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Decomposition plays a significant role in cooperative co-evolution which shows great potential in large scale black-box optimization. However, current popular decomposition algorithms generally require to sample and evaluate a large number of solutions for interdependency detection, which is very time-consuming. To address this issue, this study proposes a new decomposition algorithm named fast differential grouping (FDG). FDG first identifies the type of an instance by detecting the interdependencies of a few pairs of variable subsets selected according to certain rules, and thus can rapidly complete the decomposition of a fully separable or nonseparable instance. For an identified partially separable instance, FDG converts the key decomposition process into a search process in a binary tree by taking corresponding variable subsets as tree nodes. This enables it to directly deduce the interdependency related to a child node by reutilizing the solutions sampled for corresponding parent and brother nodes. To support the above operations, this study designs a normalized variable-subset-oriented interdependency indicator, which can adaptively generate decomposition thresholds according to its distribution and thus enhances decomposition accuracy. Computational complexity analysis and experimental results verify that FDG outperforms popular decomposition algorithms. Further tests indicate that FDG embedded in a cooperative co-evolution framework can achieve highly competitive optimization results as compared with some state-of-the-art algorithms for large scale black-box optimization.

This paper proposes a novel sliding mode control (SMC) method for a class of affine dynamic systems. In this type of systems, the high-frequency gain matrix (HFGM), which is the matrix multiplying the control vector in the dynamic equation of the sliding variables vector, is neither deterministic nor positive definite. This case has rarely been covered by general SMC methods, which perform well under the condition that the HFGM is certain or uncertain but positive definite. In this study, the control law is determined by solving a nonlinear vector equation instead of the conventional algebraic expression, which is not applicable when the HFGM is uncertain and non-positive definite. Theorems with some relaxed system parametric uncertainty assumptions are proposed to guarantee the existence and uniqueness of the solution, and proofs of them, based on the principle of the convex cone set, are given in the text. The proposed control strategy can be easily applied in practice, and the chattering caused by the discontinuous control can be suppressed, as it can in general SMCs. The proposed controller was used in two affine dynamic systems, and the simulation results demonstrate its effectiveness.

Cooperative coevolution (CC) has shown great potential in solving large scale optimization problems (LSOPs). However, traditional CC algorithms often waste part of computation resource (CR) as they equally allocate CR among all the subproblems. The recently developed contribution-based CC (CBCC) algorithms improve the traditional ones to a certain extent by adaptively allocating CR according to some heuristic rules. Different from existing works, this study explicitly constructs a mathematical model for the CR allocation (CRA) problem in CC and proposes a novel fine-grained CRA (FCRA) strategy by fully considering both the theoretically optimal solution of the CRA model and the evolution characteristics of CC. FCRA takes a single iteration as a basic CRA unit and always selects the subproblem which is most likely to make the largest contribution to the total fitness improvement to undergo a new iteration, where the contribution of a subproblem at a new iteration is estimated according to its current contribution, current evolution status as well as the estimation for its current contribution. We verified the efficiency of FCRA by combining it with SHADE which is an excellent differential evolution variant but has never been employed in the CC framework. Experimental results on two benchmark suites for LSOPs demonstrate that FCRA significantly outperforms existing CRA strategies and the resultant CC algorithm is highly competitive in solving LSOPs., Comment: We changed the experiments in this manuscript that it has to be rewritten. With the agreement of my co-authors, we would like to withdraw this manuscript

As a typical model-based evolutionary algorithm (EA), estimation of distribution algorithm (EDA) possesses unique characteristics and has been widely applied to global optimization. However, the common-used Gaussian EDA (GEDA) usually suffers from premature convergence which severely limits its search efficiency. This study first systematically analyses the reasons for the deficiency of the traditional GEDA, then tries to enhance its performance by exploiting its evolution direction, and finally develops a new GEDA variant named EDA2. Instead of only utilizing some good solutions produced in the current generation when estimating the Gaussian model, EDA2 preserves a certain number of high-quality solutions generated in previous generations into an archive and takes advantage of these historical solutions to assist estimating the covariance matrix of Gaussian model. By this means, the evolution direction information hidden in the archive is naturally integrated into the estimated model which in turn can guide EDA2 towards more promising solution regions. Moreover, the new estimation method significantly reduces the population size of EDA2 since it needs fewer individuals in the current population for model estimation. As a result, a fast convergence can be achieved. To verify the efficiency of EDA2, we tested it on a variety of benchmark functions and compared it with several state-of-the-art EAs, including IPOP-CMAES, AMaLGaM, three high-powered DE algorithms, and a new PSO algorithm. The experimental results demonstrate that EDA2 is efficient and competitive., Comment: We modified some experiments in this manuscript that it has to be rewritten. With the agreement of my co-authors, we would like to withdraw this manuscript

In this paper, the two-stage orienteering problem with stochastic weights (OPSW) is considered, where the first-stage problem is to plan a path under the uncertain environment and the second-stage problem is recourse action to make sure that the length constraint is satisfied after the uncertainty is realized. Two recourse models are introduced based on two different uncertainty realization ways, one is based on sequentially realized weights which leads to the recourse model proposed by Evers et al. (2014) and the other is based on concurrently realized weights which leads to a new recourse model with less variables and less constraints and is computationally more efficient. Subsequently two two-stage robust models are introduced for OPSW based on the two different recourse models, and the relationships between the two-stage robust models and their corresponding static robsut models are investigated. Theoretical conclusions are drawn which show that the two-stage robust models are equivalent to their corresponding static robust models with the box uncertainty set defined, and the two two-stage robust models are also equivalent to each other even though they are based on different recourse models. A case study is presented by comparing the two-stage robust models with an one-stage robust model for OPSW. The numerical results of the comparative studies show the effectiveness and superiority of the proposed two-stage robust models for dealing with the two-stage OPSW., Comment: 24 pages, 3 figures