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However, for such carefully selected categories of test problems, a critical analysis of the performance of each of the three algorithms requires cautious treatment. This is necessary because although all the three test problems are nonlinear, scalable, and multimodal in nature, they are quite different in complexity w... |
Beginning with the Rastrigin function (figure 5.3 mid row), the observed remarkable performance of all the three algorithms on this test problem may be attributed to its moderate level of dispersion. This could be why it seems easy to the three different global optimization approaches. Nonetheless, it can be noticed th... |
With regard to the Ackley function (figure 5.3 top row), one might notice that while the hybrid EC/SQP and CMA-ES algorithms clearly outperform the standard EC algorithm, the excellent efficiency exhibited by the CMA-ES algorithm (figure 5.3 (a)) seem to deteriorate following the [MATH] times increase in the problem si... |
It is interesting to note that for all the test cases considered in this experiment (figure 5.3 ), the standard EC algorithm always lags behind the other two algorithms except on the Schwefel function (bottom row). Surprisingly, this is where the CMA-ES algorithm exhibited least performance. This rather difficult to in... |
It remains unclear, though, why high dispersion problems like Shwefel function seem to be easy for the standard EC algorithm (see figure 5.3 bottom row). But one possible reason for the sudden decline in the performance of the CMA-ES algorithm may be due to its excessive evaluation of infeasible solutions during the ea... |
These results corroborate the findings of some previous studies in this field. An investigation on the CMA-ES algorithm by revealed that the adaptive step-size heuristic, called cumulation , does not function as intended when the best regions of the search space are too spread out (such as in a high dispersion problems... |
, that although CMA-ES algorithm may need more than [MATH] function evaluations for such high dispersion problems, it will at some point converge to the true global solution if sufficient evaluations are granted. |
Noteworthy, from virtually all the results in these experiments (see figure 5.3 ), the moment at which the switching from the global algorithm (EC) to the local algorithm (SQP) took place might easily be noticed. Thus, one could appreciate the remarkable contribution of the local algorithm towards the overall success o... |
Ultimately, the results presented herein illustrate that the proposed hybrid EC/SQP optimization method is a robust and efficient novel approach that can effectively complement the traditional global optimization methods. |
5.5 Contribution This first attempt to realize the proposed hybrid optimization algorithm was possible following an intuitive amalgamation of the previously presented evolutionary (EC) algorithm and the gradient based (SQP) local search method. Initially, we focus the design of the individual algorithms in such a way t... |
The new convergence detection method (proposed in chapter , section 3.5 ) that detects convergence of the EC algorithm, and in due course, shifts control from the EC to the SQP algorithm adaptively, and |
The incorporation of the novel validation routine (section 5.3 ) which essentially further substantiates the quality of the solution returned by the local algorithm. |
But above all, is the fact that the experimental results presented in this chapter have given us a glimpse of what to expect in the future when the proposed method is further refined. |
5.6 Remarks This chapter has presented a broad review of various techniques of hybridizing evolutionary algorithms. Particularly, a multidisciplinary survey of their applications in the recent years was presented. At a glance, an investigation of some of the essential qualities of a good global optimization test proble... |
Chapter 6 Discussion and Conclusions 6.1 Development of Ideas Robust optimization requires systems that can not only widely explore the problem space (such as global algorithms like EAs), but also effectively exploit the high quality regions. However, the struggle to establish optimum balance between exploration and ex... |
Investigations revealed that hybridization of EAs with local search methods popularly known as memetic algorithms often yield systems that are more robust than a mere combination of approximate algorithms. The proposal made in chapter 5 is an attempt to take a step further in this direction. |
It was noticed that the challenges in designing good and generic hybrid algorithms go much beyond the choice of the individual algorithms. And if given the required attention, improving the methodologies in which these algorithms individually operate and interact could certainly play a major role towards the overall su... |
It thus became apparent that there is a great deal of open research in this direction. Any further investigation with the aim of; sorting out the key features of the individual optimization algorithms that could be enhanced or added (as treated in chapter ); determining how best to improve the individual algorithms for... |
6.2 Discussion Several convergence detection methods are available in the literature most of which are based on some distance measures such as the Hamming distance. In a somewhat radical perspective, rather than directly monitoring the similarity among the solutions, the proposed convergence measure is designed to dete... |
in quantitative genetics. Our investigations reveal that convergence of evolution can effectively be measured via monitoring the effect of genetic operators (specifically, the crossover operator) on fitness progress in a population. Empirical results have shown that the value of the proposed convergence threshold param... |
, our investigation suggests that the new convergence threshold parameter could be bounded in [MATH] for crossover and mutation rates of [MATH] to [MATH] and [MATH] or [MATH] respectively; given the empirical investigation we have undertaken. Further investigations could provide more definitive evidence regarding any p... |
Turning to the local search method, gradient based local search algorithms are known for their suitability in deriving local optimal solution from virtually any given solution point. However, these algorithms rely on information about the slope of the problem under consideration to estimate their descent directions. Th... |
Amongst others, the proposed way of hybridizing evolutionary computation (EC) algorithm with the sequential quadratic programming (SQP) local search method (chapter ) would lessen the strong dependence of global search methods on a particular parameter setting for each problem category. In other words, the proposed met... |
6.3 Conclusions This report has investigated the reasons for the recent wide acceptance of hybrid algorithms in the field of global optimization. Particularly, the use of evolutionary algorithms together with gradient based local search method has been thoroughly examined. Then, a hybrid system composing of an evolutio... |
Prior to that an investigation on the parameterization aspect of the evolutionary algorithm was conducted and a new convergence threshold parameter was proposed for the EAs. Appropriate bound for the new convergence parameter was determined with regards to standard settings for other important EA parameters like crosso... |
Thereafter, a Newton based local search algorithm (SQP) was modified to use interior point method (IPM) instead of the conventional active set method (ASM) for solving its quadratic programming subproblems. Unlike in ASM approach where the search follows the boundary of the feasible region, IPM progresses by searching ... |
Returning to the hypotheses presented at the beginning of this study, based on the preliminary results obtained so far, it seems justifiable to infer that a skilful hybridization of evolutionary algorithms with a suitable local search method could yield a robust and efficient means of solving wide range of global optim... |
6.4 Research Contributions This study provides new understandings of the concepts and challenges behind global optimization approaches. Thus far, the following additions are made to the fast growing literature in the areas of local, global and hybrid optimization. |
1. An Adaptive Elitist Strategy: The broad survey conducted on various aspects of evolutionary algorithms gives valuable insight into the key parameterization issues upon which development of successful global optimization methods relied. Most importantly, it leads to the development of new ideas that give birth to the... |
2. A Convergence Detection via Monitoring Crossover: Using extended Price’s theorem , a critical convergence analysis of evolutionary algorithm that aids understanding of the interactions among genetic operators in an evolutionary search was conducted. As a result, a visual means for investigating the individual roles ... |
3. A Vectorized Forward Accumulation AD: The gradient based local search algorithms, particularly the standard sequential quadratic programming (SQP) algorithm were thoroughly analyzed. Then, the convergence characteristics of the SQP algorithm were upgraded to that of a [MATH] order algorithm that searches by taking f... |
4. A Validation Strategy: As a first attempt, we combine the global and local algorithms in a collaborative manner and the design seems to greatly enhance the balance in exploration and exploitation of search space which is a necessary ingredient for robust optimization. Nonetheless, the novel validation routine, among... |
6.5 Further work The results of the investigations undertaken have so far supported the core objective of this work. They have, in essence, opened several avenues for further investigation. Therefore, the other key objectives of this work can in no way be met without broadening the research in the following areas: |
1. Sensitivity Analysis: Further sensitivity analysis on the newly proposed automatic convergence detection parameter is necessary to determine all sorts of parameter settings where its application will be suitable in addition to the standard EA parameter settings (i.e. population size, crossover and mutation) explored... |
2. Hamming distance [MATH] Hamming distance between individual solutions should be estimated and the plots for the delta [MATH] changes in the Hamming distance can be compared side-by-side to the Price’s plots so as to validate the current findings. Use of cross correlation measurements could aid verify the hypothesis ... |
3. Other Seeding Potentials: Pertaining to the newly introduced validation routine, which is invoked at the end of every run of the local algorithm, the reported improvement is as a result of using single copies of the best solution and its inverted version to seed the creation of a new population which will mainly con... |
4. Extension to Algorithmic Portfolio: It would be interesting to assess the possibility of extending the proposed hybridization system to mimic the design of portfolios of algorithms |
. Thus, instead of forcing a halt at the end of every validation routine, the system might be left to decide whether to embark on a fresh run of the global and local algorithms. Although from the design of the proposed hybrid system the schedule for the portfolios can be self-adapted, the implications of utilizing best... |
5. Representation Sensitivity Considerations: The EC algorithm employed in this investigation utilizes binary representation of candidate solutions. Considering the implications of the role of representation on the overall performance of any evolutionary algorithm |
, a more equitable performance evaluation with other global optimization methods that are based on real valued representation may require restructuring the proposed method to use real-valued representation. This will aid further comparison with other well known hybrid metaheuristics. |
6. Accounting for Constraints and Dynamism: Further research needs to be done to enhance the proposed system to handle global optimization problems having not only parameter bounds, but also constraints. This is crucial to facilitate further investigations on its applicability to practical optimization problems that ma... |
7. Mapping to Feedback Control Systems: In the long run and if time permits, the proposed hybrid system could be utilized in online and offline parameter optimization of feedback control systems (such as in parameter tuning of the proportional-integral-derivative (PID) controllers) which are widely used in industries. |
8. Mapping to ANN Problem Domain: Additionally, we could map the proposed hybrid optimization technique to evolving-then-training ANNs. The EC algorithm could conduct initial search of the weights state-space and then switch to the local algorithm which would then do the final tuning of the weights. The system could th... |
# Source: arxiv 1304.1842 # Title: Information Measures of Complexity, Emergence, Self-organization, Homeostasis, and Autopoiesis # Sections: all # Downloaded: 2026-03-03T01:58:16.285061+00:00 |
Information Measures of Complexity, Emergence, Self-organization, Homeostasis, and Autopoiesis Abstract This chapter reviews measures of emergence, self-organization, complexity, homeostasis, and autopoiesis based on information theory. These measures are derived from proposed axioms and tested in two case studies: ran... |
Emergence is defined as the information a system or process produces. Self-organization is defined as the opposite of emergence, while complexity is defined as the balance between emergence and self-organization. Homeostasis reflects the stability of a system. Autopoiesis is defined as the ratio between the complexity ... |
Introduction In recent decades, the scientific study of complex systems (Bar-Yam, 1997 ; Mitchell, 2009 has demanded a paradigm shift in our worldviews (Gershenson et al 2007 ; Heylighen et al 2007 . Traditionally, science has been reductionistic. Still, complexity occurs when components are difficult to separate, due ... |
It can be said that this novel information is emergent , since it is not in the components, but produced by their interactions. Interactions can also be used by components to self-organize , i.e. produce a global pattern from local dynamics. Interactions are also key for feedback control loops, which help systems regul... |
We can see that reductionism is limited for describing such concepts as complexity, emergence, self-organization, and life. In the wake of the fall of reductionism as a dominant worldview (Morin, 2007 , a plethora of definitions, notions, and measures of these concepts has been proposed. Still, their diversity seems to... |
Measures are precise and formal. Measures are simple enough to be used and understood by people without a strong mathematical background. |
Measures can help clarify the meaning of the concepts they describe. Measures can be applied to any phenomenon, as anything can be described in terms of information (Gershenson, 2012b |
This chapter is organized as follows: In the next section, background concepts are presented, covering briefly complexity, emergence, self-organization, homeostasis, autopoiesis, information theory, random Boolean networks, and limnology. Section presents axioms and derives measures for emergence, self-organization, co... |
Background 2.1 Complexity There are dozens of notions and measures of complexity, proposed in different areas with different purposes (Edmonds, 1999 ; Lloyd, 2001 . Etymologically, complexity comes from the Latin plexus , which means interwoven. Thus, something complex is difficult to separate. This means that its comp... |
(Gershenson, 2013 . Thus, studying the components in isolation—as reductionistic approaches attempt—is not sufficient to describe the dynamics of complex systems. |
Nevertheless, it would be useful to have global measures of complexity, just as temperature characterizes the properties of kinetic energy of molecules or photons. Each component can have a different kinetic energy, but the statistical average is represented in the temperature. For complex systems, particular interacti... |
A useful measure of complexity should enable us to answer questions such as: Is a desert more or less complex than a tundra? What is the complexity of different influenza outbreaks? Which organisms are more complex: predators or preys; parasites or hosts; individual or social? What is the complexity of different music ... |
Moreover, with the recent scandalous increase of data availability in most domains, we urgently need measures to make sense of it. |
2.2 Emergence Emergence has probably been one of the most misused concepts in recent decades. The reasons for this misuse are varied and include: polysemy (multiple meanings), buzzwording, confusion, hand waving, Platonism, and even mysticism. Still, the concept of emergence can be clearly defined and understood (Ander... |
Some might perceive difficulties in describing phenomena at different scales (Gershenson, 2013 , but this is a consequence of attempting to find a single “true” description of phenomena. Phenomena do not depend on the descriptions we have of them, and we can have several different descriptions of the same phenomenon. I... |
2.3 Self-organization Self-organization has been used to describe swarms, flocks, traffic, and many other systems where the local interactions lead to a global pattern or behavior (Camazine et al 2003 ; Gershenson, 2007 Intuitively, self-organization implies that a system increases its own organization. This leads to t... |
In any case, it is convenient to have a measure of self-organization which can capture the nature of local dynamics at a global scale. This is especially relevant for the nascent field of guided self-organization (GSO) (Prokopenko, 2009 ; Ay et al 2012 ; Polani et al 2013 . GSO can be described as the steering of the s... |
(Gershenson, 2012a . This desired configuration will not always be the natural attractor of a controlled system. The mechanisms for guiding the dynamics and the design of such mechanisms will benefit from measures characterizing the dynamics of systems in a precise and concise way. |
2.4 Homeostasis Originally, the concept of homeostasis was developed to describe internal and physiological regulation of bodily functions, such as temperature or glucose levels. Probably the first person to recognize the internal maintenance of a near-constant environment as a condition for life was Bernard 1859 . Sub... |
Later, the British cybernetician William R. Ashby proposed, in an alternative form, that homeostasis implicates an adaptive reaction to maintain “essential variables” within a range (Ashby, 1947a 1960 . In order to explain the generation of behavior and learning in machines and living systems, Ashby also contributed by... |
A dynamical system has a high homeostatic capacity if it is able to maintain its dynamics close to a certain state or states (attractors). As explained above, when perturbations or environmental changes occur, the system adapts to face the changes within the viability zone, that is, without the system “breaking” (Ashby... |
2.5 Autopoiesis Autopoiesis comes from the Greek auto (self) and poiesis (creation, production) and was proposed as a concept to define the living. According to Maturana 2011 , the notion of autopoiesis was created to connote and describe the molecular processes taking place in the realization of living beings as auton... |
In the context of autopoiesis, living beings occur as discrete autonomous dynamic molecular autopoietic entities. These entities are in a continuous realization of their self-production. Thus, autopoiesis describes the internal dynamics of a living system in the molecular domain. Maturana notices that living beings are... |
For example, a bacterium may produce another bacterium by cellular division, while a virus requires a host cell to produce another virus. The production of the new bacterium is made by the interactions between the elements of another bacterium. The production of a new virus depends on interactions between elements of a... |
The concept of autopoiesis has been extended to other areas beyond biology (Luisi, 2003 ; Seidl, 2004 ; Froese and Stewart, 2010 , although no formal measure had been proposed so far. |
2.6 Information Theory Information has had a most interesting history (Gleick, 2011 Information theory was created by Claude Shannon in 1948 in the context of telecommunications. He analyzed whether it was possible to reconstruct data transmitted across a noisy channel. In his model, information is represented as a str... |
Shannon was interested in a function to measure how much information a process “produces”. Quoting Shannon ( 1948 Suppose we have a set of possible events whose probabilities of occurrence are [MATH] . These probabilities are known but that is all we know about the event that might occur. Can we find a measure of how m... |
1. should be continuous in each [MATH] 2. If all the [MATH] are equal, [MATH] , then should be a monotonic increasing function of [MATH] . With equally [MATH] likely events there is more choice, or uncertainty, when there are more possible events. |
3. If a choice be broken down into two successive choices, the original should be the weighted sum of the individual values of With these few axioms , Shannon demonstrates that the only function satisfying the three above is of the form: |
[EQUATION] where [MATH] is a positive constant. For example, if we have a string ‘0001000100010001…’, we can estimate [MATH] and [MATH] , then [MATH] . If we use [MATH] and a base 2 logarithm, then [MATH] |
Shannon used [MATH] to describe information (we are using [MATH] ) because he was thinking in the Boltzmann’s theorem when he developed the theory. Therefore, he called equation the entropy of the set of probabilities [MATH] . In modern words, is a function of a random variable [MATH] |
The unit of information is the bit ( bi nary digi ). One bit represents the information gained when a binary random variable becomes known. However, since equation is a sum of probabilities, Shannon’s information is a unitless measure. |
More details about information theory in general can be found in Ash ( 1990 , while a primer on information theory related to complexity, self-organization, and emergence is found in Prokopenko et al 2009 |
2.7 Random Boolean Networks Random Boolean networks (RBNs) are abstract computational models, originally proposed to study genetic regulatory networks (Kauffman, 1969 1993 . However, being general models, their study and use has expanded beyond biology (Aldana-González et al 2003 ; Gershenson, 2004 2012a |
A RBN is formed by [MATH] nodes linked by [MATH] connections . Each node has a Boolean state, i.e. zero or one. The future state of each node is determined by the current states of the nodes that link to it and a lookup table which specifies how the update will take place. The connectivity (which nodes affect which) an... |
RBNs have been found to have three different dynamical regimes, which have been studied extensively (Gershenson, 2004 Ordered. Most nodes are static, RBNs are robust to perturbations. |
Chaotic. Most nodes are changing, RBNs are fragile to perturbations. Critical. Some nodes are changing, RBNs have adaptive potential. |
Different parameters and properties determine the regime, which can be used to guide a particular RBN towards a desired regime (Gershenson, 2012a |
It can be said that the critical regime balances the robustness of the chaotic regime and the changeability of the chaotic regime. It has been argued that computation and life require this balance to be able to compute and adapt (Langton, 1990 ; Kauffman, 1993 |
RBNs will be used in Section 4.1 to illustrate the measures proposed in the next section. 2.8 Limnology Lakes are studied by limnology. Lakes can be divided in different zones, as shown in Figure : (i) The macrophyte zone, composed mainly of aquatic plants, which are rooted, floating or submerged. (ii) The planktonic z... |
At different zones, one or more components or subsystems can be an assessment for the ecosystem dynamics. For our case study to be presented in Section 4.2 , we considered three components: physiochemical, limiting nutrients and photosynthetic biomass for the planktonic and benthic zones. |
The physiochemical component refers to the chemical composition of water. It is affected by various conditions and processes such as geological nature, the water cycle, dispersion, dilution, solute and solids generation (e.g. photosynthesis), and sedimentation. In this component, we highlight two water variables that a... |
Related to the physiochemical component, limiting nutrients which are basic for photosynthesis are associated with the biogeochemical cycles of nitrogen, carbon, and phosphorous. These cycles permit the adsorption of gases into the water or the dilution of some limiting nutrients. |
In addition, among limnetic biota, photoautotrophic biomass is the basis for the trophic web establishment. The term autotrophs is used for organisms that increase their mass through the accumulation of proteins which they manufacture, mainly from inorganic radicals (Stumm, 2004 . This type of organisms can be found at... |
The previous basic limnology concepts will be useful to follow the case study of an Arctic lake, presented in Section 4.2 Measures |
We have recently proposed earlier versions of the measures presented in this chapter (Fernández et al 2012 ; Gershenson and Fernández, 2012 . The ones presented here are more refined and are based on axioms. The benefit of using axioms is that the discussion is not taken so much at the level of the measures, but at the... |
A comparison of the proposed measures with others can be found in Gershenson and Fernández ( 2012 . It is worth noting that all of the proposed measures are unitless. |
3.1 Emergence We mentioned that emergence refers to properties of a phenomenon which are present at one scale and are not at another scale. Scales can be temporal or spatial. If we describe phenomena in terms of information, in order to have “new” information, “old” information has to be transformed . This transformati... |
[MATH] includes both dynamic emergence and scale emergence. If we recall, Shannon proposed a quantity which measures how much information a process “produces”. Therefore, we can say that emergence is the same as Shannon’s information [MATH] From now on, we will consider the emergence of a process as the information and... |
[EQUATION] We now revise that the intuitive idea of emergence fulfills the three basic notions (axioms) that Shannon used to derive (Shannon’s [MATH] ). For the continuity axiom, it is expected of a measure not to give big jumps when small changes are made. The second axiom will be harder to show. It states that if we ... |
[MATH] , as well as [MATH] , is a probabilistic measure. [MATH] means that when any random binary variable becomes known, one bit of information emerges. If [MATH] , then no new information will emerge, even as random binary variables become “known” (they are known beforehand). Again, we emphasize that emergence can ta... |
3.1.1 Multiple Scales When Shannon defined equation , he included [MATH] which is a positive constant. This is important because we will change the value of [MATH] to normalize a measure onto the [MATH] interval. The value of [MATH] will depend on the length of the finite alphabet [MATH] we use. In the particular Boole... |
[EQUATION] where [MATH] is the length of the alphabet we use. In this way we will normalize [MATH] and measures derived from it, having a maximum of 1 and a minimum of 0. |
For example, consider the string in base 4 ‘0133013301330133…’. We can estimate [MATH] [MATH] , and [MATH] . Following equation , we have |
[MATH] . Since [MATH] [MATH] . Thus, we obtain a normalized [MATH] 3.2 Self-organization Self-organization has been correlated with an increase in order, i.e. a reduction of entropy (Gershenson and Heylighen, 2003 . If emergence implies an increase of information, which is analogous to entropy and disorder, self-organi... |
A measure of self-organization [MATH] should be a function [MATH] (where [MATH] ) with the following properties: 1. The range of [MATH] is the real interval [MATH] |
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