Phase transitions in Pareto optimal complex networks

The organization of interactions in complex systems can be described by networks connecting different units. These graphs are useful representations of the local and global complexity of the underlying systems. The origin of their topological structure can be diverse, resulting from different mechanisms including multiplicative processes and optimization. In spatial networks or in graphs where cost constraints are at work, as it occurs in a plethora of situations from power grids to the wiring of neurons in the brain, optimization plays an important part in shaping their organization. In this paper we study network designs resulting from a Pareto optimization process, where different simultaneous constraints are the targets of selection. We analyze three variations on a problem finding phase transitions of different kinds. Distinct phases are associated to different arrangements of the connections; but the need of drastic topological changes does not determine the presence, nor the nature of the phase transitions encountered. Instead, the functions under optimization do play a determinant role. This reinforces the view that phase transitions do not arise from intrinsic properties of a system alone, but from the interplay of that system with its external constraints.Comment: 14 pages, 7 figure

Seeking multiple solutions:an updated survey on niching methods and their applications

Multi-Modal Optimization (MMO) aiming to locate multiple optimal (or near-optimal) solutions in a single simulation run has practical relevance to problem solving across many fields. Population-based meta-heuristics have been shown particularly effective in solving MMO problems, if equipped with specificallydesigned diversity-preserving mechanisms, commonly known as niching methods. This paper provides an updated survey on niching methods. The paper first revisits the fundamental concepts about niching and its most representative schemes, then reviews the most recent development of niching methods, including novel and hybrid methods, performance measures, and benchmarks for their assessment. Furthermore, the paper surveys previous attempts at leveraging the capabilities of niching to facilitate various optimization tasks (e.g., multi-objective and dynamic optimization) and machine learning tasks (e.g., clustering, feature selection, and learning ensembles). A list of successful applications of niching methods to real-world problems is presented to demonstrate the capabilities of niching methods in providing solutions that are difficult for other optimization methods to offer. The significant practical value of niching methods is clearly exemplified through these applications. Finally, the paper poses challenges and research questions on niching that are yet to be appropriately addressed. Providing answers to these questions is crucial before we can bring more fruitful benefits of niching to real-world problem solving

A memetic particle swarm optimisation algorithm for dynamic multi-modal optimisation problems

Copyright @ 2011 Taylor & Francis.Many real-world optimisation problems are both dynamic and multi-modal, which require an optimisation algorithm not only to find as many optima under a specific environment as possible, but also to track their moving trajectory over dynamic environments. To address this requirement, this article investigates a memetic computing approach based on particle swarm optimisation for dynamic multi-modal optimisation problems (DMMOPs). Within the framework of the proposed algorithm, a new speciation method is employed to locate and track multiple peaks and an adaptive local search method is also hybridised to accelerate the exploitation of species generated by the speciation method. In addition, a memory-based re-initialisation scheme is introduced into the proposed algorithm in order to further enhance its performance in dynamic multi-modal environments. Based on the moving peaks benchmark problems, experiments are carried out to investigate the performance of the proposed algorithm in comparison with several state-of-the-art algorithms taken from the literature. The experimental results show the efficiency of the proposed algorithm for DMMOPs.This work was supported by the Key Program of National Natural Science Foundation (NNSF) of China under Grant no. 70931001, the Funds for Creative Research Groups of China under Grant no. 71021061, the National Natural Science Foundation (NNSF) of China under Grant 71001018, Grant no. 61004121 and Grant no. 70801012 and the Fundamental Research Funds for the Central Universities Grant no. N090404020, the Engineering and Physical Sciences Research Council (EPSRC) of UK under Grant no. EP/E060722/01 and Grant EP/E060722/02, and the Hong Kong Polytechnic University under Grant G-YH60

Multimodal estimation of distribution algorithms

Taking the advantage of estimation of distribution algorithms (EDAs) in preserving high diversity, this paper proposes a multimodal EDA. Integrated with clustering strategies for crowding and speciation, two versions of this algorithm are developed, which operate at the niche level. Then these two algorithms are equipped with three distinctive techniques: 1) a dynamic cluster sizing strategy; 2) an alternative utilization of Gaussian and Cauchy distributions to generate offspring; and 3) an adaptive local search. The dynamic cluster sizing affords a potential balance between exploration and exploitation and reduces the sensitivity to the cluster size in the niching methods. Taking advantages of Gaussian and Cauchy distributions, we generate the offspring at the niche level through alternatively using these two distributions. Such utilization can also potentially offer a balance between exploration and exploitation. Further, solution accuracy is enhanced through a new local search scheme probabilistically conducted around seeds of niches with probabilities determined self-adaptively according to fitness values of these seeds. Extensive experiments conducted on 20 benchmark multimodal problems confirm that both algorithms can achieve competitive performance compared with several state-of-the-art multimodal algorithms, which is supported by nonparametric tests. Especially, the proposed algorithms are very promising for complex problems with many local optima

A multiobjective optimization approach to statistical mechanics

Optimization problems have been the subject of statistical physics approximations. A specially relevant and general scenario is provided by optimization methods considering tradeoffs between cost and efficiency, where optimal solutions involve a compromise between both. The theory of Pareto (or multi objective) optimization provides a general framework to explore these problems and find the space of possible solutions compatible with the underlying tradeoffs, known as the {\em Pareto front}. Conflicts between constraints can lead to complex landscapes of Pareto optimal solutions with interesting implications in economy, engineering, or evolutionary biology. Despite their disparate nature, here we show how the structure of the Pareto front uncovers profound universal features that can be understood in the context of thermodynamics. In particular, our study reveals that different fronts are connected to different classes of phase transitions, which we can define robustly, along with critical points and thermodynamic potentials. These equivalences are illustrated with classic thermodynamic examples.Comment: 14 pages, 8 figure

A Partition-Based Random Search Method for Multimodal Optimization

Practical optimization problems are often too complex to be formulated exactly. Knowing multiple good alternatives can help decision-makers easily switch solutions when needed, such as when faced with unforeseen constraints. A multimodal optimization task aims to find multiple global optima as well as high-quality local optima of an optimization problem. Evolutionary algorithms with niching techniques are commonly used for such problems, where a rough estimate of the optima number is required to determine the population size. In this paper, a partition-based random search method is proposed, in which the entire feasible domain is partitioned into smaller and smaller subregions iteratively. Promising regions are partitioned faster than unpromising regions, thus, promising areas will be exploited earlier than unpromising areas. All promising areas are exploited in parallel, which allows multiple good solutions to be found in a single run. The proposed method does not require prior knowledge about the optima number and it is not sensitive to the distance parameter. By cooperating with local search to refine the obtained solutions, the proposed method demonstrates good performance in many benchmark functions with multiple global optima. In addition, in problems with numerous local optima, high-quality local optima are captured earlier than low-quality local optima

Derating NichePSO

The search for multiple solutions is applicable to many fields (Engineering [54][67], Science [75][80][79][84][86], Economics [13][59], and others [51]). Multiple solutions allow for human judgement to select the best solution from a group of solutions that best match the search criteria. Finding multiple solutions to an optimisation problem has shown to be difficult to solve. Evolutionary computation (EC) and more recently Particle Swarm Optimisation (PSO) algorithms have been used in this field to locate and maintain multiple solutions with fair success. This thesis develops and empirically analyses a new method to find multiple solutions within a convoluted search space. The method is a hybrid of the NichePSO [14] and the sequential niche technique (SNT)[8]. The original SNT was developed using a Genetic Algorithm (GA). It included restrictions such as knowing or approximating the number of solutions that exist. A further pitfall of the SNT is that it introduces false optima after modifying the search space, thereby reducing the accuracy of the solutions. However, this can be resolved with a local search in the unmodified search space. Other sequential niching algorithms require that the search be repeated sequentially until all solutions are found without considering what was learned in previous iterations, resulting in a blind and wasteful search. The NichePSO has shown to be more accurate than GA based algorithms [14][15]. It does not require knowledge of the number of solutions in the search space prior to the search process. However, the NichePSO does not scale well for problems with many optima [16]. The method developed in this thesis, referred to as the derating NichePSO, combines SNT with the NichePSO. The main objective of the derating NichePSO is to eliminate the inaccuracy of SNT and to improve the scalability of the NichePSO. The derating NichePSO is compared to the NichePSO, deterministic crowding [23] and the original SNT using various multimodal functions. The performance of the derating NichePSO is analysed and it is shown that the derating NichePSO is more accurate than SNT and more scalable than the NichePSO.Dissertation (MSc)--University of Pretoria, 2007.Computer ScienceMScUnrestricte