Perturbation strength and the global structure of qap fitness landscapes

We study the effect of increasing the perturbation strength on the global structure of QAP fitness landscapes induced by Iterated Local Search (ILS). The global structure is captured with Local Optima Networks. Our analysis concentrates on the number, characteristics and distribution of funnels in the landscape, and how they change with increasing perturbation strengths. Well-known QAP instance types are considered. Our results confirm the multi-funnel structure of QAP fitness landscapes and clearly explain, visually and quantitatively, why ILS with large perturbation strengths produces better results. Moreover, we found striking differences between randomly generated and real-world instances, which warns about using synthetic benchmarks for (manual or automatic) algorithm design and tuning

Inferring Future Landscapes: Sampling the Local Optima Level

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 paper, 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

Comparing Communities of Optima with Funnels in Combinatorial Fitness Landscapes

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.Authors listed as ECOM Trac

Clarifying the Difference in Local Optima Network Sampling Algorithms

We conduct the first ever statistical comparison between two Local Optima Network (LON) sampling algorithms. These methodologies attempt to capture the connectivity in the local optima space of a fitness landscape. One sampling algorithm is based on a random-walk snowballing procedure, while the other is centred around multiple traced runs of an Iterated Local Search. Both of these are proposed for the Quadratic Assignment Problem (QAP), making this the focus of our study. It is important to note the sampling algorithm frameworks could easily be modified for other domains. In our study descriptive statistics for the obtained search space samples are contrasted and commented on. The LON features are also used in linear mixed models and random forest regression for predicting heuristic optimisation performance of two prominent heuristics for the QAP on the underlying combinatorial problems. The model results are then used to make deductions about the sampling algorithms’ utility. We also propose a specific set of LON metrics for use in future predictive models alongside previously-proposed network metrics, demonstrating the payoff in doing so

The Local Optima Level in Chemotherapy Schedule Optimisation

In this paper a multi-drug Chemotherapy Schedule Optimisation Problem (CSOP) is subject to Local Optima Network (LON) analysis. LONs capture global patterns in fitness landscapes. CSOPs have not previously been subject to fitness landscape analysis. We fill this gap: LONs are constructed and studied for meaningful structure. The CSOP formulation presents novel challenges and questions for the LON model because there are infeasible regions in the fitness landscape and an unknown global optimum; it also brings a topic from healthcare to LON analysis. Two LON Construction algorithms are proposed for sampling CSOP fitness landscapes: a Markov-Chain Construction Algorithm and a Hybrid Construction Algorithm. The results provide new insight into LONs of highly-constrained spaces, and into the proficiency of search operators on the CSOP. Iterated Local Search and Memetic Search, which are the foundations for the LON algorithms, are found to markedly out-perform a Genetic Algorithm from the literature

Anatomy of the Local Optima Level in Combinatorial Optimisation

Many situations in daily life represent complex combinatorial optimisation problems. These include issues such as efficient fuel consumption, nurse scheduling, or distribution of humanitarian aid. There are many algorithms that attempt to solve these problems but the ability to understand their likely performance on a given problem is still lacking. Fitness landscape analysis identifies some of the reasons why metaheuristic algorithms behave in a particular way. The Local Optima Network (LON) model, proposed in 2008, encodes local optima connectivity in fitness landscapes. In this approach, nodes are local optima and edges encode transitions between these optima. A LON provides a static image of the dynamics of algorithm-problem inter- play. Analysing these structures provides insights into the reactions between optimisation problems and metaheuristic search algorithms. This thesis proposes that analysis of the local optima space of combinatorial fitness landscapes encoded using a LON provides important information concerning potential search algorithm performance. It considers the question as to whether or not features of LONs can contribute to explaining or predicting the outcome of trying to optimise an associated combinatorial problem. Topological landscape features of LONs are proposed, analysed and compared. Benchmark and novel problem instances are studied; both types of problem are sampled and in some cases exhaustively-enumerated such that LONs can be extracted for analysis. Investigations into the nature and biases of LON construction algorithms are conducted and compared. Contributions include aligning fractal geometry to the study of LONs; proposals for novel ways to compute fractal dimension from these structures; comparing the power of different LON construction algorithms for explaining algorithm performances; and analysing the interplay between algorithmic operations and infeasible regions in the local optima space using LONs as a tool. Throughout the thesis, large scale structural patterns in fitness landscapes are shown to be strongly linked with metaheuristic algorithm performance. This includes arrangements of local optima funnel structures; spatial and geometric complexity in the LON (measured by their fractal dimensionality) and fitness levels in the space of local optima. These features are demonstrated to have explanatory or predictive ability with respect to algorithm performance for the underlying combinatorial problems. The results presented here indicate that large topological patterns in fitness landscapes are important during metaheuristic search algorithm design. In many cases they are incontrovertibly linked to the success of the algorithm. These results indicate that use of the suggested fitness landscape measures would be highly beneficial when considering the design of search algorithms for a given problem domain

The Fractal Geometry of Fitness Landscapes at the Local Optima Level

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.Output Status: Forthcoming/Available Onlin

Combinatorial optimization and metaheuristics

Today, combinatorial optimization is one of the youngest and most active areas of discrete mathematics. It is a branch of optimization in applied mathematics and computer science, related to operational research, algorithm theory and computational complexity theory. It sits at the intersection of several fields, including artificial intelligence, mathematics and software engineering. Its increasing interest arises for the fact that a large number of scientific and industrial problems can be formulated as abstract combinatorial optimization problems, through graphs and/or (integer) linear programs. Some of these problems have polynomial-time (“efficient”) algorithms, while most of them are NP-hard, i.e. it is not proved that they can be solved in polynomial-time. Mainly, it means that it is not possible to guarantee that an exact solution to the problem can be found and one has to settle for an approximate solution with known performance guarantees. Indeed, the goal of approximate methods is to find “quickly” (reasonable run-times), with “high” probability, provable “good” solutions (low error from the real optimal solution). In the last 20 years, a new kind of algorithm commonly called metaheuristics have emerged in this class, which basically try to combine heuristics in high level frameworks aimed at efficiently and effectively exploring the search space. This report briefly outlines the components, concepts, advantages and disadvantages of different metaheuristic approaches from a conceptual point of view, in order to analyze their similarities and differences. The two very significant forces of intensification and diversification, that mainly determine the behavior of a metaheuristic, will be pointed out. The report concludes by exploring the importance of hybridization and integration methods

An integrated White+Black box approach for designing and tuning stochastic local search algorithms

Ph.DDOCTOR OF PHILOSOPH