Optimal Transport (OT) has become a powerful tool to compare probability distributions. However, it suffers from a severe computational burden for high dimensional and continuous distributions. To this end, we develop two novel methods for the Kantorovich and Monge formulations, which are the fundamental problems in OT. First, we learn the optimal joint distribution in the Kantorovich formulation and propose an algorithm, namely Cop-OT, which transforms the primal objective of the Kantorovich problem into a tractable objective with respect to the copula parameter. Second, based on the copula formulation of the joint distribution, we learn the optimal map in the Monge problem and propose an algorithm, namely Map-OT, which describes the optimal map using a parameterized function estimated by approximating the barycentric projection of the optimal joint distribution and then obtains a tractable objective with respect to parameters of interest. Both of them can be solved by stochastic optimization with a stable optimizing process. Empirical results demonstrate that Cop-OT and Map-OT can gain more accurate approximations of the Kantorovich and Monge problems compared with the baseline methods.

Grey wolf optimization (GWO) is a widely used meta-heuristic method. It has limited searching potential when solving the majority of function optimization problems. This paper proposes a new variant of GWO, named SCGWO, which combines GWO with an improved spread strategy and a chaotic local search (CLS) mechanism to overcome these performance limitations. In detail, a spread strategy is introduced into the basic GWO to change the search agent's ability to avoid the local optima, the global exploration capability, and the individual movement's randomness. Then, a CLS mechanism is adopted to accelerate the convergence rate of the evolving agents. This method's effectiveness is illustrated by comparing the proposed SCGWO method with various algorithms, including seven GWO variants and eight well-known state-of-the-art algorithms on a comprehensive set of benchmark functions with the type of unimodal, multimodal, and composition functions. The experimental results confirmed that the established SCGWO algorithm has apparent advantages in processing unimodal, multimodal, and composition functions. Additionally, the proposed algorithm was utilized for finding the approximate optimal feature subset when applied to the feature selection problems on a set of 32 real-world data sets from the UCI machine learning repository. The results show that the binary variant also reveals a very competitive performance in dealing with feature selection. Our findings and analysis suggest that the proposed method can be a very suitable tool for realizing the optimal solutions to global optimization and wrapper-based feature selection tasks.

Gradient-based optimizer (GBO) is a metaphor-free mathematic-based algorithm proposed in recent years. Encouraged by the gradient-based Newton's method, this algorithm combines with population-based evolutionary methods. The disadvantage of the traditional GBO algorithm is that the global search ability of the algorithm is too strong, and the local search ability is too weak; accordingly, it is difficult to obtain the global optimal solution efficiently. Therefore, a new improved GBO algorithm (GOMGBO) is developed to mitigate such performance concerns by introducing a Gaussian bare-bones mechanism, an opposition-based learning mechanism, and a moth spiral mechanism enhanced GBO algorithm. The proposed GOMGBO has been compared against many famous methods and improved variants on 30 benchmark functions. The experimental results show that GOMGBO has apparent advantages in convergence speed and precision. In addition, this paper analyzes the balance and diversity of the GOMGBO algorithm and compares GOMGBO with other algorithms on several engineering problems. The experimental results show that the GOMGBO algorithm is also better than the competitive algorithm in engineering problems. This study uses the GOMGBO algorithm to optimize kernel extreme learning machine (KELM), and a new GOMGBO-KELM model is proposed. The model is used to deal with four clinical disease diagnosis problems. Compared with GBO-KELM, back propagation neural network algorithm, and other models, comparative experiments show that GOMGBO-KELM has high performance in dealing with practical cases. We invite the community to investigate further our method for solving problems more efficiently with reasonable speed and efficiency. Readers of this study can refer to https://aliasgharheidari.com for any guidance about the proposed GOMGBO method.