv 1303.3469 # Title: Hybrid Evolutionary Computation for Continuous Optimization # Sections: all # Downloaded: 2026-03-03T01:57:19.745074+00:00 |
Hybrid Evolutionary Computation for Continuous Optimization Technical Memorandum 2011-v.01 School of Computer Science, University of Manchester, Kilburn Building, Oxford Road, MANCHESTER, M13 9PL |
Author: Hassan A Bashir Supervisor: Dr. Richard Neville November 9, 2011 This material represents the opinion of the author only and does not necessarily represent the opinion of the School or the University. While every attempt has been made to ensure the accuracy of this publication, neither the author, the School or... |
Abstract Hybrid optimization algorithms have gained popularity as it has become apparent there cannot be a universal optimization strategy which is globally more beneficial than any other. Despite their popularity, hybridization frameworks require more detailed categorization regarding: the nature of the problem domain... |
This report proposes a hybrid algorithm for solving small to large-scale continuous global optimization problems. It comprises evolutionary computation (EC) algorithms and a sequential quadratic programming (SQP) algorithm; combined in a collaborative portfolio. The SQP is a gradient based local search method. To optim... |
The proposed hybrid design aim was to ensure that the two algorithms complement each other by exploring and exploiting the problem search space. Preliminary results justify that an adept hybridization of evolutionary algorithms with a suitable local search method, could yield a robust and efficient means of solving wid... |
Finally, a discussion of the outcomes of the initial investigation and a review of the associated challenges and inherent limitations of the proposed method is presented to complete the investigation. The report highlights extensive research, particularly, some potential case studies and application areas. |
Chapter 1 Hybrid Evolutionary Computation for Optimization of Continuous Problems 1.1 Introduction As a vital aspect for successful achievement of our everyday goals, optimization arises naturally in our daily lives. It deals with the task of selecting the best out of the many possible decisions encountered in a typica... |
Of interest is the fact that optimization has over the years become a subject that is widely used in sciences, engineering, management and economics, and in the industry. This has led to the growing need for thorough understanding of optimization problems and their solution methods. As a research field in particular, o... |
The recent growth in the development of new algorithmic and modelling techniques and in the theoretical background has largely led to the rapid diffusion of optimization into other disciplines. The striking emphasis on the interdisciplinary nature of the field has shifted it from being a mere tool in applied and comput... |
“Optimization is a cornerstone for the development of civilization” Formally, the subject is involved in determining optimal solutions for problems which are defined mathematically. It often requires the assessment of the problem’s optimality conditions, model construction, building the algorithmic method of solution, ... |
Although optimization problems include but not limited to the continuous problems, discrete or combinatorial problems, multiobjective optimization problems etc., the focus of this work was on the global linear/nonlinear continuous optimization problems potentially subject to some constraints and bounds. This was due to... |
Controlling a chemical process or a mechanical device to optimize performance or meet standards of robustness; Designing an investment portfolio to maximize expected return while maintaining an acceptable level of risk; |
Finding an optimal trajectory for an aircraft or a robot arm; Computing the optimal shape of an automobile or aircraft component in a manufacturing and process plant; |
Scheduling tasks such as school time tabling or operations in manufacturing plants to maximize production level within the limited available resources while meeting the required quality standards and satisfying customer demands, etc. |
Worth noting is that all these situations share the following three important aspects: 1. Objective : Also called an overall goal, it is a measure used to assess the extent to which the ultimate target in the activity is being realized and it is technically termed as the objective function which is typically modelled m... |
2. Constraints : This reflects the requirements within which the quest to optimizing the objective must be limited. It can be a limitation due to resource, time or space and or acceptable error levels or tolerance. |
3. Design variables : This constitutes the set of all possible choices that must be made to ensure successful realization of the overall objective while satisfying the constraints. These implicit choices are technically referred to as decision or design variables and are the parameters around which the optimization tas... |
Several solution techniques exist for the different types of the aforementioned optimization problems. However, the classical solution approach involves the use of numerical algorithms that have originated ever since the invention of the popular simplex algorithm for linear programming by Dantzig |
in the late 1940s. Thereafter, many numerical algorithms such as gradient-based methods, conjugate gradient methods, Newton and quasi-Newton methods have evolved into powerful techniques for solving large scale nonlinear optimization problems. This category of algorithms is classified as exact or complete methods and c... |
Subsequent to the development of the exact methods, a number of solution methods that are based on various heuristics are developed. This category of algorithms also called approximate algorithms can be successfully applied to a wide range of optimization problems with little or no modifications in order to adapt to an... |
is a generic term that was introduced to delineate a universal algorithmic framework designed to solve different optimization problems based on probabilistic decisions made during the search process. These approximate methods are usually easier to implement than their exact counterparts like the classical gradient-base... |
A large number of algorithms established on different theoretical paradigms and backgrounds such as the evolutionary computations (EC) like genetic algorithms (GA), genetic programming (GP) and evolutionary strategy (ES), simulated annealing, tabu search, ant colony optimization, artificial immune system, scatter searc... |
It has become evident that , many real-world large scale optimization problems elude acceptable solutions via simple exact methods or even the approximate metaheuristics when applied independently. Therefore, in the recent years, researchers have become increasingly interested in the concepts that are not limited to th... |
. The combination of such algorithms is what is referred to as hybrid algorithms or hybrid metaheuristics . A skilful hybridization of algorithms is believed to provide a more flexible and efficient solution method that is suitable for large scale real-world problems. |
In fact, the need for hybrid algorithms surfaces and gains popularity after competing research communities have waived their traditional stance and believe in the invincibility of some classes of algorithms and philosophies that were regarded as generally the best . It has become apparent that there cannot be a general... |
. The NFL theorem proved that on average over all possible functions/problems, the performance of all search/optimization methods that satisfy certain conditions is the same. Hence, as declared by |
among others, the primary motivation behind the notion of hybridizing algorithms was to come up with robust systems that harness the benefits of the individual algorithms while discarding their inherent weaknesses. |
1.2 Motivations Despite the growing interest in the area of hybrid algorithms, more needs to be done to address matters of crucial importance vis-à-vis: |
Establishing a proper categorization of the hybrid strategies based on the expected precision or solution quality required for any given problem instance in the intended area of application. |
Assessing based on the overall optimization goal the composition of the hybrid scheme as to whether it should comprise of algorithms from only approximate metaheuristics, exact algorithms or a mixture of the two. |
Ascertaining when and why the identified approaches should be combined in an interleaved, paralleled or sequential manner. Enhancing the capabilities of the individual algorithms prior to hybridization, specifically focusing on the identified key features of the algorithms that are expected to play major roles in the h... |
Identifying at what stages of the solution process the key features of the algorithms can effectively be exploited to optimally benefit from the hybridization scheme. For instance, ensuring proper convergence assessment and maintenance of useful level of diversity at different stages of a typical EC algorithm. |
Use of the well-known measures of problem difficulties to judge the complexity of the problem categories upon which the hybrid algorithms are expected to be applied. |
Developing hybrids that combine approximate algorithms with the state-of-the-art of exact optimization techniques like the sequential quadratic programming (SQP) algorithm . This type of hybridization scheme is also called memetic algorithms |
. It is believed that although the approach can be very successful in practice, so far not much work exists in this direction 1.3 Aims and Objectives |
The aim of this research was; to analyse and elucidate the current trend in hybridization of algorithms for optimization; to propose a novel hybrid optimization method that combines EC algorithm (for global searching) and an interior point method (IPM) based SQP algorithm (for local searching) to address large scale gl... |
The first objective of this report was to examine evolutionary computation algorithms (GA in specific) and their hybrids. We conduct an in-depth investigation on the parameterization aspect of EC algorithms. We investigate the effect of elitism in EC algorithms and propose a novel adaptive elitism method based on the o... |
Secondly, we review local search optimization algorithms giving emphasis to quasi-Newton based numerical techniques. We investigate various methods for approximating and updating the Hessian matrix. We then extend the IPM based SQP algorithm |
that uses BFGS Hessian approximation to use exact Hessians so as to effectively solve complex constrained nonlinear optimization problems. |
The third objective was to study the technique of automatic differentiation (AD) for exact gradient and Hessian calculations. We investigate both the forward and reverse accumulation methods and then design an automatic differentiation tool based on the operator overloading principle. We implement the AD tool using obj... |
Finally, we examine the current trends in the design and applications of hybridization methods for system optimization. And for the various techniques reported in the literature, we adopt the proposal in |
to devise a generalization that categorizes the hybrid systems based on the types of the combined algorithms and how they are combined in view of the overall optimization goal. We then design a hybrid system that combines the proposed global and local algorithms in a collaborative, batched and weakly-coupled manner wit... |
1.4 Hypotheses In the following, we recast the aims of this research into the following hypotheses. Therefore, our overall objective is to verify the following: |
#H1: Hybrid global and local search methodologies provide good search strategies. #H2: Specific types of local search algorithms (e.g. SQP/IPM) |
are efficient in locating local optima. #H3: Local optimization methods alone may not provide fast convergence to the global optimal solution. |
#H4: Hybridization of global and local optimization algorithms should provide fast convergence to the optimal solution. #H4.1: The global and local algorithms can serve as a means to validate each other’s result. |
1.5 Scope and Limitations The scope of this work was to deal with continuous optimization problems that are either local or global in nature. Hereby, the report is limited to problems that can be mathematically modelled in form of differentiable functions with at least second derivatives available. |
1.6 Chapter Organization and Summary Besides the introduction in this chapter, chapter 2 and 3 focus mainly on the principles and dynamics of evolutionary computation algorithms. Chapter 2 provides an in-depth review on the current trend and challenges militating against the development and simulation of evolutionary p... |
Chapter 3 further analyse the convergence characteristics of evolutionary algorithms and presents a fundamentally new way of perceiving the individual roles of evolutionary operators/processes towards the success of the evolution. It then empirically analyse the efficacy of crossover in convergence detection in evoluti... |
Chapter 4 investigates the framework of local optimization algorithms with particular emphasis on the gradient-based methods. The design of the sequential quadratic programming (SQP) algorithm and interior point method is investigated. The chapter then presents how an algorithmic approach for effective evaluation of de... |
In chapter 5, various techniques for hybridizing optimization algorithms are examined and a chronological taxonomy of various categories of hybrid algorithms is presented. The chapter presents a novel approach for hybridizing the EC algorithm with the SQP algorithm. A series of experiments undertaken to evaluate the pr... |
Finally, Chapter 6 summarises the current work, discusses, in general, the outcome of our initial investigations. The chapter then concludes by pinpointing the open questions that will guide our further research in this direction. |
Chapter 2 Evolutionary Computation Algorithms–An Overview In this part of the report, a review focusing on the foundation, development processes, mechanics and simulation of evolutionary computation (EC) algorithms will be provided. Details of parameterization aspect of EC will be investigated. Emphasis will be given o... |
2.1 Introduction Evolution is a process that originated from the biologically inspired neo-Darwinian paradigm (i.e. the principle of survival of the fittest). It is believed to be a collection of stochastic processes that act on and within populations of species. These processes include reproduction, mutation, competit... |
. In the late 1950s, evolution was understood as an optimization process that naturally shapes and maintains the balance in the existence and progress of individuals’ life. As reported in |
, a salient rule of thumb of evolution as have come to be understood is that ”Darwinian evolution is essentially an optimization technique. It is not a predictive theory, nor is it a tautology” . Thus, as in most optimization processes, the solution point(s) are discovered via a trial and error search process. |
The far reaching impact of the idea of evolution has gone beyond the classical boundaries of biological thoughts. In what is termed as evolutionary computation (EC), the process of evolution has now become an optimization tool that can be simulated and applied in solving complex engineering problems. |
Evolutionary computation algorithms are designed to mimic the intrinsic mechanisms of natural evolution and progressively yield improved solutions to a wide range of optimization problems. This is evident because, the success of these algorithms is always not directly inclined to the domain knowledge specific to any pr... |
The three popular evolutionary computation algorithms that stand out are genetic algorithm (GA), evolutionary strategies (ES) and evolutionary programming (EP). These techniques are all built around the common principles of natural evolution and rose almost independently of each other. They are strongly interrelated an... |
In the original implementation of genetic algorithms, their data structure enforces representation of candidate solutions as binary vectors. In their distinct nature, evolutionary programming algorithms use finite state machines for representing candidate solutions, whereas in evolutionary strategies solution points ar... |
, problem dependent representation of candidate solutions can significantly improve the effectiveness of the overall optimization process thereby avoiding the problem of mapping between various representations. |
As highlighted in the previous chapter, the aim of this work was to use genetic algorithms as global optimization method. Thus, subsequent treatment of the evolutionary computation literature will focus mainly on the evolution principles of genetic algorithms. |
2.2 Background and Process Dynamics of Evolutionary Computation Genetic algorithms are evolutionary based algorithms originally inspired by Holland in the 1970s and the principles of which is extensively disseminated in his book Adaptation in Natural and Artificial Systems |
. Although Holland’s contribution to the development of the original ideas has been quite remarkable, history has shown that quite a number of researchers working on the same area have also contributed immensely in the design and development of these techniques. In late 1960s, an independent work by Schewefel and Reche... |
led to their proposal of the technique of evolutionary strategies. Parallel to that Fogel and his colleagues implemented the idea of evolutionary programming which also is based on natural evolution principles. Hitherto the work of Goldberg |
who researched and extensively outlined the typical form of the genetic algorithm used today, prior proposals were mainly mutation and selection based without incorporation of the recombination operator. Detailed historical background on genetic algorithms can be found in the excellent collection by David Fogel |
Genetic algorithms have proven to provide a heuristic means of solving complex optimization problems that require a robust solution method. Recently, they have been successfully applied in the areas of computing and industrial engineering such as vehicle routing |
, scheduling and sequencing , network design and synthesis , reliability design , facility layout and location , to mention a few. |
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