The theoretical understanding of MOEAs is lagging far behind their success in practice. In particular, previous theory work considers mostly easy problems that are composed of unimodal objectives. As a first step towards a deeper understanding of how evolutionary algorithms solve multimodal multiobjective problems, we propose the OJZJ problem, a bi-objective problem composed of two objectives isomorphic to the classic jump function benchmark. We prove that SEMO with probability one does not compute the full Pareto front, regardless of the runtime. In contrast, for all problem sizes $n$ and all jump sizes ${k \in [4..\frac n2 - 1]}$, the global SEMO (GSEMO) covers the Pareto front in an expected number of $\Theta((n-2k)n^{k})$ iterations. For $k = o(n)$, we also show the tighter bound $\frac 32 e n^{k+1} \pm o(n^{k+1})$, which might be the first runtime bound for an MOEA that is tight apart from lower-order terms. We also combine the GSEMO with two approaches that showed advantages in single-objective multimodal problems. When using the GSEMO with a heavy-tailed mutation operator, the expected runtime improves by a factor of at least $k^{\Omega(k)}$. When adapting the recent stagnation-detection strategy of Rajabi and Witt (2022) to the GSEMO, the expected runtime also improves by a factor of at least $k^{\Omega(k)}$ and surpasses the heavy-tailed GSEMO by a small polynomial factor in $k$. Via an experimental analysis, we show that these asymptotic differences are visible already for small problem sizes: A factor-$5$ speed-up from heavy-tailed mutation and a factor-$10$ speed-up from stagnation detection can be observed already for jump size~$4$ and problem sizes between $10$ and $50$. Overall, our results show that the ideas recently developed to aid single-objective evolutionary algorithms to cope with local optima can be effectively employed also in multiobjective optimization.
The study of semantics in Genetic Program (GP) deals with the behaviour of a program given a set of inputs and has been widely reported in helping to promote diversity in GP for a range of complex problems ultimately improving evolutionary search. The vast majority of these studies have focused their attention in single-objective GP, with just a few exceptions where Pareto-based dominance algorithms such as NSGA-II and SPEA2 have been used as frameworks to test whether highly popular semantics-based methods, such as Semantic Similarity-based Crossover (SSC), helps or hinders evolutionary search. Surprisingly it has been reported that the benefits exhibited by SSC in SOGP are not seen in Pareto-based dominance Multi-objective GP. In this work, we are interested in studying if the same carries out in Multi-objective Evolutionary Algorithms based on Decomposition (MOEA/D). By using the MNIST dataset, a well-known dataset used in the machine learning community, we show how SSC in MOEA/D promotes semantic diversity yielding better results compared to when this is not present in canonical MOEA/D.
Heat and power have become the most indispensable resources. However, the traditional ways of generating power and heat are inefficient and cause high pollution; a CHP (Combined Heat and Power) unit can solve these problems well. In recent years, more attention has been paid to energy conservation and environmental protection, and Combined Heat and Power Economic Emission Dispatch (CHPEED) has become an important multi-objective optimization problem. In this paper, an Indicator & crowding Distance-based Evolutionary Algorithm (IDBEA) is put forward for handling this non-convex and non-linear problem. With consideration of the valve-point effects and power transmission loss, IDBEA is tested on three standard test systems with different types, including four units, five units and seven units. In the experiment, IDBEA is compared with several evolutionary algorithms, the simulation results demonstrate that IDBEA has strong stability and superiority, while the solutions show better convergence and diversity than several typical algorithms.
Target tracking problem has many practical applications in real life. In submarines, target tracking is done using, preferably, passive sensors. These sensors measure only the bearing angles between the observed target and the ownship. Therefore, this problem is generally referred as bearing only target tracking or target motion analysis. The classical approach is to use a state observer based filter, i.e. Extended Kalman Filter, to estimate the range, course and speed of the target, using only the bearings. In recent studies, the problem is solved as a global optimization problem by utilizing evolutionary algorithms with respect to some objective functions. In this study, we investigate the effect of the commonly used cost functions on the performance of the TMA algorithms. Particularly, we investigate the cost functions based on bearing differences and equidistant line segments. The simulation results show that the latter gives a sub-optimal solution to the target motion analysis problem, compared to the former.
Partial (replication) index tracking is a popular passive investment strategy. It aims to replicate the performance of a given index by constructing a tracking portfolio which contains some constituents of the index. The tracking error optimisation is quadratic and NP-hard when taking the L0 constraint into account so it is usually solved by heuristic methods such as evolutionary algorithms. This paper introduces a simple, efficient and scalable connectionist model as an alternative. We propose a novel reparametrisation method and then solve the optimisation problem with stochastic neural networks. The proposed approach is examined with S&P 500 index data for more than 10 years and compared with widely used index tracking approaches such as forward and backward selection and the largest market capitalisation methods. The empirical results show our model achieves excellent performance. Compared with the benchmarked models, our model has the lowest tracking error, across a range of portfolio sizes. Meanwhile it offers comparable performance to the others on secondary criteria such as volatility, Sharpe ratio and maximum drawdown.
Noise cancellation is one of the important signal processing functions of any communication system, as noise affects data integrity. In existing systems, traditional filters are used to cancel the noise from the received signals. These filters use fixed hardware which is capable of filtering specific frequency or a range of frequencies. However, next generation communication technologies, such as cognitive radio, will require the use of adaptive filters that can dynamically reconfigure their filtering parameters for any frequency. To this end, a few noise cancellation techniques have been proposed, including least mean squares (LMS) and its variants. However, these algorithms are susceptible to non-linear noise and fail to locate the global optimum solution for de-noising. In this paper, we investigate the efficiency of two global search optimization based algorithms, genetic algorithm and particle swarm optimization in performing noise cancellation in cognitive radio systems. These algorithms are implemented and their performances are compared to that of LMS using bit error rate and mean square error as performance evaluation metrics. Simulations are performed with additive white Gaussian noise and random nonlinear noise. Results indicate that GA and PSO perform better than LMS for the case of AWGN corrupted signal but for non-linear random noise PSO outperforms the other two algorithms.
Noise originating from several sources in a RF environment degrades the performance of communication systems. In wideband systems, such as cognitive radios, noise at the receiver can originate from non-linearity present in the RF front end, time-varying thermal noise within the receiver radio system, and noise from adjacent network nodes. Several denoising techniques have been proposed for cognitive radios, some of which are applied during spectrum sensing and others to received noisy signal during communication. Examples of some of these techniques used for noise cancellation in received signals are least mean square (LMS) and its variants. However, these algorithms have low performance with non-linear signals and cannot locate a global optimum solution for noise cancellation. Therefore, application of global search optimization techniques, such as evolutionary algorithms, is considered for noise cancellation. In this paper, particle swarm optimization (PSO) and LMS algorithms are implemented and their performances are evaluated. Extensive simulations were performed where Gaussian and non-linear random noise were added to the transmitted signal. The performance comparison was done using two metrics: bit error rate and mean square error. The results show that PSO outperforms LMS under both Gaussian and nonlinear random noise.
We propose a mathematical framework for natural selection in finite populations. Traditionally, many of the selection-based processes used to describe cultural and genetic evolution (such as imitation and birth-death models) have been studied on a case-by-case basis. Over time, these models have grown in sophistication to include population structure, differing phenotypes, and various forms of interaction asymmetry, among other features. Furthermore, many processes inspired by natural selection, such as evolutionary algorithms in computer science, possess characteristics that should fall within the realm of a "selection process," but so far there is no overarching theory encompassing these evolutionary processes. The framework of $\textit{stochastic selection processes}$ we present here provides such a theory and consists of three main components: a $\textit{population state space}$, an $\textit{aggregate payoff function}$, and an $\textit{update rule}$. A population state space is a generalization of the notion of population structure, and it can include non-spatial information such as strategy-mutation rates and phenotypes. An aggregate payoff function allows one to generically talk about the fitness of traits without explicitly specifying a method of payoff accounting or even the nature of the interactions that determine payoff/fitness. An update rule is a fitness-based function that updates a population based on its current state, and it includes as special cases the classical update mechanisms (Moran, Wright-Fisher, etc.) as well as more complicated mechanisms involving chromosomal crossover, mutation, and even complex cultural syntheses of strategies of neighboring individuals. Our framework covers models with variable population size as well as with arbitrary, measurable trait spaces.
Assessing the empirical performance of Multi-Objective Evolutionary Algorithms (MOEAs) is vital when we extensively test a set of MOEAs and aim to determine a proper ranking thereof. Multiple performance indicators, e.g., the generational distance and the hypervolume, are frequently applied when reporting the experimental data, where typically the data on each indicator is analyzed independently from other indicators. Such a treatment brings conceptual difficulties in aggregating the result on all performance indicators, and it might fail to discover significant differences among algorithms if the marginal distributions of the performance indicator overlap. Therefore, in this paper, we propose to conduct a multivariate $\mathcal{E}$-test on the joint empirical distribution of performance indicators to detect the potential difference in the data, followed by a post-hoc procedure that utilizes the linear discriminative analysis to determine the superiority between algorithms. This performance analysis's effectiveness is supported by an experimentation conducted on four algorithms, 16 problems, and 6 different numbers of objectives.
This study presents the approach to analyzing the evolution of an arbitrary complex system whose behavior is characterized by a set of different time-dependent factors. The key requirement for these factors is only that they must contain an information about the system; it does not matter at all what the nature (physical, biological, social, economic, etc.) of a complex system is. Within the framework of the presented theoretical approach, the problem of searching for non-linear regression models that express the relationship between these factors for a complex system under study is solved. It will be shown that this problem can be solved using the methodology of \emph{genetic (evolutionary)} algorithms. The resulting regression models make it possible to predict the most probable evolution of the considered system, as well as to determine the significance of some factors and, thereby, to formulate some recommendations to drive by this system. It will be shown that the presented theoretical approach can be used to analyze the data (information) characterizing the educational process in the discipline "Physics" in the secondary school, and to develop the strategies for improving academic performance in this discipline.
Discrete latent variables are considered important for real world data, which has motivated research on Variational Autoencoders (VAEs) with discrete latents. However, standard VAE training is not possible in this case, which has motivated different strategies to manipulate discrete distributions in order to train discrete VAEs similarly to conventional ones. Here we ask if it is also possible to keep the discrete nature of the latents fully intact by applying a direct discrete optimization for the encoding model. The approach is consequently strongly diverting from standard VAE-training by sidestepping sampling approximation, reparameterization trick and amortization. Discrete optimization is realized in a variational setting using truncated posteriors in conjunction with evolutionary algorithms. For VAEs with binary latents, we (A) show how such a discrete variational method ties into gradient ascent for network weights, and (B) how the decoder is used to select latent states for training. Conventional amortized training is more efficient and applicable to large neural networks. However, using smaller networks, we here find direct discrete optimization to be efficiently scalable to hundreds of latents. More importantly, we find the effectiveness of direct optimization to be highly competitive in `zero-shot' learning. In contrast to large supervised networks, the here investigated VAEs can, e.g., denoise a single image without previous training on clean data and/or training on large image datasets. More generally, the studied approach shows that training of VAEs is indeed possible without sampling-based approximation and reparameterization, which may be interesting for the analysis of VAE-training in general. For `zero-shot' settings a direct optimization, furthermore, makes VAEs competitive where they have previously been outperformed by non-generative approaches.
In this study, we review robots behavior especially warrior robots by using evolutionary algorithms. This kind of algorithms is inspired by nature that causes robots behaviors get resemble to collective behavior. Collective behavior of creatures such as bees was shown that do some functions which depended on interaction and cooperation would need to a well-organized system so that all creatures within it carry out their duty, very well. For robots which do not have any intelligence, we can define an algorithm and show the results by a simple simulation.
In this paper we consider the problem of finding Pulse Repetition Intervals allowing the best compromises mitigating range and Doppler ambiguities in a Pulsed-Doppler radar system. We revisit a problem that was proposed to the Evolutionary Computation community as a real-world case to test Many-objective Optimization algorithms. We use it as a baseline to compare several Evolutionary Algorithms for black-box optimization with different metrics. Resulting data is aggregated to build a reference set of Pareto optimal points and is the starting point for further analysis and operational use by the radar designer.
Evolutionary algorithms have long been used for optimization problems where the appropriate size of solutions is unclear a priori. The applicability of this methodology is here investigated on the problem of designing a nano-particle (NP) based drug delivery system targeting cancer tumours. Utilizing a treatment comprising of multiple types of NPs is expected to be more effective due to the higher complexity of the treatment. This paper begins by utilizing the well-known NK model to explore the effects of fitness landscape ruggedness upon the evolution of genome length and, hence, solution complexity. The size of a novel sequence and the absence or presence of sequence deletion are also considered. Results show that whilst landscape ruggedness can alter the dynamics of the process, it does not hinder the evolution of genome length. These findings are then explored within the aforementioned real-world problem. In the first known instance, treatments with multiple types of NPs are used simultaneously, via an agent-based open source physics-based cell simulator. The results suggest that utilizing multiple types of NPs is more efficient when the solution space is explored with the evolutionary techniques under a predefined computational budget.
Numerous deep reinforcement learning agents have been proposed, and each of them has its strengths and flaws. In this work, we present a Cooperative Heterogeneous Deep Reinforcement Learning (CHDRL) framework that can learn a policy by integrating the advantages of heterogeneous agents. Specifically, we propose a cooperative learning framework that classifies heterogeneous agents into two classes: global agents and local agents. Global agents are off-policy agents that can utilize experiences from the other agents. Local agents are either on-policy agents or population-based evolutionary algorithms (EAs) agents that can explore the local area effectively. We employ global agents, which are sample-efficient, to guide the learning of local agents so that local agents can benefit from sample-efficient agents and simultaneously maintain their advantages, e.g., stability. Global agents also benefit from effective local searches. Experimental studies on a range of continuous control tasks from the Mujoco benchmark show that CHDRL achieves better performance compared with state-of-the-art baselines.
In model-based evolutionary algorithms (EAs), the underlying search distribution is adapted to the problem at hand, for example based on dependencies between decision variables. Hill-valley clustering is an adaptive niching method in which a set of solutions is clustered such that each cluster corresponds to a single mode in the fitness landscape. This can be used to adapt the search distribution of an EA to the number of modes, exploring each mode separately. Especially in a black-box setting, where the number of modes is a priori unknown, an adaptive approach is essential for good performance. In this work, we introduce multi-objective hill-valley clustering and combine it with MAMaLGaM, a multi-objective EA, into the multi-objective hill-valley EA (MO-HillVallEA). We empirically show that MO-HillVallEA outperforms MAMaLGaM and other well-known multi-objective optimization algorithms on a set of benchmark functions. Furthermore, and perhaps most important, we show that MO-HillVallEA is capable of obtaining and maintaining multiple approximation sets simultaneously over time.
In the past, evolutionary algorithms (EAs) that use probabilistic modeling of the best solutions incorporated latent or hidden vari- ables to the models as a more accurate way to represent the search distributions. Recently, a number of neural-network models that compute approximations of posterior (latent variable) distributions have been introduced. In this paper, we investigate the use of the variational autoencoder (VAE), a class of neural-network based generative model, for modeling and sampling search distributions as part of an estimation of distribution algorithm. We show that VAE can capture dependencies between decision variables and ob- jectives. This feature is proven to improve the sampling capacity of model based EAs. Furthermore, we extend the original VAE model by adding a new, fitness-approximating network component. We show that it is possible to adapt the architecture of these models and we present evidence of how to extend VAEs to better fulfill the requirements of probabilistic modeling in EAs. While our results are not yet competitive with state of the art probabilistic-based optimizers, they represent a promising direction for the application of generative models within EDAs.
We study evolutionary algorithms in a dynamic setting, where for each generation a different fitness function is chosen, and selection is performed with respect to the current fitness function. Specifically, we consider Dynamic BinVal, in which the fitness functions for each generation is given by the linear function BinVal, but in each generation the order of bits is randomly permuted. For the (1+1)-EA it was known that there is an efficiency threshold $c_0$ for the mutation parameter, at which the runtime switches from quasilinear to exponential. A previous empirical evidence suggested that for larger population size $\mu$, the threshold may increase. We prove that this is at least the case in an $\varepsilon$-neighborhood around the optimum: the threshold of the (\mu+1)-EA becomes arbitrarily large if the $\mu$ is chosen large enough. However, the most surprising result is obtained by a second order analysis for $\mu=2$: the threshold INcreases with increasing proximity to the optimum. In particular, the hardest region for optimization is NOT around the optimum.
We study the task of replicating the functionality of black-box neural models, for which we only know the output class probabilities provided for a set of input images. We assume back-propagation through the black-box model is not possible and its training images are not available, e.g. the model could be exposed only through an API. In this context, we present a teacher-student framework that can distill the black-box (teacher) model into a student model with minimal accuracy loss. To generate useful data samples for training the student, our framework (i) learns to generate images on a proxy data set (with images and classes different from those used to train the black-box) and (ii) applies an evolutionary strategy to make sure that each generated data sample exhibits a high response for a specific class when given as input to the black box. Our framework is compared with several baseline and state-of-the-art methods on three benchmark data sets. The empirical evidence indicates that our model is superior to the considered baselines. Although our method does not back-propagate through the black-box network, it generally surpasses state-of-the-art methods that regard the teacher as a glass-box model. Our code is available at: https://github.com/antoniobarbalau/black-box-ripper.
In the area of evolutionary computation the calculation of diverse sets of high-quality solutions to a given optimization problem has gained momentum in recent years under the term evolutionary diversity optimization. Theoretical insights into the working principles of baseline evolutionary algorithms for diversity optimization are still rare. In this paper we study the well-known Minimum Spanning Tree problem (MST) in the context of diversity optimization where population diversity is measured by the sum of pairwise edge overlaps. Theoretical results provide insights into the fitness landscape of the MST diversity optimization problem pointing out that even for a population of $\mu=2$ fitness plateaus (of constant length) can be reached, but nevertheless diverse sets can be calculated in polynomial time. We supplement our theoretical results with a series of experiments for the unconstrained and constraint case where all solutions need to fulfill a minimal quality threshold. Our results show that a simple $(\mu+1)$-EA can effectively compute a diversified population of spanning trees of high quality.