LCS & GBML Central

The ultimate goal of multiobjective optimization is to help a decision maker (DM) identify solution(s) of interest (SOI) achieving satisfactory tradeoffs among multiple conflicting criteria. This can be realized by leveraging DM’s preference information in evolutionary multiobjective optimization (EMO). No consensus has been reached on the effectiveness brought by incorporating preference in EMO (either a priori or interactively) versus a posteriori decision making after a complete run of an EMO algorithm. Bearing this consideration in mind, this article: 1) provides a pragmatic overview of the existing developments of preference-based EMO (PBEMO) and 2) conducts a series of experiments to investigate the effectiveness brought by preference incorporation in EMO for approximating various SOI. In particular, the DM’s preference information is elicited as a reference point, which represents her/his aspirations for different objectives. The experimental results demonstrate that preference incorporation in EMO does not always lead to a desirable approximation of SOI if the DM’s preference information is not well utilized, nor does the DM elicit invalid preference information, which is not uncommon when encountering a black-box system. To a certain extent, this issue can be remedied through an interactive preference elicitation. Last but not the least, we find that a PBEMO algorithm is able to be generalized to approximate the whole PF given an appropriate setup of preference information.

It is our pleasure to introduce this special issue on the recent advances in the theoretical foundations of evolutionary computation (EC). While in the early days of this field, theoretical analyses inevitably focused on simplified models of evolutionary algorithms (EAs), the continuous progress made in the development of suitable mathematical techniques for the analysis now allows to derive proven statements regarding the performance of off-the-shelf metaheuristics, such as standard generational and steady-state genetic algorithms with no algorithmic simplifications. Comparisons between the performance of a given EA with the best possible one can also be made nowadays, allowing to assess whether a given algorithm may be improved upon or whether its performance is optimal for a given class of problems. Such understanding often provides insights for the design of new EAs which provably have better performance for given problems. We are glad that examples of results of this kind are present within this special issue. A total of 27 papers were submitted which were the subject of at least three independent reviews, and six manuscripts of the highest quality were selected for publication in the special issue. In the following, we provide a brief summary of these manuscripts.

It is known that the (1 + 1)-EA with mutation rate $c/n$ optimizes every monotone function efficiently if $c < 1$ , and needs exponential time on some monotone functions (HotTopic functions) if $cgeq 2.2$ . We study the same question for a large variety of algorithms, particularly for the $(1 + lambda)$ -EA, $(mu + 1)$ -EA, $(mu + 1)$ -GA, their “fast” counterparts, and for the $(1 + (lambda,lambda))$ -GA. We find that all considered mutation-based algorithms show a similar dichotomy for HotTopic functions, or even for all monotone functions. For the $(1 + (lambda,lambda))$ -GA, this dichotomy is in the parameter $cgamma $ , which is the expected number of bit flips in an individual after mutation and crossover, neglecting selection. For the fast algorithms, the dichotomy is in $m_{2}/m_{1}$ , where $m_{1}$ and $m_{2}$ are the first and second falling moment of the number of bit flips. Surprisingly, the range of efficient parameters is not affected by either population size $mu $ –
/inline-formula> nor by the offspring population size $lambda $ . The picture changes completely if crossover is allowed. The genetic algorithms $(mu + 1)$ -GA and $(mu + 1)$ -fGA are efficient for arbitrary mutations strengths if $mu $ is large enough.

We propose a new black-box complexity model for search algorithms evaluating $lambda $ search points in parallel. The parallel unary unbiased black-box complexity gives lower bounds on the number of function evaluations every parallel unary unbiased black-box algorithm needs to optimize a given problem. It captures the inertia caused by offspring populations in evolutionary algorithms and the total computational effort in parallel metaheuristics.¹ We present complexity results for LeadingOnes and OneMax. Our main result is a general performance limit: we prove that on every function every $lambda $ -parallel unary unbiased algorithm needs at least a certain number of evaluations (a function of problem size and $lambda $ ) to find any desired target set of up to exponential size, with an overwhelming probability. This yields lower bounds for the typical optimization time on unimodal and multimodal problems, for the time to find any local optimum, and for the time to even get close to any optimum. The power and versatility of this approach is shown for a wide range of illustrative problems from combinatorial optimization. Our performance limits can guide parameter choice and algorithm design; we demonstrate the latter by presenting an optimal $lambda $ -parallel algorithm for OneMax that uses parallelism most effectively.¹

This article significantly extends preliminary results which appeared in [1].

We theoretically analyze the information geometric optimization (IGO), which is a unified framework of stochastic search algorithms for black-box optimization. The IGO framework has two parameters: 1) the learning rate and 2) the sample size, and they influence the behavior of the algorithm. We investigate the strategy parameters of the IGO with the family of isotropic Gaussian distributions on a general convex quadratic function. Compared to the previous theoretical works, where an infinite sample size is assumed and the deterministic algorithm dynamics is studied, we investigate the expected improvement of the algorithm with a finite sample size. The analysis finds that the relative decrease rates of the distance from the distribution mean to the landscape optimum and the distribution standard deviation must be the same, which we observe in practice, while the analysis based on an infinite sample size failed to obtain. We derive these rates explicitly as a function of the eigenvalues of the Hessian of the objective function and the strategy parameters. We also derive the stable value of the ratio of the square distance to the optimum over the distribution variance, as well as the conditions that the stable value exists. These theoretical values coincide with our numerical simulations.

Estimation-of-distribution algorithms (EDAs) are randomized search heuristics that create a probabilistic model of the solution space, which is updated iteratively, based on the quality of the solutions sampled according to the model. As previous works show, this iteration-based perspective can lead to erratic updates of the model, in particular, to bit-frequencies approaching a random boundary value. In order to overcome this problem, we propose a new EDA based on the classic compact genetic algorithm (cGA) that takes into account a longer history of samples and updates its model only with respect to information which it classifies as statistically significant. We prove that this significance-based cGA (sig-cGA) optimizes the commonly regarded benchmark functions OneMax (OM), LeadingOnes, and BinVal all in quasilinear time, a result shown for no other EDA or evolutionary algorithm so far. For the recently proposed stable compact genetic algorithm—an EDA that tries to prevent erratic model updates by imposing a bias to the uniformly distributed model—we prove that it optimizes OM only in a time exponential in its hypothetical population size. Similarly, we show that the convex search algorithm cannot optimize OM in polynomial time.

We expose and contrast the impact of landscape characteristics on the performance of search heuristics for black-box multiobjective combinatorial optimization problems. A sound and concise summary of features characterizing the structure of an arbitrary problem instance is identified and related to the expected performance of global and local dominance-based multiobjective optimization algorithms. We provide a critical review of existing features tailored to multiobjective combinatorial optimization problems, and we propose additional ones that do not require any global knowledge from the landscape, making them suitable for large-size problem instances. Their intercorrelation and their association with algorithm performance are also analyzed. This allows us to assess the individual and the joint effect of problem features on algorithm performance, and to highlight the main difficulties encountered by such search heuristics. By providing effective tools for multiobjective landscape analysis, we highlight that multiple features are required to capture problem difficulty, and we provide further insights into the importance of ruggedness and multimodality to characterize multiobjective combinatorial landscapes.

This paper discusses the genetic diversity of real-coded populations processed by an evolutionary algorithm (EA). Diversity is expressed as a variance or a covariance matrix of individuals contained in the population, in one- or multi-dimensional cases, respectively. We focus on the exploration stage of the optimization, therefore, the fitness function is modeled as noise. We prove that the expected value of genetic diversity achieves a level proportional to the mutation covariance matrix. The proportionality coefficient depends solely on the EA parameters. Formulas are derived to predict the diversity for fitness proportionate, tournament, and truncation selection, with and without arithmetic crossover and with Gaussian mutation. Experimental validation of the multidimensional case shows that prediction accuracy is satisfactory in a broad spectrum of settings of EA parameters.