LCS & GBML Central

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This paper introduces sparse representation into spectral clustering and provides a sparse spectral clustering framework via a multiobjective evolutionary algorithm. In contrast to conventional spectral clustering, the main contribution of this paper is to construct the similarity matrix using a sparse representation approach by modeling spectral clustering as a constrained multiobjective optimization problem. Specific operators are designed to obtain a set of high quality solutions in the optimization process. Furthermore, we design a method to select a tradeoff solution from the Pareto front using a measurement called ratio cut based on an adjacency matrix constructed by all the nondominated solutions. We also extend the framework to the semi-supervised clustering field by using the semi-supervised information brought by the labeled samples to set some constraints or to guide the searching process. Experiments on commonly used datasets show that our approach outperforms four well-known similarity matrix construction methods in spectral clustering, and one multiobjective clustering algorithm. A practical application in image segmentation also demonstrates the efficiency of the proposed algorithm.

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Multiobjective evolutionary algorithms (MOEAs) are typically proposed, studied, and applied as monolithic blocks with a few numerical parameters that need to be set. Few works have studied how the algorithmic components of these evolutionary algorithms can be classified and combined to produce new algorithmic designs. The motivation for studies of this latter type stem from the development of flexible software frameworks and the usage of automatic algorithm configuration methods to find novel algorithm designs. In this paper, we propose anMOEA template and a new conceptual view of its components that surpasses existing frameworks in both number of algorithms that can be instantiated from the template and flexibility to produce novel algorithmic designs. We empirically demonstrate the flexibility of our proposed framework by automatically designing MOEAs for continuous and combinatorial optimization problems. The automatically designed algorithms are often able to outperform six traditional MOEAs from the literature, even after tuning their numerical parameters.

A decomposition approach decomposes a multiobjective optimization problem into a number of scalar objective optimization subproblems. It plays a key role in decomposition-based multiobjective evolutionary algorithms. However, many widely used decomposition approaches, originally proposed for mathematical programming algorithms, may not be very suitable for evolutionary algorithms. To help decomposition-based multiobjective evolutionary algorithms balance the population diversity and convergence in an appropriate manner, this letter proposes to impose some constraints on the subproblems. Experiments have been conducted to demonstrate that our proposed constrained decomposition approach works well on most test instances. We further propose a strategy for adaptively adjusting constraints by using information collected from the search. Experimental results show that it can significantly improve the algorithm performance.

Visualization of population in a high-dimensional objective space throughout the evolution process presents an attractive feature that could be well exploited in designing many-objective evolutionary algorithms (MaOEAs). In this paper, a new visualization method is proposed. It maps individuals from a high-dimensional objective space into a 2-D polar coordinate plot while preserving Pareto dominance relationship, retaining shape and location of the Pareto front, and maintaining distribution of individuals. From it, a decision-maker can observe the evolution process, estimate location, range, and distribution of Pareto front, assess quality of the approximated front and tradeoff between objectives, and easily select preferred solutions. Furthermore, its applications can be scalable to any dimensions, handle a large number of individuals on front, and simultaneously visualize multiple fronts for comparison. Based on this visualization tool, a performance metric, named polar-metric, is designed. The convergence of the approximate front is measured by radial values of all population members on that front. Meanwhile, the diversity performance is mainly determined by niche count of each subregion in a high-dimensional objective space. Experimental results show that it can provide a comprehensive and reliable comparison among MaOEAs.

Stability analysis is an important research direction in evolutionary game theory. Evolutionarily stable states have a close relationship with Nash equilibria of repeated games, which are characterized by the folk theorem. When applying the folk theorem, one needs to compute the minimax profile of the game in order to find Nash equilibria. Computing the minimax profile is an NP-hard problem. In this paper, we investigate a new methodology to compute evolutionary stable states based on the level- ${k}$ equilibrium, a new refinement of Nash equilibrium in repeated games. A level- ${k}$ equilibrium is implemented by a group of players who adopt reactive strategies and who have no incentive to deviate from their strategies simultaneously. Computing the level- ${k}$ equilibria is tractable because the minimax payoffs and strategies are not needed. As an application, this paper develops a tractable algorithm to compute the evolutionarily stable states and the Pareto front of ${n}$ -player symmetric games. Three games, including the iterated prisoner’s dilemma, are analyzed by means of the proposed methodology.

In this paper, we investigate three important properties (stability, local convergence, and transformation invariance) of a variant of particle swarm optimization (PSO) called standard PSO 2011 (SPSO2011). Through some experiments, we identify boundaries of coefficients for this algorithm that ensure particles converge to their equilibrium. Our experiments show that these convergence boundaries for this algorithm are: 1) dependent on the number of dimensions of the problem; 2) different from that of some other PSO variants; and 3) not affected by the stagnation assumption. We also determine boundaries for coefficients associated with different behaviors, e.g., nonoscillatory and zigzagging, of particles before convergence through analysis of particle positions in the frequency domain. In addition, we investigate the local convergence property of this algorithm and we prove that it is not locally convergent. We provide a sufficient condition and related proofs for local convergence for a formulation that represents updating rules of a large class of PSO variants. We modify the SPSO2011 in such a way that it satisfies that sufficient condition; hence, the modified algorithm is locally convergent. Also, we prove that the original standard PSO algorithm is not sensitive to rotation, scaling, and translation of the search space.

Presents the table of contents for this issue of the publication.

Traditionally, evolutionary algorithms (EAs) have been systematically developed to solve mono-, multi-, and many-objective optimization problems, in this order. Despite some efforts in unifying different types of mono-objective evolutionary and non-EAs, researchers are not interested enough in unifying all three types of optimization problems together. Such a unified algorithm will allow users to work with a single software enabling one-time implementation of solution representation, operators, objectives, and constraints formulations across several objective dimensions. For the first time, we propose a unified evolutionary optimization algorithm for solving all three classes of problems specified above, based on the recently proposed elitist, guided nondominated sorting procedure, developed for solving many-objectives problems. Using a new niching-based selection procedure, our proposed unified algorithm automatically degenerates to an efficient equivalent population-based algorithm for each class. No extra parameters are needed. Extensive simulations are performed on unconstrained and constrained test problems having single-, two-, multi-, and many-objectives and on two engineering optimization design problems. Performance of the unified approach is compared to suitable population-based counterparts at each dimensional level. Results amply demonstrate the merit of our proposed unified approach and motivate similar studies for a richer understanding of the development of optimization algorithms.