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1.
The paper describes methods for solving very large overdetermined algebraic polynomial systems on an example that appears from a classification of all integrable 3-dimensional scalar discrete quasilinear equations Q 3=0 on an elementary cubic cell of the lattice ℤ3. The overdetermined polynomial algebraic system that has to be solved is far too large to be formulated. A “probing” technique, which replaces independent variables by random integers or zero, allows to formulate subsets of this system. An automatic alteration of equation formulating steps and equation solving steps leads to an iteration process that solves the computational problem. The text was submitted by the author in English.  相似文献   

2.
Engineering with Computers - Multi-objective optimization has been rising in popularity, especially within an industrial environment, where several cost functions often need to be considered during...  相似文献   

3.
Multi-objective optimization of simulated stochastic systems aims at estimating a representative set of Pareto optimal solutions and a common approach is to rely on metamodels to alleviate computational costs of the optimization process. In this article, both the objective and constraint functions are assumed to be smooth, highly non-linear and computationally expensive and are emulated by stochastic Kriging models. Then a novel global optimization algorithm, combing the expected hypervolume improvement of approximated Pareto front and the probability of feasibility of new points, is proposed to identify the Pareto front (set) with a minimal number of expensive simulations. The algorithm is suitable for the situations of having disconnected feasible regions and of having no feasible solution in initial design. Then, we also quantify the variability of estimated Pareto front caused by the intrinsic uncertainty of stochastic simulation using nonparametric bootstrapping method to better support decision making. One test function and an (s, S) inventory system experiment illustrate the potential and efficiency of the proposed sequential optimization algorithm for constrained multi-objective optimization problems in stochastic simulation, which is especially useful in Operations Research and Management Science.  相似文献   

4.
This paper presents a new method that effectively determines a Pareto front for bi-objective optimization with potential application to multiple objectives. A traditional method for multiobjective optimization is the weighted-sum method, which seeks Pareto optimal solutions one by one by systematically changing the weights among the objective functions. Previous research has shown that this method often produces poorly distributed solutions along a Pareto front, and that it does not find Pareto optimal solutions in non-convex regions. The proposed adaptive weighted sum method focuses on unexplored regions by changing the weights adaptively rather than by using a priori weight selections and by specifying additional inequality constraints. It is demonstrated that the adaptive weighted sum method produces well-distributed solutions, finds Pareto optimal solutions in non-convex regions, and neglects non-Pareto optimal solutions. This last point can be a potential liability of Normal Boundary Intersection, an otherwise successful multiobjective method, which is mainly caused by its reliance on equality constraints. The promise of this robust algorithm is demonstrated with two numerical examples and a simple structural optimization problem.  相似文献   

5.
多目标强度Pareto 混沌差分进化算法   总被引:1,自引:0,他引:1  
提出一种多目标强度Pareto混沌差分进化算法(SPCDE).首先利用Tent映射进行种群的混沌初始化,采用一种基于均匀排挤机制的截断排挤操作和混沌替换操作进行种群的环境选择操作;然后基于一种变缩放因子的差分变异策略进行变异操作,通过计算支配关系得到变异个体;最后通过支配关系的计算和环境选择操作进行进化选择操作并得到子代个体.以上操作不仅提高了算法的收敛性能,而且保证了Pareto最优解的均匀分布性.数值实验结果表明了该算法的有效性.  相似文献   

6.
We introduce a simple evolution scheme for multiobjective optimization problems, called the Pareto Archived Evolution Strategy (PAES). We argue that PAES may represent the simplest possible nontrivial algorithm capable of generating diverse solutions in the Pareto optimal set. The algorithm, in its simplest form, is a (1 + 1) evolution strategy employing local search but using a reference archive of previously found solutions in order to identify the approximate dominance ranking of the current and candidate solution vectors. (1 + 1)-PAES is intended to be a baseline approach against which more involved methods may be compared. It may also serve well in some real-world applications when local search seems superior to or competitive with population-based methods. We introduce (1 + lambda) and (mu + lambda) variants of PAES as extensions to the basic algorithm. Six variants of PAES are compared to variants of the Niched Pareto Genetic Algorithm and the Nondominated Sorting Genetic Algorithm over a diverse suite of six test functions. Results are analyzed and presented using techniques that reduce the attainment surfaces generated from several optimization runs into a set of univariate distributions. This allows standard statistical analysis to be carried out for comparative purposes. Our results provide strong evidence that PAES performs consistently well on a range of multiobjective optimization tasks.  相似文献   

7.
This paper attempts to argue that most adaptive systems, such as evolutionary or learning systems, have inherently multiple objectives to deal with. Very often, there is no single solution that can optimize all the objectives. In this case, the concept of Pareto optimality is key to analyzing these systems. To support this argument, we first present an example that considers the robustness and evolvability trade-off in a redundant genetic representation for simulated evolution. It is well known that robustness is critical for biological evolution, since without a sufficient degree of mutational robustness, it is impossible for evolution to create new functionalities. On the other hand, the genetic representation should also provide the chance to find new phenotypes, i.e., the ability to innovate. This example shows quantitatively that a trade-off between robustness and innovation does exist in the studied redundant representation. Interesting results will also be given to show that new insights into learning problems can be gained when the concept of Pareto optimality is applied to machine learning. In the first example, a Pareto-based multi-objective approach is employed to alleviate catastrophic forgetting in neural network learning. We show that learning new information and memorizing learned knowledge are two conflicting objectives, and a major part of both information can be memorized when the multi-objective learning approach is adopted. In the second example, we demonstrate that a Pareto-based approach can address neural network regularizationmore elegantly. By analyzing the Pareto-optimal solutions, it is possible to identifying interesting solutions on the Pareto front.  相似文献   

8.
9.
Evolutionary Algorithms (EAs) have been recognized to be well suited to approximate the Pareto front of Multi-objective Optimization Problems (MOPs). In reality, the Decision Maker (DM) is not interested in discovering the whole Pareto front rather than finding only the portion(s) of the front that matches at most his/her preferences. Recently, several studies have addressed the decision-making task to assist the DM in choosing the final alternative. Knee regions are potential parts of the Pareto front presenting the maximal trade-offs between objectives. Solutions residing in knee regions are characterized by the fact that a small improvement in either objective will cause a large deterioration in at least another one which makes moving in either direction not attractive. Thus, in the absence of explicit DM’s preferences, we suppose that knee regions represent the DM’s preferences themselves. Recently, few works were proposed to find knee regions. This paper represents a further study in this direction. Hence, we propose a new evolutionary method, denoted TKR-NSGA-II, to discover knee regions of the Pareto front. In this method, the population is guided gradually by means of a set of mobile reference points. Since the reference points are updated based on trade-off information, the population converges towards knee region centers which allows the construction of a neighborhood of solutions in each knee. The performance assessment of the proposed algorithm is done on two- and three-objective knee-based test problems. The obtained results show the ability of the algorithm to: (1) find the Pareto optimal knee regions, (2) control the extent (We mean by extent the breadth/spread of the obtained knee region.) of the obtained regions independently of the geometry of the front and (3) provide competitive and better results when compared to other recently proposed methods. Moreover, we propose an interactive version of TKR-NSGA-II which is useful when the DM has no a priori information about the number of existing knees in the Pareto optimal front.  相似文献   

10.
Corner characterization by differential geometry techniques   总被引:2,自引:0,他引:2  
An algorithm is presented for the characterization of the corner points in an image by means of three parameters of easily understandable physical meaning, namely the amplitude, aperture, and smoothness of the blurred wedge that best fits the data. Formulas are also given for the computation of the gradients and curvatures for the surface obtained convolving the step wedge of amplitude A and aperture φ with the bidimensional Gaussian function with variance σ2.  相似文献   

11.
This paper attempts to argue that most adaptive systems, such as evolutionary or learning systems, have inherently multiple objectives to deal with. Very often, there is no single solution that can optimize all the objectives. In this case, the concept of Pareto optimality is key to analyzing these systems. To support this argument, we first present an example that considers the robustness and evolvability trade-off in a redundant genetic representation for simulated evolution. It is well known that robustness is critical for biological evolution, since without a sufficient degree of mutational robustness, it is impossible for evolution to create new functionalities. On the other hand, the genetic representation should also provide the chance to find new phenotypes, i.e., the ability to innovate. This example shows quantitatively that a trade-off between robustness and innovation does exist in the studied redundant representation. Interesting results will also be given to show that new insights into learning problems can be gained when the concept of Pareto optimality is applied to machine learning. In the first example, a Pareto-based multi-objective approach is employed to alleviate catastrophic forgetting in neural network learning. We show that learning new information and memorizing learned knowledge are two conflicting objectives, and a major part of both information can be memorized when the multi-objective learning approach is adopted. In the second example, we demonstrate that a Pareto-based approach can address neural network regularizationmore elegantly. By analyzing the Pareto-optimal solutions, it is possible to identifying interesting solutions on the Pareto front.  相似文献   

12.
An algorithm for theorem proving in differential geometry based on the calculation of the differential dimension of differential quasi-algebraic sets is shown. In the case in which only ordinary differential equations are involved, an algorithm for such computation is presented. Different notions of validity for differential geometry statements are also compared.This paper was supported by Italian M.P.I. (40% 1985).  相似文献   

13.
14.
In this paper, the diversity information included by dominating number is analyzed, and the probabilistic relationship between dominating number and diversity in the space of objective function is proved. A ranking method based on dominating number is proposed to build the Pareto front. Without increasing basic Pareto method’s computation complexity and introducing new parameters, a new multiobjective genetic algorithm based on proposed ranking method (MOGA-DN) is presented. Simulation results on function optimization and parameters optimization of control system verify the efficiency of MOGA-DN.  相似文献   

15.
New challenges in engineering design lead to multiobjective (multicriteria) problems. In this context, the Pareto front supplies a set of solutions where the designer (decision-maker) has to look for the best choice according to his preferences. Visualization techniques often play a key role in helping decision-makers, but they have important restrictions for more than two-dimensional Pareto fronts. In this work, a new graphical representation, called Level Diagrams, for n-dimensional Pareto front analysis is proposed. Level Diagrams consists of representing each objective and design parameter on separate diagrams. This new technique is based on two key points: classification of Pareto front points according to their proximity to ideal points measured with a specific norm of normalized objectives (several norms can be used); and synchronization of objective and parameter diagrams. Some of the new possibilities for analyzing Pareto fronts are shown. Additionally, in order to introduce designer preferences, Level Diagrams can be coloured, so establishing a visual representation of preferences that can help the decision-maker. Finally, an example of a robust control design is presented - a benchmark proposed at the American Control Conference. This design is set as a six-dimensional multiobjective problem.  相似文献   

16.
17.
In this paper, the problem of the determination of Pareto optimal solutions for certain large-scale systems with multiple conflicting objectives is considered. As a consequence, a two-level hierarchical method is proposed, where the global problem is decomposed into smaller multiobjective problems (lower level) which are coordinated by an upper level that has to take into account the relative importance assigned to each subsystem. The scheme that has been developed is an iterative one, so that a continuous information exchange is carried out between both levels in order to obtain efficient solutions for the initial global problem. The practical implementation of the developed scheme allows us to prove its efficiency in terms of processing time.Scope and PurposeMany are the problems that can arise when attempting to modelize and solve real problems using mathematical techniques. Among them, two questions must be pointed out. First, decisions are usually taken according to several criteria which are in conflict among them, rather than as the result of the optimization of a single objective. This fact has been faced by the Multiple Criteria Decision Analysis in its many aspects (see, for example, Ignizio, Goal Programming and Extensions, Lexington Books, Massachusets, 1976 or Steuer, Multiple Criteria Optimization: Theory, Computation and Application, Wiley, New York, 1986 for an overview of the problems and techniques). Second, real problems are usually very large and complex, in the sense that many variables and constraints are involved, and complex relations hold among them. In particular, many companies have a hierarchical structure with different decision levels. Such models have been studied in the literature (see Singh, Titli, Systems: Decomposition, Optimization and Control, Pergamon Press, New York, 1978). This paper follows the line of others like Haimes et al. (Hierarchical-Multiobjective Analysis of Large-Scale Systems, Hemisphere, New York, 1990), where both aspects are combined. Namely, an algorithm is designed to generate non-dominated solutions for a hierarchical multiple objective model.  相似文献   

18.
Truong  Duy Tin  Battiti  Roberto 《Machine Learning》2015,98(1-2):57-91

Supervised alternative clustering is the problem of finding a set of clusterings which are of high quality and different from a given negative clustering. The task is therefore a clear multi-objective optimization problem. Optimizing two conflicting objectives at the same time requires dealing with trade-offs. Most approaches in the literature optimize these objectives sequentially (one objective after another one) or indirectly (by some heuristic combination of the objectives). Solving a multi-objective optimization problem in these ways can result in solutions which are dominated, and not Pareto-optimal. We develop a direct algorithm, called COGNAC, which fully acknowledges the multiple objectives, optimizes them directly and simultaneously, and produces solutions approximating the Pareto front. COGNAC performs the recombination operator at the cluster level instead of at the object level, as in the traditional genetic algorithms. It can accept arbitrary clustering quality and dissimilarity objectives and provides solutions dominating those obtained by other state-of-the-art algorithms. Based on COGNAC, we propose another algorithm called SGAC for the sequential generation of alternative clusterings where each newly found alternative clustering is guaranteed to be different from all previous ones. The experimental results on widely used benchmarks demonstrate the advantages of our approach.

  相似文献   

19.
In evolutionary multi-objective optimization (EMO), the convergence to the Pareto set of a multi-objective optimization problem (MOP) and the diversity of the final approximation of the Pareto front are two important issues. In the existing definitions and analyses of convergence in multi-objective evolutionary algorithms (MOEAs), convergence with probability is easily obtained because diversity is not considered. However, diversity cannot be guaranteed. By combining the convergence with diversity, this paper presents a new definition for the finite representation of a Pareto set, the B-Pareto set, and a convergence metric for MOEAs. Based on a new archive-updating strategy, the convergence of one such MOEA to the B-Pareto sets of MOPs is proved. Numerical results show that the obtained B-Pareto front is uniformly distributed along the Pareto front when, according to the new definition of convergence, the algorithm is convergent.  相似文献   

20.
在多目标进化算法中,时间复杂度过高是普遍的问题,特别是三个目标函数以上时,解的等级分配占用了过多运算时间。针对三目标问题,利用帕累托支配关系,对解的等级分配进行研究,发现经典的等级排序及分配方法存在一定冗余操作,需对全部的解先排序后,才能再分配等级并选择下一代,造成部分不必要的运算。为减少该冗余,利用帕累托非支配关系结合差分进化,实现高效三目标进化算法。算法每次迭代对种群中最高等级的个体进行计算,在分配等级同时进行选择后代个体操作,当后代种群生成时便跳出计算,从而减少个体的计算数量,降低运算量,同时给出该方法的相关理论分析和证明过程。然后,针对一系列三目标优化问题,将提出方法与著名排序方法NSGAII,及近年来优秀的ENS方法进行对比实验。仿真实验结果表明,提出方法在时间复杂度和收敛速度上优于经典方法,稍差于ENS方法。在标准测试函数DTLZ1-DTLZ6的性能上,提出方法近似于ENS方法,优于NSGAII算法,从而验证了提出方法的有效性和正确性。  相似文献   

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