Particle swarm optimization : stability analysis using N-informers under arbitrary coefficient distributions

This paper derives, under minimal modelling assumptions, a simple to use theorem for obtaining both order-1 and order-2 stability criteria for a common class of particle swarm optimization (PSO) variants. Specifically, PSO variants that can be rewritten as a finite sum of stochastically weighted difference vectors between a particle’s position and swarm informers are covered by the theorem. Additionally, the use of the derived theorem allows a PSO practitioner to obtain stability criteria that contains no artificial restriction on the relationship between control coefficients. The majority of previous stability results for PSO variants provided stability criteria under the restriction that certain control coefficients are equal; such restrictions are not present when using the derived theorem. Using the derived theorem, as demonstration of its ease of use, stability criteria are derived without the imposed restriction on the relation between the control coefficients for four popular PSO variants.http://www.elsevier.com/locate/swevohj2023Mathematics and Applied Mathematic

Understanding parameter spaces using local optima networks: a case study on particle swarm optimization

A major challenge with utilizing a metaheuristic is finding optimal or near optimal parameters for a given problem instance. It is well known that the best performing control parameters are often problem dependent, with poorly chosen parameters even leading to algorithm failure. What is not obvious is how strongly the complexity of the parameter landscape itself is coupled with the underlying objective function the metaheuristic is attempting to solve. In this paper local optima networks (LONs) are utilized to visualize and analyze the parameter landscapes of particle swarm optimization (PSO) over an array of objective functions. It was found that the structure of the parameter landscape is affected by the underlying objective function, and in some cases by a considerable degree across multiple metrics. Furthermore, despite PSO's parameter landscape having a relatively simple macro structure, the LONs demonstrate that there is actually a considerable amount of complexity at a micro level; making parameter tuning harder for PSO than would have been initially thought. Apart from the PSO specific findings this paper also provides a formalism of parameter landscapes and demonstrates that LONs can be used as an effective tool in the analysis and visualization of parameter landscapes of metaheuristics

A Markov chain model for geographical accessibility

Accessibility analyses are conducted for a variety of applications, including urban planning and public health studies. These applications may aggregate data at the level of administrative units, such as provinces or municipalities. Accessibility between administrative units can be quantified by travel distance. However, modelling the distances between all administrative units in a region is computationally expensive if a large number of administrative units is considered. We propose a methodology to model accessibility between administrative units as a homogeneous Markov chain, where the administrative units are states and standardised inverse travel distances act as transition probabilities. Single transitions are allowed only between adjacent administrative units, resulting in a sparse one-step transition probability matrix (TPM). Powers of the TPM are taken to obtain transition probabilities between non-adjacent units. The methodology assumes that the Markov property holds for travel between units. We apply the methodology to administrative units within Tshwane, South Africa, considering only major roads for the sake of computation. The results are compared to those obtained using Euclidean distance, and we show that using network distance yields more reasonable results. The proposed methodology is computationally efficient and can be used to estimate accessibility between any set of administrative units connected by a road network.In part by the National Research Foundation of South Africa and the NRF-SASA Academic Statistics Grant.http://www.elsevier.com/locate/spastaam2024StatisticsNon

A Local Optima Network Analysis of the Feedforward Neural Architecture Space

This study investigates the use of local optima network (LON) analysis, a derivative of the fitness landscape of candidate solutions, to characterise and visualise the neural architecture space. The search space of feedforward neural network architectures with up to three layers, each with up to 10 neurons, is fully enumerated by evaluating trained model performance on a selection of data sets. Extracted LONs, while heterogeneous across data sets, all exhibit simple global structures, with single global funnels in all cases but one. These results yield early indication that LONs may provide a viable paradigm by which to analyse and optimise neural architectures.Comment: A version of this paper has been accepted for publication at IJCNN'2

Critical considerations on angle modulated particle swarm optimisers

This article investigates various aspects of angle modulated particle swarm optimisers (AMPSO). Previous attempts at improving the algorithm have only been able to produce better results in a handful of test cases. With no clear understanding of when and why the algorithm fails, improving the algorithm’s performance has proved to be a difficult and sometimes blind undertaking. Therefore, the aim of this study is to identify the circumstances under which the algorithm might fail, and to understand and provide evidence for such cases. It is shown that the general assumption that good solutions are grouped together in the search space does not hold for the standard AMPSO algorithm or any of its existing variants. The problem is explained by specific characteristics of the generating function used in AMPSO. Furthermore, it is shown that the generating function also prevents particle velocities from decreasing, hindering the algorithm’s ability to exploit the binary solution space. Methods are proposed to both confirm and potentially solve the problems found in this study. In particular, this study addresses the problem of finding suitable generating functions for the first time. It is shown that the potential of a generating function to solve arbitrary binary optimisation problems can be quantified. It is further shown that a novel generating function with a single coefficient is able to generate solutions to binary optimisation problems with fewer than four dimensions. The use of ensemble generating functions is proposed as a method to solve binary optimisation problems with more than 16 dimensions.http://link.springer.com/journal/117212016-12-31hb201

Particle swarm variants: standardized convergence analysis

This paper presents an objective function specially designed for the convergence analysis of a number of particle swarm optimization (PSO) variants. It was found that using a specially designed objective function for convergence analysis is both a simple and valid method for performing assumption free convergence analysis. It was also found that the canonical particle swarm's topology did not have an impact on the parameter region needed to ensure convergence. The parameter region needed to ensure convergent particle behavior was empirically obtained for the fully informed PSO, the bare bones PSO, and the standard PSO 2011 algorithm. In the case of the bare bones PSO and the standard PSO 2011 the region needed to ensure convergent particle behavior di ers from previous theoretical work. The di erence in the obtained regions in the bare bones PSO is a direct result of the previous theoretical work relying on simplifying assumptions, speci - cally the stagnation assumption. A number of possible causes for the discrepancy in the obtained convergent region for the standard PSO 2011 are given.http://link.springer.com/journal/117212016-09-30hb201