## 2 Analysis of Two-Step Method
### Optimal Parameters  
In this subsection, we discuss the solution of minimization problems (19) and (20) and the following problem, which is stated in order to analyze the effect of parameter \(\beta_{2}\):  
\[F(\beta_{1},h)=\max(r(h,\beta_{1},\beta_{2},I),r(h,\beta_{1},\beta_{2},L)) \rightarrow\min_{\Delta}, \tag{21}\]  
where  
\[\Delta=\bigg{\{}(\beta_{1},h):\beta_{1}\in[0,1),h\in\bigg{(}0,\frac{2(1+\beta _{1})}{L(1+2\beta_{2})}\bigg{]}\bigg{\}}.\]  
So in (21) \(\beta_{2}\) is treated as an external parameter, which can be varied. In our computations, problem (21) is solved using the following approach: in the first stage, we obtain three 'good' initial points in \(\Delta\) by random search, and in the second stage, we apply the Nelder-Mead method in order to obtain the optimal point more precisely than in the first stage. For computations at any value of \(\beta_{2}\), we use \(10^{5}\) random points in \(\Delta\) and the accuracy \(10^{-5}\) for the Nelder-Mead method. The use of a large number of random points provides the possibility of obtaining the initial points in the small neighborhood of the optimal point, and the points obtained with the Nelder-Mead method do not leave \(\Delta\). This approach to solving the problem is very simple to realize and eliminates the need to use methods of unconstrained optimization. All computations were realized with the use of codes implemented in Matlab 2021a.  
In Figure 1, the plots of optimal values of \(F\) are presented for the cases of interval \(\beta_{2}\in[0,1]\) (Figure 1a) and \(\beta_{2}\in[0,100]\) (Figure 1b) for four values of \(\kappa\): \(10\), \(10^{2}\), \(10^{3}\), and \(10^{5}\). As can be seen, for both intervals and all considered values of \(\kappa\), the minimum values of \(F_{opt}\) takes place for \(\beta_{2}=0\). The value of \(F_{opt}\) becomes smaller at smaller values of \(\kappa\). The last feature is also mentioned for the multi-parametric method of [18].  
In addition, we try to compare the optimal convergence rate as a function of \(\kappa\) for method (4) with the optimal rates for the GD method (2), the HBM (3), and the following Nesterov methods:  
(1) [PERSON]'s accelerated gradient method for \(f\in\mathcal{F}_{l,L}^{1,1}\) (Nesterov1) [6; 28]:  
\[x^{k+1}=y^{k}-h\
abla f(y^{k}),\quad y^{k}=x^{k}+\beta(x^{k}-x^{k-1}),\]  
Figure 1: Plots of the dependence of optimal values of \(F\) on the value of \(\beta_{2}\): (**a**) \(\beta_{2}\in[0,1]\); (**b**) \(\beta_{2}\in[0,100]\).  
\[h_{opt}=\frac{1}{L},\quad\beta_{opt}=\frac{\sqrt{\kappa}-1}{\sqrt{\kappa}+1}, \quad\rho_{opt}=1-\frac{1}{\sqrt{\kappa}}.\]
2. [PERSON]'s accelerated gradient method for a strongly convex quadratic function (Nesterov2) [28]: \[x^{k+1}=y^{k}-h\
abla f(y^{k}),\quad y^{k}=x^{k}+\beta(x^{k}-x^{k-1}),\] \[h_{opt}=\frac{4}{3L+1},\quad\beta_{opt}=\frac{\sqrt{3\kappa+1}-2}{\sqrt{3 \kappa+1}+2},\quad\rho_{opt}=1-\frac{2}{\sqrt{3\kappa+1}}.\]  
The numerical solution to problem (19) is realized using the same method as for problem (21), but for the Nelder-Mead method, four 'good' points are obtained with a random search. The interval on \(\beta_{2}\geq 0\) is bounded by 0.5, according to the behavior, illustrated in Figure 1.  
Plots of \(\rho_{opt}\) are presented in Figure 2. As can be seen, the minimum values of \(\rho_{opt}\) took place for methods (3) and (4), and they were very close. So, from the results of the computations, the following conclusion can be drawn: for the _quadratic_ function \(f(x)\), parameter \(\beta_{2}\) does not provide an additional acceleration effect in comparison with the standard HBM (3).