## 4 Conclusions  
In the presented paper, we tried to perform an analysis of the properties of method (4) in theory and practice. Despite the results of the investigations presented in [28; 29], this method requires further analysis, so we tried to realize this in the presented paper.  
The following new results were obtained:  
1. It was demonstrated that, in the case of the quadratic function, method (4) can be easily investigated using the first Lyapunov method. As a result of its application, the convergence conditions presented in Theorem 1 were obtained. Such conditions led to the conditions for the HBM (3) in the case of \(\beta_{2}=0\) (see [7]). For functions from \(\mathcal{F}_{l,L}^{2,1}\), such conditions can be treated as the conditions of local convergence.
2. In comparison with the HBM, optimal parameters for method (4) can only be obtained numerically by the solution of the 3D constrained problems (19) and (20). As demonstrated, for the _quadratic_ case, the optimal value of \(\beta_{2}\) was equal to zero, so method (4) did not provide additional acceleration in comparison to the standard HBM.
3. The'mechanical' role of \(\beta_{2}\) was demonstrated by the consideration of the ODE (25), which is equivalent to (4) in the 1D case. This ODE describes the descent process in the neighborhood of \(x^{*}\). As can be seen from (25), the presence of \(\beta_{2}\) realized an additional damping of oscillations associated with non-monotone convergence of the HBM [13].
4. In numerical examples from different applications, it was demonstrated that, with the use of proper values of \(\beta_{2}\), a decrease in oscillation amplitudes typical of the HBM can be realized.  
The following remarks on future investigations can be made:  
1. In this paper, a local convergence analysis was presented. For \(f(x)\in\mathcal{F}_{l,L}^{1,1}\), global convergence for a specific choice of the parameters was demonstrated in [29]. It is imperative to obtain the general conditions for the parameters that guarantee global convergence.As is known for the HBM (e.g., see [28]), the convergence conditions obtained for strongly convex quadratic functions can lead to a lack of global convergence for \(f(x)\in\mathcal{F}_{l,L}^{1,1}\).
2. An analysis of method (4) was performed for the case of constant values of \(\beta_{1}\) and \(\beta_{2}\). But as known [18], it is effective to use methods with adaptive momentum, whose value is dependent on \(k\) in order to improve the convergence. Thus, the construction of extensions of method (4) to the case of adaptive parameters is a perspective for future research.
3. In this paper, all methods were considered in their deterministic formulations. However, in modern problems, especially those arising in machine learning, stochastic gradient methods are used according to the size of the datasets. Therefore, the extension of method (4) and its modifications for stochastic optimization has potential for future investigation, especially for applications in machine learning.  
Conceptualization, G.V.K.; methodology, G.V.K.; software, G.V.K. and V.Y.S.; validation, G.V.K. and V.Y.S.; formal analysis, G.V.K.; investigation, G.V.K. and V.Y.S.; writing--original draft preparation, G.V.K.; writing--review and editing, G.V.K. and V.Y.S.; visualization, G.V.K. and V.Y.S.; supervision, G.V.K. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Data Availability StatementData are included in the article.  
The authors wish to thank anonymous reviewers for their useful comments and discussions.  
The authors declare no conflicts of interest.