This book will leave readers inspired to develop simplicial versions of other algorithms for global optimization and even use other non-rectangular partitions for special applications. Product Details Table of Contents. Table of Contents 1. Average Review. Write a Review. Related Searches.
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Algorithms and Classification in Combinatorial Group Theory. The papers in this volume are the result of a workshop held in January The papers in this volume are the result of a workshop held in January at the Mathematical Sciences Research Institute. Topics covered include decision problems, finitely presented simple groups, combinatorial geometry and homology, and automatic groups and related topics. View Product. Compatible Spatial Discretizations. This volume contains This volume contains original contributions based on the material presented there.
A unique feature is the inclusion of work that is representative of the recent developments in compatible One of the most effective ways to stimulate students to enjoy intellectual efforts is the One of the most effective ways to stimulate students to enjoy intellectual efforts is the scientific competition. In the Hungarian Mathematical and Physical Society introduced a mathematical competition for high school students. The success of high school competitions led Current Topics in Medical Mycology. This latest volume in the Current Topics in Medical Mycology series brings together internationally recognized This latest volume in the Current Topics in Medical Mycology series brings together internationally recognized researchers to summarize current topics of interest to medical mycologists and other scientists who are working in microbiology and immunology.
A blend of contemporary, authoritative Readers will see techniques applied to medical imaging such as Radon transformation, image reconstruction, image rendering, image enhancement Ecological Genetics: The Interface. Traditionally, studies in ecological genetics have involved both field observations and laboratory genetic analyses. Factorization and Primality Testing. Due to the hardness of global optimization problems and different nature of methods from these two groups, the problem of their comparison is very difficult and methods are collated on some dozens of test functions 1 , 2 , 15 , 16 , 19 , 20 giving so a poor information and non reliable results.
In order to bridge the gap between the two communities we propose a new efficient visual technique for a systematic comparison of global optimization algorithms having different nature. More than , runs on randomly generated multidimensional test problems have been performed to compare five popular stochastic metaheuristics and three deterministic methods giving so a new level of understanding the tested algorithms. The test problems 21 have been chosen because, after they have been randomly generated, the optimizer is provided with locations of the global minimum and of all local minimizers this property has made the generator of these test problems very popular—it is used nowadays in more than 40 countries of the world.
The knowledge of the global solution gives the possibility to check whether the tested method has found the global optimum. Since in practical problems the global solution is unknown and, therefore, it is not possible to check the quality of the obtained solution, it is very important to see how different methods are close to the global solution after their stopping rule has been satisfied. It is well-known that a general continuous global optimization problem 1 is NP-hard 22 — This is true also, in particular, for problems 1 where the objective function f x satisfies the Lipschitz condition.
This condition means that any limited change in the parameters yields some limited changes in the values of the objective function. The assumption 2 can be justified by the fact that in technical systems the energy of change is always limited. In fact, this kind of problems can be very frequently met in practice see 1 — 3 , 5 , 7 , 18 , in particular, in many engineering applications in which observations of the produced values of f x can be made, but analytical expressions of the functions are not available.
On Simplicial Longest Edge Bisection in Lipschitz Global Optimization
For example, the values of the objective function f x can be obtained by running some computationally expensive numerical models, by performing a set of experiments, and so on. One may refer, for instance, to various decision-making problems in automatic control and robotics, structural optimization, engineering design, etc. In the traditional local optimization 9 , where strong assumptions on the structure of the objective function such as convexity, continuity, differentiability, etc.
In these cases, the dimensionality of the solved problem is often a measure of the goodness of optimization algorithms. Under the latter statement, not the dimension of the problem that is important in local optimization but the number of allowed function evaluations often called budget becomes critical. In other words, when one has the possibility to evaluate f x M times these evaluations are called trials hereinafter in the global optimization problem of the dimension 5, 10 or , then the quality of the found solution after M evaluations is crucial and not the dimensionality of f x.
This happens because it is not possible to adequately explore the multi-dimensional search region D at this limited budget of expensive evaluations of f x. This means that one million of trials is not sufficient not only to explore well the whole region D but even to evaluate f x at all vertices of D.
Thus, the statement P2 makes sense both because in practice the budget is always limited and because the problem under consideration is NP-hard. They mean just that the found solution was better than solutions found by other competitors this is especially true for highly dimensional global optimization problems where the global solutions are unknown.
That is why the possibility to compare the found solutions with the known global optimum offered by the generator of classes of test functions 21 is very precious.
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Let us describe now two groups of methods used in different communities and studied here. Metaheuristic algorithms widely used to solve in sense of the statement P2 discussed above real-life global optimization problems have a number of attractive properties that have ensured their success among engineers and practitioners. First, they have limpid nature-inspired interpretations explaining how these algorithms simulate behavior of populations of individuals.
Algorithms of this type studied here are: Particle Swarm Optimization PSO simulating fish schools 11 , Firefly Algorithm FA simulating the flashing behavior of the fireflies 13 , Artificial Bee Colony ABC representing a colony of bees in searching the food sources 14 , Differential Evolution DE and Genetic algorithms GA simulating the evolution on a phenotype and genotype level, respectively 4 , 6.
Other reasons that have led to a wide spread of metaheuristics are the following: it is not required to have a high level mathematical preparation to understand them; their implementation usually is simple and many codes are available for free; finally, they do not need a lot of memory working at each moment with only a limited population of points in the search domain. In fact, populations used by these methods can degenerate prematurely, returning only a locally optimal solution instead of the global one or even non locally optimal point if it has been obtained at one of the last evaluations of f x and the budget of M evaluations has not allowed to proceed with an improvement of the obtained solution.
Deterministic algorithms belonging to the second group of methods studied here are based on the knowledge that the objective function f x satisfies the Lipschitz condition 2. Lipschitz global optimization algorithms is a well-studied class of deterministic methods 1 — 3 , 5 , 7 , These methods are usually technically more sophisticated than metaheuristics, their implementation is not so easy, they require more memory and a higher mathematical preparation is necessary to understand and to use them.
Commonly, they have a strong theory ensuring convergence to the global solution and a small number of control parameters allowing so their users to configure the search easily. Even though the Lipschitz constant L can be unknown, there exist several strategies for its estimation 2 , 3 , 5 , 7 , 18 and one of the most frequently used techniques 16 works with all possible values of L from zero to infinity simultaneously.
All deterministic algorithms considered here use it. How can one compare these two groups of methods? On the one hand, there exist several approaches for a visual comparison of deterministic algorithms see, e. However, they do not allow one to compare stochastic methods. On the other hand, comparison of metaheuristics often is performed on different collections of single benchmark problems 15 , 20 , As a result, the difficulty of test problems in collections can vary significantly leading sometimes to non homogeneous and, as a consequence, non reliable results.
An additional difficulty consists of the fact that, due to a stochastic nature of metaheuristics, the obtained results cannot be repeated and have a character of some averages. Thus, the difficulties existing in performing a reliable comparison of these two groups of methods constitutes a serious gap between the respective communities. The goal of this paper is to start a dialog between them by proposing a methodology allowing one to compare numerically deterministic algorithms and stochastic metaheuristics using the problem statement P1.
Instead of traditional comparisons executed just on several dozens of tests 1 , 2 , 15 , 16 , 19 , 20 in this contribution more than , runs on randomly generated test problems 21 have been performed for a systematic comparison of the methods. In order to make this comparison more reliable, parameters of all tested algorithms were fixed following recommendations of their authors and then were used in all the experiments. One known and two novel methodologies for comparing global optimization algorithms are applied here: Operational Characteristics 25 for comparing deterministic algorithms and new Operational Zones and Aggregated Operational Zones generalizing ideas of operational characteristics to collate multidimensional stochastic algorithms.
An operational characteristic 25 constructed on a class of randomly generated test functions is a graph showing the number of solved problems in dependence on the number of executed evaluations of the objective function f x. To construct classes of test functions required to build operational characteristics, the popular GKLS generator 21 of multidimensional, multiextremal test functions was used. This generator allows one to generate randomly classes of test problems having the same dimension, number of local minima, and difficulty. The property making this generator especially attractive consists of the fact that for each function a complete information of coordinates and values of all local minima including the global one is provided.
Here, 8 different classes from 18 were used see supplementary materials for their description and for definition of what does it mean that a problem has been solved.
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These classes and the respective search accuracies have been taken since they represent a well established tool used frequently to compare deterministic global optimization algorithms 18 , 28 — Higher is a characteristic of a method with respect to characteristics of its competitors better is the behavior of this method. Operational characteristics allow us also to see the best performers in dependence on the available budget of evaluations of f x. For instance, it can be seen from Fig.
Construction of operational characteristics for deterministic methods and of the operational zone for metaheuristic Firefly Algorithm FA built on the hard 5-dimensional class of GKLS test functions. The upper and the lower boundaries of the zone are shown as dark blue curves. Since operational characteristics cannot be used to compare stochastic methods, we propose in this paper a new methodology called operational zones that can be used for collating stochastic algorithms. Then, each run of a tested metaheuristic was considered as a particular method and its operational characteristic was constructed.
The totality of all operational characteristics form the respective operational zone see Fig. Then, the upper and the lower boundaries of the zone shown in Fig. The graph for the average performance within the zone can be also depicted see Fig. The joint representation of operational zones together with characteristics offers a lot of visual information. It can be seen, for example, in Fig.
If the budget is less than 30, trials see Fig. If the budget is higher than 40, trials than ADC behaves better since its characteristic is higher than the upper boundary of this FA zone. Notice also that Fig. For the same two test classes, Fig.
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One can see also that in many runs metaheuristics got trapped into local minima and were not able to exit from their attraction regions producing so operational zones with long horizontal parts see, e. This means that increasing the number of trials does not improve results in this case and it is necessary to restart metaheuristics. Aggregated operational zones proposed in this paper show what happens in this case. They are constructed as follows. Then, for non-solved problems the algorithm is launched again with the same number n max of allowed trials.
In this way, T runs are executed to complete the aggregated characteristic. The lower and upper boundaries are defined analogously. Results of the experiments. For each test class the average number of trials required to solve all problems is presented for each deterministic algorithm. In this case, the maximal number of trials set to 10 6 was used to calculate the average number of trials m. It should be stressed that the aggregated operational zones allow one to emphasize better the potential of nature-inspired metaheuristics.
It can be seen from Fig. In contrast, the aggregated zone of ABC is higher than the characteristics of both deterministic methods, i. Notice that for deterministic methods and metaheuristics, due to the stochastic nature of the latter ones, different averages should be used: for metaheuristics the results on 10, runs for each class are used, whereas for the deterministic algorithms results on runs one run for each of functions. This creates difficulties in comparing. To see the detailed results, larger tables with hundreds of rows and columns should be used, complicating so the visual analysis of the results.
In contrast, operational zones very well present visually performance of tested methods giving the entire panorama of their behavior for different budgets. The average, the best, and the worst cases for each metaheuristic can be easily obtained from the graphs for any chosen number of trials. Let us see now another way for a statistical comparison of the two groups of algorithms using the same data. Let X A C be a random variable describing the consumed percentage of the computational budget N max performed by an algorithm A for solving a problem from the test class C.
Then, after the construction of the cumulative distribution functions F X A C x , one can obtain the sampled distribution quantiles of X A C. These results can be interpreted as follows. For each algorithm, quantiles Q 25 , Q 50 , Q 75 and Q 90 for the number of trials for simple test classes are presented. For each algorithm, quantiles Q 25 , Q 50 , Q 75 , and Q 90 for the number of trials for hard test classes are presented.