Accurate and stable natural gas price forecasts are essential for effective management of energy systems. However, due to the mixed frequency of data and the inherent nonlinear fluctuation characteristics of natural gas price changes, it is difficult to achieve satisfactory forecasting performance. In order to effectively improve the forecasting results of mixed frequency data. In this study, the MIDAS regression model and the machine learning MIDAS models are successfully combined to form a novel combination forecasting model. Moreover, the extreme learning machine with the multi-objective grey-wolf algorithm is used to combine univariate MIDAS results and further improve the forecasting accuracy. In the empirical analysis, the weekly natural gas futures prices of the Intercontinental Exchange UK NBP are used to generate real-time forecasts to evaluate the forecasting performance of the proposed combination model. The experimental results show that the comprehensive forecasting accuracy of the novel combination MIDAS model is 26.35%, 8.82% and 12.91% higher than that of the benchmark MIDAS regression models, the combination MIDAS models and the multivariate MIDAS models, respectively. Based on the forecasting results, the non-linear, non-stationary and irregular natural gas futures prices can be effectively managed, which provides better investment and management tools.

We proposed a three-term gradient descent method that can be well applied to address the optimization problems in this article. The search direction of the obtained method is generated in a specific subspace. Specifically, a quadratic approximation model is applied in the process of generating the search direction. In order to reduce the amount of calculation and make the best use of the existing information, the subspace was made up of the gradient of the current and prior iteration point and the previous search direction. By using the subspace-based optimization technology, the global convergence result is established under Wolfe line search. The results of numerical experiments show that the new method is effective and robust.

In this paper, a three-parameter subspace conjugate gradient method is proposed for solving large-scale unconstrained optimization problems. By minimizing the quadratic approximate model of the objective function on a new special three-dimensional subspace, the embedded parameters are determined and the corresponding algorithm is obtained. The global convergence result of a given method for general nonlinear functions is established under mild assumptions. In numerical experiments, the proposed algorithm is compared with SMCG_NLS and SMCG_Conic, which shows that the given algorithm is robust and efficient.