# Application of the Parabola Method in Nonconvex Optimization
## 3 Accelerated Two-Stage Approach  
In this section, we propose a modification of the two-stage approach described in the previous section. We set the integer parameter \(N\) in the TPP procedure in advance and do not change it. When the TPP procedure is finished, \(N+1\) values of the objective function at points \(x_{k}=\alpha+k\Delta,\ k=0,\ldots,N\) with \(\Delta=\frac{\beta-\alpha}{N}\) are available. We determine the record value \(f^{Rec}=\min\{f(x_{i}):i=0,\ldots,N\}\) and a corresponding record point \(x^{Rec}\in\operatorname{Arg\,min}\{f(x_{i}):i=0,\ldots,N\}\). If the number \(k_{p}\) of the TPP subintervals is positive, \(k_{p}>0\), then we choose the TPP subinterval, which contains the record point \(x^{Rec}\), and run the Parabola Method over this TPP subinterval. Let \(x^{PM}\) be a point found by the Parabola Method. We define the point \(x^{*}=\operatorname{arg\,min}\{f(x^{Rec}),f(x^{PM})\) and the corresponding objective value \(f^{*}=f(x^{*})\). We deliver the pair \((x^{*},f^{*})\) as the result of the Accelerated Two-Stage Approach. If the number \(k_{p}\) is equal to zero, i.e., no TPP subintervals were detected, then we determine  
\[m\in\{0,\ldots,N-1\}:\ x_{m}\leq x^{Rec}\leq x_{m+1}, \tag{4}\]  
and run the Golden Section Method over the interval \([x_{m},x_{m+1}]\). Let \(x^{GSM}\) be the corresponding point determined by the Golden Section Method. As in the previous case, we define the point \(x^{*}=\operatorname{arg\,min}\{f(x^{GSM}),f(xRec)\) and the value \(f^{*}=f(x^{*})\), and we deliver the pair \(x^{*},f^{*}\), as a result of the Accelerated Two-Stage Approach (ATSA). Let us now give the description of the ATSA procedure. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \begin{tabular}{c} **Problem** \\ **No.** \\ \end{tabular} & \begin{tabular}{c} **Minima** \\ **Structure** \\ \end{tabular} & \begin{tabular}{c} **TPP** \\ **\(k_{f}^{TPP}\)** \\ \end{tabular} & \begin{tabular}{c} **Subintervals** \\ \end{tabular} & \begin{tabular}{c} **\(k_{f}^{GSM}\)** \\ \end{tabular} & \
\begin{tabular}{c} **\(k_{f}^{PM}\)** \\ \end{tabular} &
\begin{tabular}{c} g/l \\ \end{tabular} \\ \hline \multirow{4}{*}{6.} & 17 local & 96 & \([-9.583,-9.167]\) & 13 & 3 & 1 \\ \cline{3-6}  & 3 global & & \([-8.541,-8.125]\) & 13 & 3 & 1 \\ \cline{3-6}  & & \([-7.291,-6.875]\) & 13 & 3 & g \\ \cline{3-6}  & & \([-6.250,-5.833]\) & 13 & 4 & 1 \\ \cline{3-6}  & & \([-5.208,-4.791]\) & 13 & 4 & 1 \\ \cline{3-6}  & & \([-4.166,-3.750]\) & 13 & 4 & 1 \\ \cline{3-6}  & & \([-3.125,-2.708]\) & 13 & 4 & 1 \\ \cline{3-6}  & & \([-2.291,-1.875]\) & 13 & 4 & 1 \\ \cline{3-6}  & & \([-1.042,-0.625]\) & 13 & 4 & g \\ \hline \([0.208,0.625]\) & 13 & 4 & 1 \\ \cline{3-6}  & & \([1.042,1.458]\) & 13 & 4 & 1 \\ \cline{3-6}  & & \([2.083,2.500]\) & 13 & 3 & 1 \\ \cline{3-6}  & & \([3.125,3.542]\) & 13 & 4 & 1 \\ \cline{3-6}  & & \([4.166,4.583]\) & 13 & 2 & 1 \\ \cline{3-6}  & & \([5.208,5.625]\) & 13 & 4 & g \\ \cline{3-6}  & & \([6.458,6.875]\) & 13 & 3 & 1 \\ \cline{3-6}  & & \([7.500,7.917]\) & 13 & 2 & 1 \\ \cline{3-6}  & & \([8.333,8.750]\) & 13 & 4 & 1 \\ \cline{3-6}  & & \([9.375,9.792]\) & 13 & 2 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Application of the two-stage approach to problem 6. The ATSA Procedure  
1. Apply the TPP procedure. Let \(k_{p}\) and \(\Delta\) be parameters calculated by the TPP procedure. Let \(f^{Rcc}\) be the record value over all calculated points and \(x^{Rcc}:f(x^{Rcc})=f^{Rcc}\). 2. If \(k_{p}>0\), then select the subinterval containing \(x^{Rcc}\) and run the Parabola Method over the selected subinterval, obtaining the point \(x^{PM}\). Define the point \(x^{*}=\arg\min\{f(x^{Rcc}),f(x^{PM})\}\) and value \(f^{*}=f(x^{*})\). Stop. 3. If \(k_{p}=0\), then determine the subinterval \([x_{m},x_{m+1}]\) according to (4) and run the Golden Section Method, obtaining the point \(x^{CSM}\). Define the point \(x^{*}=\arg\min\{f(x^{Rcc}),f(x^{CSM})\}\) and value \(f^{*}=f(x^{*})\). Stop.