Selection hyper-heuristics (SHHs) select from a set of low-level heuristics which to apply during the optimisation process. One such approach, namely the random gradient SHH, which continues to apply a randomly selected heuristic as long as it remains successful, has been shown to be able to effectively select heuristics leading to optimal expected runtimes on a range of unimodal functions. In this work, we extend the analysis of the random gradient SHH to multimodal optimisation problems to assess their performance at escaping from local optima. We consider the TwoRates benchmark function which includes several consecutive local optima separated by gaps of two alternating different sizes. The function was recently introduced to assess the performance of the flex-EA that uses an archive to store and re-apply the two most suitable Randomized Local Search (RLSk) operators to make the jumps of different lengths. We show that the SHH can optimise the function considerably faster by identifying and consecutively re-applying the single best heuristic to overcome all of the local optima. This performance also holds when the set of low-level heuristics contains all the n possible RLSk operators, where n is the problem size.

In the field of neuroevolution (NE), evolutionary algorithms are used to update the weights, biases and topologies of artificial neural networks (ANNs). A recent theoretical work presented the first runtime analysis of NE in a simple setting, considering a single neuron and intuitive benchmark function classes. However, this work was limited by the unrealistic settings with regard to activation functions and fitness measurements.

Support vector machines (SVMs) are a powerful machine learning paradigm, performing supervised learning for classification and regression analysis. A number of SVM models in the literature have made use of advances in mixed-integer linear programming (MILP) techniques in order to perform this task efficiently. In this work, we present three new models for SVMs that make use of piecewise linear (PWL) functions. This allows effective separation of data points where a simple linear SVM model may not be sufficient. The models we present make use of binary variables to assign data points to SVM segments, and hence fit within a recently presented framework for machine learning MILP models. Alongside presenting an inbuilt feature selection operator, we show that the models can benefit from robust inbuilt outlier detection. Experimental results show when each of the presented models is effective, and we present guidelines on which of the models are preferable in different scenarios.

Selection hyper-heuristics (HHs) are automated algorithm selection methodologies that choose between different heuristics during the optimisation process. Recently, selection HHs choosing between a collection of elitist randomised local search heuristics with different neighbourhood sizes have been shown to optimise standard unimodal benchmark functions from evolutionary computation in the optimal expected runtime achievable with the available low-level heuristics. In this paper, we extend our understanding of the performance of HHs to the domain of multimodal optimisation by considering a Move Acceptance HH (MAHH) from the literature that can switch between elitist and non-elitist heuristics during the run. In essence, MAHH is a non-elitist search heuristic that differs from other search heuristics in the source of non-elitism. We first identify the range of parameters that allow MAHH to hillclimb efficiently and prove that it can optimise the standard hillclimbing benchmark function OneMax in the best expected asymptotic time achievable by unbiased mutation-based randomised search heuristics. Afterwards, we use standard multimodal benchmark functions to highlight function characteristics where MAHH outperforms elitist evolutionary algorithms and the well-known Metropolis non-elitist algorithm by quickly escaping local optima, and ones where it does not. Since MAHH is essentially a non-elitist random local search heuristic, the paper is of independent interest to researchers in the fields of artificial intelligence and randomised search heuristics.

Clustering and regression are two of the most important problems in data analysis and machine learning. Recently, mixed-integer linear programs (MILPs) have been presented in the literature to solve these problems. By modelling the problems as MILPs, they are able to be solved very quickly by commercial solvers. In particular, MILPs for bivariate clusterwise linear regression (CLR) and (continuous) piecewise linear regression (PWLR) have recently appeared. These MILP models make use of binary variables and logical implications modelled through big-M constraints. In this paper, we present these models in the context of a unifying MILP framework for bivariate clustering and regression problems. We then present two new formulations within this framework, the first for ordered CLR, and the second for clusterwise piecewise linear regression (CPWLR). The CPWLR problem concerns simultaneously clustering discrete data, while modelling each cluster with a continuous PWL function. Extending upon the framework, we discuss how outlier detection can be implemented within the models, and how specific decomposition methods can be used to find speedups in the runtime. Experimental results show when each model is the most effective.

The problem of fitting continuous piecewise linear (PWL) functions to discrete data has applications in pattern recognition and engineering, amongst many other fields. To find an optimal PWL function, the positioning of the breakpoints connecting adjacent linear segments must not be constrained and should be allowed to be placed freely. Although the univariate PWL fitting problem has often been approached from a global optimisation perspective, recently, two mixed-integer linear programming approaches have been presented that solve for optimal PWL functions. In this paper, we compare the two approaches: the first was presented by Rebennack and Krasko [Rebennack S, Krasko V (2020) Piecewise linear function fitting via mixed-integer linear programming. INFORMS J. Comput. 32(2):507–530] and the second by Kong and Maravelias [Kong L, Maravelias CT (2020) On the derivation of continuous piecewise linear approximating functions. INFORMS J. Comput. 32(3):531–546]. Both formulations are similar in that they use binary variables and logical implications modelled by big-[Formula: see text] constructs to ensure the continuity of the PWL function, yet the former model uses fewer binary variables. We present experimental results comparing the time taken to find optimal PWL functions with differing numbers of breakpoints across 10 data sets for three different objective functions. Although neither of the two formulations is superior on all data sets, the presented computational results suggest that the formulation presented by Rebennack and Krasko is faster. This might be explained by the fact that it contains fewer complicating binary variables and sparser constraints. Summary of Contribution: This paper presents a comparison of the mixed-integer linear programming models presented in two recent studies published in the INFORMS Journal on Computing. Because of the similarity of the formulations of the two models, it is not clear which one is preferable. We present a detailed comparison of the two formulations, including a series of comparative experimental results across 10 data sets that appeared across both papers. We hope that our results will allow readers to take an objective view as to which implementation they should use.

Recent analyses have shown that a random gradient hyper-heuristic (HH) using randomised local search (RLSk) low-level heuristics with different neighbourhood sizes k can optimise the unimodal benchmark function LeadingOnes in the best expected time achievable with the available heuristics, if sufficiently long learning periods τ are employed. In this paper, we examine the impact of the learning period on the performance of the hyper-heuristic for standard unimodal benchmark functions with different characteristics: Ridge, where the HH has to learn that RLS1 is always the best low-level heuristic, and OneMax, where different low-level heuristics are preferable in different areas of the search space. We rigorously prove that super-linear learning periods τ are required for the HH to achieve optimal expected runtime for Ridge. Conversely, a sub-logarithmic learning period is the best static choice for OneMax, while using super-linear values for τ increases the expected runtime above the asymptotic unary unbiased black box complexity of the problem. We prove that a random gradient HH which automatically adapts the learning period throughout the run has optimal asymptotic expected runtime for both OneMax and Ridge. Additionally, we show experimentally that it outperforms any static learning period for realistic problem sizes.

Selection hyper-heuristics (HHs) are randomised search methodologies which choose and execute heuristics during the optimisation process from a set of low-level heuristics. A machine learning mechanism is generally used to decide which low-level heuristic should be applied in each decision step. In this article, we analyse whether sophisticated learning mechanisms are always necessary for HHs to perform well. To this end we consider the most simple HHs from the literature and rigorously analyse their performance for the LeadingOnes benchmark function. Our analysis shows that the standard Simple Random, Permutation, Greedy, and Random Gradient HHs show no signs of learning. While the former HHs do not attempt to learn from the past performance of low-level heuristics, the idea behind the Random Gradient HH is to continue to exploit the currently selected heuristic as long as it is successful. Hence, it is embedded with a reinforcement learning mechanism with the shortest possible memory. However, the probability that a promising heuristic is successful in the next step is relatively low when perturbing a reasonable solution to a combinatorial optimisation problem. We generalise the “simple” Random Gradient HH so success can be measured over a fixed period of time [Formula: see text], instead of a single iteration. For LeadingOnes we prove that the Generalised Random Gradient (GRG) HH can learn to adapt the neighbourhood size of Randomised Local Search to optimality during the run. As a result, we prove it has the best possible performance achievable with the low-level heuristics (Randomised Local Search with different neighbourhood sizes), up to lower-order terms. We also prove that the performance of the HH improves as the number of low-level local search heuristics to choose from increases. In particular, with access to [Formula: see text] low-level local search heuristics, it outperforms the best-possible algorithm using any subset of the [Formula: see text] heuristics. Finally, we show that the advantages of GRG over Randomised Local Search and Evolutionary Algorithms using standard bit mutation increase if the anytime performance is considered (i.e., the performance gap is larger if approximate solutions are sought rather than exact ones). Experimental analyses confirm these results for different problem sizes (up to [Formula: see text]) and shed some light on the best choices for the parameter [Formula: see text] in various situations.

Selection hyper-heuristics are automated algorithm selection methodologies that choose between different heuristics during the optimisation process. Recently selection hyperheuristics choosing between a collection of elitist randomised local search heuristics with different neighbourhood sizes have been shown to optimise a standard unimodal benchmark function from evolutionary computation in the optimal expected runtime achievable with the available low-level heuristics. In this paper we extend our understanding to the domain of multimodal optimisation by considering a hyper-heuristic from the literature that can switch between elitist and nonelitist heuristics during the run. We first identify the range of parameters that allow the hyper-heuristic to hillclimb efficiently and prove that it can optimise a standard hillclimbing benchmark function in the best expected asymptotic time achievable by unbiased mutation-based randomised search heuristics. Afterwards, we use standard multimodal benchmark functions to highlight function characteristics where the hyper-heuristic is efficient by swiftly escaping local optima and ones where it is not. For a function class called CLIFFd where a new gradient of increasing fitness can be identified after escaping local optima, the hyper-heuristic is extremely efficient while a wide range of established elitist and non-elitist algorithms are not, including the well-studied Metropolis algorithm. We complete the picture with an analysis of another standard benchmark function called JUMPd as an example to highlight problem characteristics where the hyper-heuristic is inefficient. Yet, it still outperforms the wellestablished non-elitist Metropolis algorithm.

Selection hyper-heuristics are randomised optimisation techniques that select from a set of low-level heuristics which one should be applied in the next step of the optimisation process. Recently it has been proven that a Random Gradient hyper-heuristic optimises the LeadingOnes benchmark function in the best runtime achievable with any combination of its low-level heuristics, up to lower order terms. To achieve this runtime, the learning period τ, used to evaluate the performance of the currently chosen heuristic, should be set appropriately, i.e., super-linear in the problem size but not excessively larger. In this paper we automate the hyper-heuristic further by allowing it to self-adjust the learning period τ during the run. To achieve this we equip the algorithm with a simple self-adjusting mechanism, called 1 - o(1) rule, inspired by the 1/5 rule traditionally used in continuous optimisation. We rigorously prove that the resulting hyper-heuristic solves LeadingOnes in optimal runtime by automatically adapting τ and achieving a 1 - o(1) ratio of the desired behaviour. Complementary experiments for realistic problem sizes show the value of τ adapting as desired and that the hyper-heuristic with adaptive learning period outperforms the hyper-heuristic with fixed learning periods.

Selection hyper-heuristics are randomised search methodologies which choose and execute heuristics from a set of low-level heuristics. Recent time complexity analyses for the LeadingOnes benchmark function have shown that the standard simple random, permutation, random gradient, greedy and reinforcement learning selection mechanisms show no effects of learning. The idea behind the learning mechanisms is to continue to exploit the currently selected heuristic as long as it is successful. However, the probability that a promising heuristic is successful in the next step is relatively low when perturbing a reasonable solution to a combinatorial optimisation problem. In this paper we generalise the classical selection-perturbation mechanisms so success can be measured over some fixed period of length r, rather than in a single iteration. We present a benchmark function where it is necessary to learn to exploit a particular low-level heuristic, rigorously proving that it makes the difference between an efficient and an inefficient algorithm. For LeadingOnes we prove that the generalised random gradient mechanism approaches optimal performance while generalised greedy, although not as fast, still outperforms random local search. An experimental analysis shows that combining the two generalised mechanisms leads to even better performance.