This paper explores the enhancement of solution diversity in evolutionary algorithms (EAs) for the maximum matching problem, concentrating on complete bipartite graphs and paths. We adopt binary string encoding for matchings and use Hamming distance to measure diversity, aiming for its maximization. Our study centers on the $(\mu+1)$-EA and $2P-EA_D$, which are applied to optimize diversity. We provide a rigorous theoretical and empirical analysis of these algorithms. For complete bipartite graphs, our runtime analysis shows that, with a reasonably small $\mu$, the $(\mu+1)$-EA achieves maximal diversity with an expected runtime of $O(\mu^2 m^4 \log(m))$ for the small gap case (where the population size $\mu$ is less than the difference in the sizes of the bipartite partitions) and $O(\mu^2 m^2 \log(m))$ otherwise. For paths, we establish an upper runtime bound of $O(\mu^3 m^3)$. The $2P-EA_D$ displays stronger performance, with bounds of $O(\mu^2 m^2 \log(m))$ for the small gap case, $O(\mu^2 n^2 \log(n))$ otherwise, and $O(\mu^3 m^2)$ for paths. Here, $n$ represents the total number of vertices and $m$ the number of edges. Our empirical studies, which examine the scaling behavior with respect to $m$ and $\mu$, complement these theoretical insights and suggest potential for further refinement of the runtime bounds.
A class of metaheuristic techniques called estimation-of-distribution algorithms (EDAs) are employed in optimization as more sophisticated substitutes for traditional strategies like evolutionary algorithms. EDAs generally drive the search for the optimum by creating explicit probabilistic models of potential candidate solutions through repeated sampling and selection from the underlying search space. Most theoretical research on EDAs has focused on pseudo-Boolean optimization. Jedidia et al. (GECCO 2023) proposed the first EDAs for optimizing problems involving multi-valued decision variables. By building a framework, they have analyzed the runtime of a multi-valued UMDA on the r-valued LeadingOnes function. Using their framework, here we focus on the multi-valued compact genetic algorithm (r-cGA) and provide a first runtime analysis of a generalized OneMax function. To prove our results, we investigate the effect of genetic drift and progress of the probabilistic model towards the optimum. After finding the right algorithm parameters, we prove that the r-cGA solves this r-valued OneMax problem efficiently. We show that with high probability, the runtime bound is O(r2 n log2 r log3 n). At the end of experiments, we state one conjecture related to the expected runtime of another variant of multi-valued OneMax function.
The diversity optimization is the class of optimization problems, in which we aim at finding a diverse set of good solutions. One of the frequently used approaches to solve such problems is to use evolutionary algorithms which evolve a desired diverse population. This approach is called evolutionary diversity optimization (EDO). In this paper, we analyse EDO on a 3-objective function LOTZ$_k$, which is a modification of the 2-objective benchmark function (LeadingOnes, TrailingZeros). We prove that the GSEMO computes a set of all Pareto-optimal solutions in $O(kn^3)$ expected iterations. We also analyze the runtime of the GSEMO$_D$ (a modification of the GSEMO for diversity optimization) until it finds a population with the best possible diversity for two different diversity measures, the total imbalance and the sorted imbalances vector. For the first measure we show that the GSEMO$_D$ optimizes it asymptotically faster than it finds a Pareto-optimal population, in $O(kn^2\log(n))$ expected iterations, and for the second measure we show an upper bound of $O(k^2n^3\log(n))$ expected iterations. We complement our theoretical analysis with an empirical study, which shows a very similar behavior for both diversity measures that is close to the theory predictions.
Ensuring adequate wireless coverage in upcoming communication technologies such as 6G is expected to be challenging. This is because user demands of higher datarate require an increase in carrier frequencies, which in turn reduce the diffraction effects (and hence coverage) in complex multipath environments. Intelligent reflecting surfaces have been proposed as a way of restoring coverage by adaptively reflecting incoming electromagnetic waves in desired directions. This is accomplished by judiciously adding extra phases at different points on the surface. In practice, these extra phases are only available in discrete quantities due to hardware constraints. Computing these extra phases is computationally challenging when they can only be picked from a discrete distribution, and existing approaches for solving this problem were either heuristic or based on evolutionary algorithms. We solve this problem by proposing fast algorithms with provably optimal solutions. Our algorithms have linear complexity, and are presented with rigorous proofs for their optimality. We show that the proposed algorithms exhibit better performance. We analyze situations when unwanted grating lobes arise in the radiation pattern, and discuss mitigation strategies, such as the use of triangular lattices and prephasing techniques, to eliminate them. We also demonstrate how our algorithms can leverage these techniques to deliver optimum beamforming solutions.
This review explores the application of intelligent optimization algorithms to Multi-Objective Optimal Power Flow (MOPF) in enhancing modern power systems. It delves into the challenges posed by the integration of renewables, smart grids, and increasing energy demands, focusing on evolutionary algorithms, swarm intelligence, and deep reinforcement learning. The effectiveness, scalability, and application of these algorithms are analyzed, with findings suggesting that algorithm selection is contingent on the specific MOPF problem at hand, and hybrid approaches offer significant promise. The importance of standard test systems for verifying solutions and the role of software tools in facilitating analysis are emphasized. Future research is directed towards exploiting machine learning for dynamic optimization, embracing decentralized energy systems, and adapting to evolving policy frameworks to improve power system efficiency and sustainability. This review aims to advance MOPF research by highlighting state-of-the-art methodologies and encouraging the development of innovative solutions for future energy challenges.
Evolutionary Algorithms (EAs) play a crucial role in the architectural configuration and training of Artificial Deep Neural Networks (DNNs), a process known as neuroevolution. However, neuroevolution is hindered by its inherent computational expense, requiring multiple generations, a large population, and numerous epochs. The most computationally intensive aspect lies in evaluating the fitness function of a single candidate solution. To address this challenge, we employ Surrogate-assisted EAs (SAEAs). While a few SAEAs approaches have been proposed in neuroevolution, none have been applied to truly large DNNs due to issues like intractable information usage. In this work, drawing inspiration from Genetic Programming semantics, we use phenotypic distance vectors, outputted from DNNs, alongside Kriging Partial Least Squares (KPLS), an approach that is effective in handling these large vectors, making them suitable for search. Our proposed approach, named Neuro-Linear Genetic Programming surrogate model (NeuroLGP-SM), efficiently and accurately estimates DNN fitness without the need for complete evaluations. NeuroLGP-SM demonstrates competitive or superior results compared to 12 other methods, including NeuroLGP without SM, convolutional neural networks, support vector machines, and autoencoders. Additionally, it is worth noting that NeuroLGP-SM is 25% more energy-efficient than its NeuroLGP counterpart. This efficiency advantage adds to the overall appeal of our proposed NeuroLGP-SM in optimising the configuration of large DNNs.
Evolutionary algorithms are particularly effective for optimisation problems with dynamic and stochastic components. We propose multi-objective evolutionary approaches for the knapsack problem with stochastic profits under static and dynamic weight constraints. The chance-constrained problem model allows us to effectively capture the stochastic profits and associate a confidence level to the solutions' profits. We consider a bi-objective formulation that maximises expected profit and minimises variance, which allows optimising the problem independent of a specific confidence level on the profit. We derive a three-objective formulation by relaxing the weight constraint into an additional objective. We consider the GSEMO algorithm with standard and a sliding window-based parent selection to evaluate the objective formulations. Moreover, we modify fitness formulations and algorithms for the dynamic problem variant to store some infeasible solutions to cater to future changes. We conduct experimental investigations on both problems using the proposed problem formulations and algorithms. Our results show that three-objective approaches outperform approaches that use bi-objective formulations, and they further improve when GSEMO uses sliding window selection.
When addressing the challenge of complex multi-objective optimization problems, particularly those with non-convex and non-uniform Pareto fronts, Decomposition-based Multi-Objective Evolutionary Algorithms (MOEADs) often converge to local optima, thereby limiting solution diversity. Despite its significance, this issue has received limited theoretical exploration. Through a comprehensive geometric analysis, we identify that the traditional method of Reference Point (RP) selection fundamentally contributes to this challenge. In response, we introduce an innovative RP selection strategy, the Weight Vector-Guided and Gaussian-Hybrid method, designed to overcome the local optima issue. This approach employs a novel RP type that aligns with weight vector directions and integrates a Gaussian distribution to combine three distinct RP categories. Our research comprises two main experimental components: an ablation study involving 14 algorithms within the MOEADs framework, spanning from 2014 to 2022, to validate our theoretical framework, and a series of empirical tests to evaluate the effectiveness of our proposed method against both traditional and cutting-edge alternatives. Results demonstrate that our method achieves remarkable improvements in both population diversity and convergence.
Recently, Pareto Set Learning (PSL) has been proposed for learning the entire Pareto set using a neural network. PSL employs preference vectors to scalarize multiple objectives, facilitating the learning of mappings from preference vectors to specific Pareto optimal solutions. Previous PSL methods have shown their effectiveness in solving artificial multi-objective optimization problems (MOPs) with uniform preference vector sampling. The quality of the learned Pareto set is influenced by the sampling strategy of the preference vector, and the sampling of the preference vector needs to be decided based on the Pareto front shape. However, a fixed preference sampling strategy cannot simultaneously adapt the Pareto front of multiple MOPs. To address this limitation, this paper proposes an Evolutionary Preference Sampling (EPS) strategy to efficiently sample preference vectors. Inspired by evolutionary algorithms, we consider preference sampling as an evolutionary process to generate preference vectors for neural network training. We integrate the EPS strategy into five advanced PSL methods. Extensive experiments demonstrate that our proposed method has a faster convergence speed than baseline algorithms on 7 testing problems. Our implementation is available at https://github.com/rG223/EPS.
The JUMP$_k$ benchmark was the first problem for which crossover was proven to give a speed-up over mutation-only evolutionary algorithms. Jansen and Wegener (2002) proved an upper bound of $O(\text{poly}(n) + 4^k/p_c)$ for the ($\mu$+1) Genetic Algorithm ($(\mu+1)$ GA), but only for unrealistically small crossover probabilities $p_c$. To this date, it remains an open problem to prove similar upper bounds for realistic $p_c$; the best known runtime bound, in terms of function evaluations, for $p_c = \Omega(1)$ is $O((n/\chi)^{k-1})$, $\chi$ a positive constant. We provide a novel approach and analyse the evolution of the population diversity, measured as sum of pairwise Hamming distances, for a variant of the $(\mu+1)$ GA on JUMP$_k$. The $(\mu+1)$-$\lambda_c$-GA creates one offspring in each generation either by applying mutation to one parent or by applying crossover $\lambda_c$ times to the same two parents (followed by mutation), to amplify the probability of creating an accepted offspring in generations with crossover. We show that population diversity in the $(\mu+1)$-$\lambda_c$-GA converges to an equilibrium of near-perfect diversity. This yields an improved time bound of $O(\mu n \log(\mu) + 4^k)$ function evaluations for a range of $k$ under the mild assumptions $p_c = O(1/k)$ and $\mu \in \Omega(kn)$. For all constant $k$, the restriction is satisfied for some $p_c = \Omega(1)$ and it implies that the expected runtime for all constant $k$ and an appropriate $\mu = \Theta(kn)$ is bounded by $O(n^2 \log n)$, irrespective of $k$. For larger $k$, the expected time of the $(\mu+1)$-$\lambda_c$-GA is $\Theta(4^k)$, which is tight for a large class of unbiased black-box algorithms and faster than the original $(\mu+1)$ GA by a factor of $\Omega(1/p_c)$. We also show that our analysis can be extended to other unitation functions such as JUMP$_{k, \delta}$ and HURDLE.
The application of evolutionary algorithms (EAs) to multi-objective optimization problems has been widespread. However, the EA research community has not paid much attention to large-scale multi-objective optimization problems arising from real-world applications. Especially, Food-Energy-Water systems are intricately linked among food, energy and water that impact each other. They usually involve a huge number of decision variables and many conflicting objectives to be optimized. Solving their related optimization problems is essentially important to sustain the high-quality life of human beings. Their solution space size expands exponentially with the number of decision variables. Searching in such a vast space is challenging because of such large numbers of decision variables and objective functions. In recent years, a number of large-scale many-objectives optimization evolutionary algorithms have been proposed. In this paper, we solve a Food-Energy-Water optimization problem by using the state-of-art intelligent optimization methods and compare their performance. Our results conclude that the algorithm based on an inverse model outperforms the others. This work should be highly useful for practitioners to select the most suitable method for their particular large-scale engineering optimization problems.
Co-evolution is a powerful problem-solving approach. However, fitness evaluation in co-evolutionary algorithms can be computationally expensive, as the quality of an individual in one population is defined by its interactions with many (or all) members of one or more other populations. To accelerate co-evolutionary systems, we introduce phylogeny-informed interaction estimation, which uses runtime phylogenetic analysis to estimate interaction outcomes between individuals based on how their relatives performed against each other. We test our interaction estimation method with three distinct co-evolutionary systems: two systems focused on measuring problem-solving success and one focused on measuring evolutionary open-endedness. We find that phylogeny-informed estimation can substantially reduce the computation required to solve problems, particularly at the beginning of long-term evolutionary runs. Additionally, we find that our estimation method initially jump-starts the evolution of neural complexity in our open-ended domain, but estimation-free systems eventually "catch-up" if given enough time. More broadly, continued refinements to these phylogeny-informed interaction estimation methods offers a promising path to reducing the computational cost of running co-evolutionary systems while maintaining their open-endedness.
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.
Real-world optimization problems often involve stochastic and dynamic components. Evolutionary algorithms are particularly effective in these scenarios, as they can easily adapt to uncertain and changing environments but often uncertainty and dynamic changes are studied in isolation. In this paper, we explore the use of 3-objective evolutionary algorithms for the chance constrained knapsack problem with dynamic constraints. In our setting, the weights of the items are stochastic and the knapsack's capacity changes over time. We introduce a 3-objective formulation that is able to deal with the stochastic and dynamic components at the same time and is independent of the confidence level required for the constraint. This new approach is then compared to the 2-objective formulation which is limited to a single confidence level. We evaluate the approach using two different multi-objective evolutionary algorithms (MOEAs), namely the global simple evolutionary multi-objective optimizer (GSEMO) and the multi-objective evolutionary algorithm based on decomposition (MOEA/D), across various benchmark scenarios. Our analysis highlights the advantages of the 3-objective formulation over the 2-objective formulation in addressing the dynamic chance constrained knapsack problem.
Parent selection plays an important role in evolutionary algorithms, and many strategies exist to select the parent pool before breeding the next generation. Methods often rely on average error over the entire dataset as a criterion to select the parents, which can lead to an information loss due to aggregation of all test cases. Under epsilon-lexicase selection, the population goes to a selection pool that is iteratively reduced by using each test individually, discarding individuals with an error higher than the elite error plus the median absolute deviation (MAD) of errors for that particular test case. In an attempt to better capture differences in performance of individuals on cases, we propose a new criteria that splits errors into two partitions that minimize the total variance within partitions. Our method was embedded into the FEAT symbolic regression algorithm, and evaluated with the SRBench framework, containing 122 black-box synthetic and real-world regression problems. The empirical results show a better performance of our approach compared to traditional epsilon-lexicase selection in the real-world datasets while showing equivalent performance on the synthetic dataset.
We propose a new, flexible approach for dynamically maintaining successful mutation rates in evolutionary algorithms using $k$-bit flip mutations. The algorithm adds successful mutation rates to an archive of promising rates that are favored in subsequent steps. Rates expire when their number of unsuccessful trials has exceeded a threshold, while rates currently not present in the archive can enter it in two ways: (i) via user-defined minimum selection probabilities for rates combined with a successful step or (ii) via a stagnation detection mechanism increasing the value for a promising rate after the current bit-flip neighborhood has been explored with high probability. For the minimum selection probabilities, we suggest different options, including heavy-tailed distributions. We conduct rigorous runtime analysis of the flexible evolutionary algorithm on the OneMax and Jump functions, on general unimodal functions, on minimum spanning trees, and on a class of hurdle-like functions with varying hurdle width that benefit particularly from the archive of promising mutation rates. In all cases, the runtime bounds are close to or even outperform the best known results for both stagnation detection and heavy-tailed mutations.
Evolutionary Algorithms (EAs) are often challenging to apply in real-world settings since evolutionary computations involve a large number of evaluations of a typically expensive fitness function. For example, an evaluation could involve training a new machine learning model. An approximation (also known as meta-model or a surrogate) of the true function can be used in such applications to alleviate the computation cost. In this paper, we propose a two-stage surrogate-assisted evolutionary approach to address the computational issues arising from using Genetic Algorithm (GA) for feature selection in a wrapper setting for large datasets. We define 'Approximation Usefulness' to capture the necessary conditions to ensure correctness of the EA computations when an approximation is used. Based on this definition, we propose a procedure to construct a lightweight qualitative meta-model by the active selection of data instances. We then use a meta-model to carry out the feature selection task. We apply this procedure to the GA-based algorithm CHC (Cross generational elitist selection, Heterogeneous recombination and Cataclysmic mutation) to create a Qualitative approXimations variant, CHCQX. We show that CHCQX converges faster to feature subset solutions of significantly higher accuracy (as compared to CHC), particularly for large datasets with over 100K instances. We also demonstrate the applicability of the thinking behind our approach more broadly to Swarm Intelligence (SI), another branch of the Evolutionary Computation (EC) paradigm with results of PSOQX, a qualitative approximation adaptation of the Particle Swarm Optimization (PSO) method. A GitHub repository with the complete implementation is available.
Experience shows that typical evolutionary algorithms can cope well with stochastic disturbances such as noisy function evaluations. In this first mathematical runtime analysis of the $(1+\lambda)$ and $(1,\lambda)$ evolutionary algorithms in the presence of prior bit-wise noise, we show that both algorithms can tolerate constant noise probabilities without increasing the asymptotic runtime on the OneMax benchmark. For this, a population size $\lambda$ suffices that is at least logarithmic in the problem size $n$. The only previous result in this direction regarded the less realistic one-bit noise model, required a population size super-linear in the problem size, and proved a runtime guarantee roughly cubic in the noiseless runtime for the OneMax benchmark. Our significantly stronger results are based on the novel proof argument that the noiseless offspring can be seen as a biased uniform crossover between the parent and the noisy offspring. We are optimistic that the technical lemmas resulting from this insight will find applications also in future mathematical runtime analyses of evolutionary algorithms.
Evolutionary algorithms are increasingly recognised as a viable computational approach for the automated optimisation of deep neural networks (DNNs) within artificial intelligence. This method extends to the training of DNNs, an approach known as neuroevolution. However, neuroevolution is an inherently resource-intensive process, with certain studies reporting the consumption of thousands of GPU days for refining and training a single DNN network. To address the computational challenges associated with neuroevolution while still attaining good DNN accuracy, surrogate models emerge as a pragmatic solution. Despite their potential, the integration of surrogate models into neuroevolution is still in its early stages, hindered by factors such as the effective use of high-dimensional data and the representation employed in neuroevolution. In this context, we address these challenges by employing a suitable representation based on Linear Genetic Programming, denoted as NeuroLGP, and leveraging Kriging Partial Least Squares. The amalgamation of these two techniques culminates in our proposed methodology known as the NeuroLGP-Surrogate Model (NeuroLGP-SM). For comparison purposes, we also code and use a baseline approach incorporating a repair mechanism, a common practice in neuroevolution. Notably, the baseline approach surpasses the renowned VGG-16 model in accuracy. Given the computational intensity inherent in DNN operations, a singular run is typically the norm. To evaluate the efficacy of our proposed approach, we conducted 96 independent runs. Significantly, our methodologies consistently outperform the baseline, with the SM model demonstrating superior accuracy or comparable results to the NeuroLGP approach. Noteworthy is the additional advantage that the SM approach exhibits a 25% reduction in computational requirements, further emphasising its efficiency for neuroevolution.
The paper provides a comprehensive overview of Neural Architecture Search (NAS), emphasizing its evolution from manual design to automated, computationally-driven approaches. It covers the inception and growth of NAS, highlighting its application across various domains, including medical imaging and natural language processing. The document details the shift from expert-driven design to algorithm-driven processes, exploring initial methodologies like reinforcement learning and evolutionary algorithms. It also discusses the challenges of computational demands and the emergence of efficient NAS methodologies, such as Differentiable Architecture Search and hardware-aware NAS. The paper further elaborates on NAS's application in computer vision, NLP, and beyond, demonstrating its versatility and potential for optimizing neural network architectures across different tasks. Future directions and challenges, including computational efficiency and the integration with emerging AI domains, are addressed, showcasing NAS's dynamic nature and its continued evolution towards more sophisticated and efficient architecture search methods.