Advancements in fitness tracking systems have revolutionised cycling by making health fitness related data more accessible. The cycling power is an essential indicator when it comes to actual measurement of absolute efforts and performance of athletes and coaches. Unfortunately, cycling power requires a specialist power meter sensor to measure, which adds significant cost to an already expensive sport. This study investigates the potential of machine learning techniques to predict average cycling power without the need for a power meter. Unlike traditional physics-based models, which rely on fixed equations, regression models can learn complex, data-driven patterns to improve accuracy and generalisability. Five regression models — multiple linear regression, random forest regression, XGBoost regression, support vector regression, and bayesian ridge regression were trained on two distinct yet comparable datasets. One dataset was from an online source and the other sourced from cycling clubs local to city of Edinburgh, Scotland. Feature selection and hyper-parameter tuning were performed to optimise each model. Support vector regression model, which has previously missing from literature in similar applications, emerged as the best performing model — achieving a mean R2score of 0.91. These results advance the field of cycling power prediction by identifying a more accurate regression model while also providing a comparative analysis with other regression techniques.

Local optima networks (LONs) capture fitness landscape information. They are typically constructed in a black-box manner; information about the problem structure is not utilised. This also applies to the analysis of LONs: knowledge about the problem, such as interaction between variables, is not considered. We challenge this status-quo with an alternative approach: we consider how LON analysis can be improved by incorporating subfunction-based information — this can either be known a-priori or learned during search. To this end, LONs are constructed for several benchmark pseudo-boolean problems using three approaches: firstly, the standard algorithm; a second algorithm which uses deterministic grey-box crossover; and a third algorithm which selects perturbations based on learned information about variable interactions. Metrics related to subfunction changes in a LON are proposed and compared with metrics from previous literature which capture other aspects of a LON. Incorporating problem structure in LON construction and analysing it can bring enriched insight into optimisation dynamics. Such information may be crucial to understanding the difficulty of solving a given problem with state-of-the-art linkage learning optimisers. In light of the results, we suggest incorporation of problem structure as an alternative paradigm in landscape analysis for problems with known or suspected subfunction structure.

We propose a new way of looking at local optima networks (LONs). LONs represent fitness landscapes; the nodes are local optima, and the edges are search transitions between them. Many metrics computed on LONs have been proposed and shown to be linked to metaheuristic search difficulty. These have typically considered LONs as describing static structures. In contrast to this, Laplacian dynamics (LD) is an approach to consider the information flow across a network as a dynamical process. We adapt and apply LD to the context of LONs. As a testbed, we consider instances from the quadratic assignment problem (QAP) library. Metrics related to LD are proposed and these are compared with existing LON metrics. The results show that certain LD metrics are strong predictors of metaheuristic performance for iterated local search and tabu search.

Many real-world problems have expensive-to-compute fitness functions and are multi-objective in nature. Surrogate-assisted evolutionary algorithms are often used to tackle such problems. Despite this, literature about analysing the fitness landscapes induced by surrogate models is limited, and even non-existent for multi-objective problems. This study addresses this critical gap by comparing landscapes of the true fitness function with those of surrogate models for multi-objective functions. Moreover, it does so temporally by examining landscape features at different points in time during optimisation, in the vicinity of the population at that point in time. We consider the BBOB bi-objective benchmark functions in our experiments. The results of the fitness landscape analysis reveals significant differences between true and surrogate features at different time points during optimisation. Despite these differences, the true and surrogate landscape features still show high correlations between each other. Furthermore, this study identifies which landscape features are related to search and demonstrates that both surrogate and true landscape features are capable of predicting algorithm performance. These findings indicate that temporal analysis of the landscape features may help to facilitate the design of surrogate switching approaches to improve performance in multi-objective optimisation.

Morpho-Evolution (ME) refers to the simultaneous optimisation of a robot's design and controller to maximise performance given a task and environment. Many genetic encodings have been proposed which are capable of representing design and control. Previous research has provided empirical comparisons between encodings in terms of their performance with respect to an objective function and the diversity of designs that are evaluated, however there has been no attempt to explain the observed findings. We address this by applying Local Optima Network (LON) analysis to investigate the structure of the fitness landscapes induced by three different encodings when evolving a robot for a locomotion task, shedding new light on the ease by which different fitness landscapes can be traversed by a search process. This is the first time LON analysis has been applied in the field of ME despite its popularity in combinatorial optimisation domains; the findings will facilitate design of new algorithms or operators that are customised to ME landscapes in the future.

The Quadratic Assignment Problem (QAP) is one of the classical N P-hard tasks from operations research with a history of more than 65 years. It is often approached with heuristic algorithms and over the years, a multitude of such methods has been applied. All of them have in common that they tend to prefer better solutions over worse ones. We approach the QAP with Frequency Fitness Assignment (FFA), an algorithm module that can be plugged into arbitrary iterative heuristics and that removes this bias. One would expect that a heuristic that does not care whether a new solution is better or worse compared to the current one should not perform very well. We plug FFA into a simple randomized local search (RLS) and yield the FRLS, which surprisingly outperforms RLS on the vast majority of the instances of the well-known QAPLIB benchmark set.

The Frequency Fitness Assignment (FFA) method steers evolutionary algorithms by objective rareness instead of objective goodness. Does this mean the size of the combinatorial search space influences its performance when compared to more traditional evolutionary algorithms? Our results suggest it does. To address to which extent the search space size matters for the effectiveness of the FFA-principle, we compare the algorithms on 420 Job Shop Scheduling Problem (JSSP) instances systematically generated in gridwise sizes. The comparison of the FFA-hillclimber and the standard hillclimber is done in both EQ setting, accepting equally good (or fitness-frequent) solutions, and NOEQ setting, only accepting improvement. FFA-hillclimbers are more successful than standard hillclimbers on smaller problem instances, but not on larger ones. It seems that the ratio between jobs and machines, influences the success of the respective algorithms for fixed computational budgets.

Abstract To guide the design of better iterative optimisation heuristics, it is imperative to understand how inherent structural biases within algorithm components affect the performance on a wide variety of search landscapes. This study explores the impact of structural bias in the modular Covariance Matrix Adaptation Evolution Strategy (modCMA), focusing on the roles of various modulars within the algorithm. Through an extensive investigation involving $$435\,456$$ 435 456 configurations of modCMA, we identified key modules that significantly influence structural bias of various classes. Our analysis utilized the Deep-BIAS toolbox for structural bias detection and classification, complemented by SHAP analysis for quantifying module contributions. The performance of these configurations was tested on a sequence of affine-recombined functions, maintaining fixed optimum locations while gradually varying the landscape features. Our results demonstrate an interplay between module-induced structural bias and algorithm performance across different landscape characteristics.

We consider search spaces associated with neural network channel configuration. Architectures and their accuracy are visualised using low-dimensional Euclidean embedding (LDEE). Optimisation dynamics are captured using local optima networks (LONs). LONs are a compression of a fitness landscape: the nodes are local optima and the edges are search transitions between them. Several neural architecture search algorithms are tested on the search space and we discover that iterated local search (ILS) is a competitive algorithm for neural channel configuration. We additionally implement a landscape-aware ILS which performs well. Observations from the search and landscape space analyses bring visual clarity and insight to the science of neural network channel design: the results indicate that a high number of channels, kept constant throughout the network, is beneficial.

Genetic algorithms might not be able to solve the HP-protein folding problem because creating random individuals for an initial population is very hard, if not impossible. The reason for this, is that the expected number of constraint violations increases with instance size when randomly sampling individuals, as we will show in an experiment. Thereby, the probability of randomly sampling a valid individual decreases exponentially with instance size. This immediately prohibits resampling, and repair mechanisms might also be non-applicable. Backtracking could generate a valid random individual, but it runs in exponential time, and is therefore also unsuitable. No wonder that previous approaches do not report how (often) random samples are created, and only address small instances. We contrast our findings with TSP, which is also NP-hard, but does not have these problems.

In this work We consider and discuss the problems which come with trying to explain human and machine intelligence.How explainable artificial intelligence research is being carried out, the pitfalls and limitations of current approaches and the bigger question of whether we need explanations for trusting inherently complex and large intelligent systems, whether artificial or not.

Unlike other NP-hard problems, the constraints on the traveling tournament problem are so pressing that it’s hardly possible to randomly generate a valid solution, for example, to use in a genetic algorithm’s initial population. In this study, we randomly generate solutions, assess the numbers of constraint violations, and extrapolate the results to predict the required number of samples for obtaining a single valid solution for any reasonable instance size. As it turns out, these numbers are astronomical, and we finish the study by discussing the feasibility of efficient sampling of valid solutions to various NP-hard problems.

We propose looking at the phenomenon of fitness landscape funnels in terms of their depth. In particular, we examine how the depth of funnels in Local Optima Networks (LONs) of benchmark Quadratic Assignment Problem instances relate to metaheuristic performance. Three distinct iterated local search (ILS) acceptance strategies are considered: better-or-equal (standard), annealing-like, and restart. Funnel measurements are analysed for their connection to ILS performance on the underlying combinatorial problems. We communicate the findings through hierarchical clustering of LONs, network visualisations, subgroup analysis, correlation analysis, and Random Forest regression models. The results show that funnel depth is associated with search difficulty, and that there is an interplay between funnel structure and acceptance strategy. Standard and annealing acceptance work better than restart on both deep-funnel and shallow-funnel problems; standard acceptance is the best strategy when optimal funnel(s) are deep, while annealing acceptance is superior when they are shallow. Regression models including funnel depth measurements could explain up to 96% of ILS runtime variance (with annealing-like acceptance). The runtime of ILS with restarts was less explainable using funnel features.

In 2021, evolutionary algorithms found the hardest-known yes and no instances for the Hamiltonian cycle problem. These instances, which show regularity patterns, require a very high number of recursions for the best exact backtracking algorithm (Vandegriend-Culberson), but don't show up in large randomized instance ensembles. In this paper, we will demonstrate that these evolutionarily found instances of the Hamiltonian cycle problem are hard for all major backtracking algorithms, not just the Vandegriend-Culberson. We compare performance of these six algorithms on an ensemble of 91,000 randomized instances plus the evolutionar-ily found instances. These results present a first glance at universal hardness for this NP-complete problem. Algorithms, source code, and input data are all publicly supplied to the community.

Abstract A local optima network (LON) encodes local optima connectivity in the fitness landscape of a combinatorial optimisation problem. Recently, LONs have been studied for their fractal dimension. Fractal dimension is a complexity index where a non-integer dimension can be assigned to a pattern. This paper investigates the fractal nature of LONs and how that nature relates to metaheuristic performance on the underlying problem. We use visual analysis, correlation analysis, and machine learning techniques to demonstrate that relationships exist and that fractal features of LONs can contribute to explaining and predicting algorithm performance. The results show that the extent of multifractality and high fractal dimensions in the LON can contribute in this way when placed in regression models with other predictors. Features are also individually correlated with search performance, and visual analysis of LONs shows insight into this relationship.

Connection patterns among Local Optima Networks (LONs) can inform heuristic design for optimisation. LON research has predominantly required complete enumeration of a fitness landscape, thereby restricting analysis to problems diminutive in size compared to real-life situations. LON sampling algorithms are therefore important. In this article, we study LON construction algorithms for the Quadratic Assignment Problem (QAP). Using machine learning, we use estimated LON features to predict search performance for competitive heuristics used in the QAP domain. The results show that by using random forest regression, LON construction algorithms produce fitness landscape features which can explain almost all search variance. We find that LON samples better relate to search than enumerated LONs do. The importance of fitness levels of sampled LONs in search predictions is crystallised. Features from LONs produced by different algorithms are combined in predictions for the first time, with promising results for this “super-sampling”: a model to predict tabu search success explained 99% of variance. Arguments are made for the use-case of each LON algorithm and for combining the exploitative process of one with the exploratory optimisation of the other.

We conduct a study of local optima networks (LONs) in a search space using fractal dimensions. The fractal dimension (FD) of these networks is a complexity index which assigns a non-integer dimension to an object. We propose a fine-grained approach to obtaining the FD of LONs, using the probabilistic search transitions encoded in LON edge weights. We then apply multi-fractal calculations to LONs for the first time, comparing with mono-fractal analysis. For complex systems such as LONs, the dimensionality may be different between two sub-systems and multi-fractal analysis is needed. Here we focus on the Quadratic Assignment Problem (QAP), conducting fractal analyses on sampled LONs of reasonable size for the first time. We also include fully enumerated LONs of smaller size. Our results show that local optima spaces can be multi-fractal and that valuable information regarding probabilistic self-similarity is encoded in the edge weights of local optima networks. Links are drawn between these phenomena and the performance of two competitive metaheuristic algorithms.

The existence of sub-optimal funnels in combinatorial fitness landscapes has been linked to search difficulty. The exact nature of these structures --- and how commonly they appear --- is not yet fully understood. Improving our understanding of funnels could help with designing effective diversification mechanisms for a 'smoothing' effect, making optimisation easier. We model fitness landscapes as local optima networks. The relationship between communities of local optima found by network clustering algorithms and funnels is explored. Funnels are identified using the notion of monotonic sequences from the study of energy landscapes in theoretical chemistry. NK Landscapes and the Quadratic Assignment Problem are used as case studies. Our results show that communities are linked to funnels. The analysis exhibits relationships between these landscape structures and the performance of trajectory-based metaheuristics such as Simulated Annealing (SA) and Iterated Local Search (ILS). In particular, ILS gets trapped in funnels, and modular communities of optima slow it down. The funnels contribute to lower success for SA. We show that increasing the strength of ILS perturbation helps to 'smooth' the funnels and improves performance in multi-funnel landscapes.