The quality of solution sets generated by decomposition-based evolutionary multiobjective optimisation (EMO) algorithms depends heavily on the consistency between a given problem's Pareto front shape and the specified weights' distribution. A set of weights distributed uniformly in a simplex often lead to a set of well-distributed solutions on a Pareto front with a simplex-like shape, but may fail on other Pareto front shapes. It is an open problem on how to specify a set of appropriate weights without the information of the problem's Pareto front beforehand. In this paper, we propose an approach to adapt the weights during the evolutionary process (called AdaW). AdaW progressively seeks a suitable distribution of weights for the given problem by elaborating five parts in the weight adaptation --- weight generation, weight addition, weight deletion, archive maintenance, and weight update frequency. Experimental results have shown the effectiveness of the proposed approach. AdaW works well for Pareto fronts with very different shapes: 1) the simplex-like, 2) the inverted simplex-like, 3) the highly nonlinear, 4) the disconnect, 5) the degenerated, 6) the badly-scaled, and 7) the high-dimensional.
An intuitive solution to this problem is to adaptively update the weight vectors during the optimisation process. Several interesting attempts have been made along this line. In early studies @cite_25 @cite_56 @cite_22 , researchers considered randomly-generated weight vectors (search directions) at each generation. This can make more computational cost allocated on the area around the nondominated solutions found so far @cite_54 . Recently, @cite_54 presented that this strategy could be helpful in dealing with MOPs having an irregular Pareto front. They introduced an external population, which is used to store promising solutions by the @math dominance relation @cite_21 , to help generating the random weights.
@cite_22 @cite_54 @cite_21 @cite_56 @cite_25
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Many-objective problems (MAPs) have put forward a number of challenges to classical Pareto-dominance based multi-objective evolutionary algorithms (MOEAs) for the past few years. Recently, researchers have suggested that MOEA D (multi-objective evolutionary algorithm based on decomposition) can work for MAPs. However, there exist two difficulties in applying MOEA D to solve MAPs directly. One is that the number of constructed weight vectors is not arbitrary and the weight vectors are mainly distributed on the boundary of weight space for MAPs. The other is that the relationship between the optimal solution of subproblem and its weight vector is nonlinear for the Tchebycheff decomposition approach used by MOEA D. To deal with these two difficulties, we propose an improved MOEA D with uniform decomposition measurement and the modified Tchebycheff decomposition approach (MOEA D-UDM) in this paper. Firstly, a novel weight vectors initialization method based on the uniform decomposition measurement is introduced to obtain uniform weight vectors in any amount, which is one of great merits to use our proposed algorithm. The modified Tchebycheff decomposition approach, instead of the Tchebycheff decomposition approach, is used in MOEA D-UDM to alleviate the inconsistency between the weight vector of subproblem and the direction of its optimal solution in the Tchebycheff decomposition approach. The proposed MOEA D-UDM is compared with two state-of-the-art MOEAs, namely MOEA D and UMOEA D on a number of MAPs. Experimental results suggest that the proposed MOEA D-UDM outperforms or performs similarly to the other compared algorithms in terms of hypervolume and inverted generational distance metrics on different types of problems. The effects of uniform weight vector initializing method and the modified Tchebycheff decomposition are also studied separately. In most kernel based online learning algorithms, when an incoming instance is misclassified, it will be added into the pool of support vectors and assigned with a weight, which often remains unchanged during the rest of the learning process. This is clearly insufficient since when a new support vector is added, we generally expect the weights of the other existing support vectors to be updated in order to reflect the influence of the added support vector. In this paper, we propose a new online learning method, termed Double Updating Online Learning, or DUOL for short, that explicitly addresses this problem. Instead of only assigning a fixed weight to the misclassified example received at the current trial, the proposed online learning algorithm also tries to update the weight for one of the existing support vectors. We show that the mistake bound can be improved by the proposed online learning method. We conduct an extensive set of empirical evaluations for both binary and multi-class online learning tasks. The experimental results show that the proposed technique is considerably more effective than the state-of-the-art online learning algorithms. The source code is available to public at http: www.cais.ntu.edu.sg chhoi DUOL . Given a large overcomplete dictionary of basis vectors, the goal is to simultaneously represent L>1 signal vectors using coefficient expansions marked by a common sparsity profile. This generalizes the standard sparse representation problem to the case where multiple responses exist that were putatively generated by the same small subset of features. Ideally, the associated sparse generating weights should be recovered, which can have physical significance in many applications (e.g., source localization). The generic solution to this problem is intractable and, therefore, approximate procedures are sought. Based on the concept of automatic relevance determination, this paper uses an empirical Bayesian prior to estimate a convenient posterior distribution over candidate basis vectors. This particular approximation enforces a common sparsity profile and consistently places its prominent posterior mass on the appropriate region of weight-space necessary for simultaneous sparse recovery. The resultant algorithm is then compared with multiple response extensions of matching pursuit, basis pursuit, FOCUSS, and Jeffreys prior-based Bayesian methods, finding that it often outperforms the others. Additional motivation for this particular choice of cost function is also provided, including the analysis of global and local minima and a variational derivation that highlights the similarities and differences between the proposed algorithm and previous approaches. The minimum weight-dominating set (MWDS) problem is NP-hard and has a lot of applications in the real world. Several metaheuristic methods have been developed for solving the problem effectively, but suffering from high CPU time on large-scale instances. In this paper, we design an effective hybrid memetic algorithm (HMA) for the MWDS problem. First, the MWDS problem is formulated as a constrained 0–1 programming problem and is converted to an equivalent unconstrained 0–1 problem using an adaptive penalty function. Then, we develop a memetic algorithm for the resulting problem, which contains a greedy randomized adaptive construction procedure, a tabu local search procedure, a crossover operator, a population-updating method, and a path-relinking procedure. These strategies make a good tradeoff between intensification and diversification. A number of experiments were carried out on three types of instances from the literature. Compared with existing algorithms, HMA is able to find high-quality solutions in much less CPU time. Specifically, HMA is at least six times faster than existing algorithms on the tested instances. With increasing instance size, the CPU time required by HMA increases much more slowly than required by existing algorithms.