In most multi-objective test problems, the dimensionality of the Pareto set is the same as the dimensionality of the Pareto front. That is, an m-objective test problem with n decision variables usually has an (m-1)-dimensional Pareto front and an (m-1)-dimensional Pareto set. Thanks to this property, the mapping between the Pareto front and the Pareto set is usually a one-to-one mapping. In this paper, we formulate two-objective distance minimization problems in an n-dimensional decision space using Manhattan distance. The two objectives are defined by two target points in the decision space. That is, each objective is to minimize the Manhattan distance from the solution to each target point. The main characteristic feature of our test problems is that the dimensionality of the Pareto set can be arbitrarily specified between 1 and n by the locations of the two target points. For example, the Pareto set dimensionality in a 20-dimensional decision space is an arbitrarily specified integer between 1 and 20. These test problems have multimodality since many points in the Pareto set are mapped to the same single point on the Pareto front. In this paper, we first explain the relation between the locations of the two target points and the Pareto set dimensionality. Then, through computational experiments, we show that our simple test problems pose difficult challenges for both multi-objective algorithms and multi-modal multi-objective algorithms. We also discuss the scalability of the proposed test problems with respect to the number of equivalent Pareto sets and the number of objectives.
In this paper, we discuss the optimal distributions of solutions for maximizing the minimum crowding distance on linear Pareto fronts of two- and three-objective problems. Our discussions are twofold. One is theoretical discussions to analytically derive the optimal distributions. The other is empirical discussions, which are based on a newly implemented indicator-based algorithm to maximize the minimum crowding distance. First, we show the upper bound on the minimum crowding distance, which is derived from the upper bound on the total crowding distance. Next, we theoretically derive the optimal distributions for maximizing the minimum crowding distance on two-objective linear Pareto fronts. Our analysis shows that two solutions always overlap in the optimal distributions when the population size is an even number. For an odd number population size, the optimal distribution cannot be uniquely specified except for some anchor solutions (i.e., many different distributions are optimal). The theoretically obtained optimal distributions are compared with experimental results by our indicator-based algorithm. Then, we show an example of optimal distributions for a three-objective linear Pareto front where the population size is a multiple of three. Our indicator-based algorithm is also compared with the standard NSGA-II algorithm and its steady-state variant.
The performance of surrogate model assisted algorithms for black-ox optimization is impacted by two factors: algorithmic design choices on how to use the surrogate model, and the ability of the model to accurately represent the true objective function. In an effort to better understand the potential of surrogate model assisted algorithms, we propose to decouple those factors by studying the performance of the algorithms assuming perfect models. As a result, we obtain a natural performance bound that algorithms that use real models can be compared against, and that also provides an indication of the goodness of those models. We employ that performance bound in order to analytically evaluate a surrogate model assisted (1 + 1)-ES. Using the same bound, we also investigate the potential of performing incremental updates of Gaussian process surrogate models in an attempt to reduce algorithm internal computational costs and find that significant savings can be achieved at the cost of a small deterioration of model accuracy.
Clustering is the process of grouping similar data points into distinct clusters. Graph theoretic techniques for data clustering attempt to find the most efficient way to convert a precomputed similarity graph into a cluster graph, which is a collection of disjoint cliques. In contrast, most existing evolutionary approaches to clustering are based on partitioning data points according to feature vectors. In this paper we present an evolutionary algorithm for tackling data clustering from the graph theoretic perspective. In particular, we adapt a genetic algorithm (the SubPopGA) that maintains a population of solved subgraphs to solve the NP-hard k-CLUSTER DELETION and k-CLUSTER VERTEX DELETION problems. This adaptation is nontrivial, as it requires modifying the technique to work on induced subgraphs and designing a novel template parent mechanism for uniform crossover. We prove that the resulting SubPopGA has a fixed-parameter tractable running time on both problems. Given an arbitrary graph on n vertices and m edges, we prove that it solves k-CLUSTER DELETION in 0(n3(4k) + mn(2) log n) generations in expectation, and the k-CLUSTER VERTEX DELETION problem in 0(n3(5k) + n(3) log n) in expectation. We also present results from a number of computational experiments that measure the running time of the SubPopGA on real-world biological data sets coming from protein-protein interaction networks.
Drift analysis is a great tool for proving that optimization algorithms work the way we think they do, and for analyzing them, potentially in great detail. In this talk, I will discuss drift analysis for evolution strategies. These algorithms exhibit linear convergence on a wide range of problems, which corresponds to a linear decrease of the logarithmic distance of the best-so-far sample from the optimum, giving rise to simple additive drift. That behavior is enabled by online adaptation of the step size, which decays at the same rate as the distance to the optimum. Moreover, modern evolution strategies like CMA-ES adapt not only the step size, but rather the full covariance matrix of their sampling distribution. The mechanism enables convergence at a problem-independent rate that depends only on the dimension of the search space. The primary challenge of proving the convergence of CMA-ES lies in establishing the stability of the adaptation process, which was recently achieved by analyzing the invariant Markov chain that describes the parameter adaptation process. Yet, a drift-based analysis is still desirable because it can yield much more fine-grained results. For instance, it can provide details about the transient adaptation phase, which often takes up the lion's share of the time for solving the problem. To achieve this, we need a potential function that appropriately penalizes unsuitable parameter configurations, or more precisely, configurations the algorithm tends to move away from. Designing a potential function that captures the dynamics of covariance matrix adaptation is an ongoing challenge. I will present our recent research efforts towards this goal and emphasize why relatively simple additive drift offers a powerful framework for achieving it.
