## 3 Determination of \(a_{i}\), \(b_{i}\), and \(\mu_{i}\)
\begin{array}{ll}S_{2}=\frac{\eta_{11}\eta_{22}V_{11}V_{22}}{41398560}&(378075 0\varphi_{5}^{2}\eta_{11}^{6}+3677800\varphi_{5}\varphi_{6}\eta_{11}^{5}\eta_{ 22}+1226624\varphi_{6}^{2}\eta_{11}^{4}\eta_{22}^{2}\\ &+2206680\varphi_{5}\varphi_{7}\eta_{11}^{4}\eta_{22}^{2}+1724940\varphi_{6} \varphi_{7}\eta_{11}^{3}\eta_{22}^{3}+689976\varphi_{7}^{2}\eta_{11}^{2}\eta_{ 22}^{4}\\ &+4957575\varphi_{5}\
u_{2}\eta_{11}^{3}V_{11}+2755760\varphi_{6}\
u_{2}\eta_{ 11}^{2}\eta_{22}V_{11}+2066820\varphi_{7}\
u_{2}\eta_{11}\eta_{22}^{2}V_{11}\\ &+4952920\
u_{2}^{2}V_{11}^{2}),\end{array}  \tag{18}\]  
Notice that the resulting polynomial expressions, (17) and (18), provide convex, continuous functions in the relative interior of their domains which depend on the values of \(\eta_{i}\), \(\eta_{ii}\), \(V_{i}\), and \(V_{ii}\)[43, 44]. These functions, \(S_{i}\), have a unique minimum value within their bounded domain intervals, as discussed in [45, 46] and references cited therein. Thus, the values of \(\eta_{i}\), \(\eta_{ii}\), \(V_{i}\), and \(V_{ii}\) that provide the minimum values of \(U_{i}\) could be computed by following a classical minimization algorithm in which the partial derivatives of Equations (17) and (18) must be determined, respectively, with respect to the unknowns: \(\eta_{i}\), \(\eta_{ii}\), \(V_{i}\), and \(V_{ii}\). This step provides the following equations:  
\[\frac{\partial S_{1}}{\partial\eta_{1}}=\frac{121\eta_{2}V_{1}V_{2}}{258741} \left(12\varphi_{4}^{2}\eta_{2}^{6}+21\varphi_{4}\eta_{2}^{3}v_{2}V_{2}+28 \imath_{2}^{2}V_{2}^{2}\right), \tag{19}\]  
\[\frac{\partial S_{1}}{\partial\eta_{2}}=\frac{484\eta_{1}V_{1}V_{2}}{369663} \Big{(}3\varphi_{4}^{2}\eta_{2}^{6}+3\varphi_{4}\eta_{2}^{3}v_{2}V_{2}+v_{2} ^{2}V_{2}^{2}\Big{)}, \tag{20}\]  
\[ \begin{array}{ll}\frac{\partial S_{1}}{\partial V_{1}}=\frac{121\eta_{1}\eta _{2}V_{2}}{258741}\left(12\varphi_{4}^{2}\eta_{2}^{6}+21\varphi_{4}\eta_{2}^ {3}v_{2}V_{2}+28\imath_{2}^{2}V_{2}^{2}\right),\end{array}  \tag{21}\]  
\[ \begin{array}{ll}\frac{\partial S_{1}}{\partial V_{2}}=\frac{242\eta_{1}\eta _{2}V_{1}}{86247}\left(2\varphi_{4}^{2}\eta_{2}^{6}+7\varphi_{4}\eta_{2}^{3} v_{2}V_{2}+14\imath_{2}^{2}V_{2}^{2}\right),\end{array}  \tag{22}\]  
\[ \begin{array}{ll}\frac{\partial S_{2}}{\partial\eta_{11}}=\frac{\eta_{22}V_{ 11}V_{22}}{2957040}&(1890375\varphi_{5}^{2}\eta_{11}^{6}+1576200\varphi_{5} \varphi_{6}\eta_{11}^{5}\eta_{22}+438080\varphi_{6}^{2}\eta_{11}^{4}\eta_{22} ^{2}\\ &+788100\varphi_{5}\varphi_{7}\eta_{11}^{4}\eta_{22}^{2}+492840\varphi_{6} \varphi_{7}\eta_{11}^{3}\eta_{22}^{2}+147852\varphi_{7}^{2}\eta_{11}^{2}\eta_{ 22}^{4}\\ &+1416450\varphi_{5}\varphi_{5}\eta_{5}^{2}\eta_{11}^{3}V_{11}+590520\varphi_{6 }\
u_{2}\eta_{11}^{2}\eta_{22}V_{11}+295260\varphi_{7}\imath_{2}\eta_{11}\eta_ {22}^{2}V_{11}\\ &+353780\
u_{2}^{2}V_{11}^{2}),\end{array}  \tag{23}\]  
\[ \begin{array}{ll}\frac{\partial S_{2}}{\partial\eta_{22}}=\frac{\eta_{11}V_{ 11}V_{22}}{41398560}&(3780750\varphi_{5}^{2}\eta_{11}^{6}+7355600\varphi_{5} \varphi_{6}\eta_{11}^{5}\eta_{22}+3679872\varrho_{6}^{2}\eta_{11}^{4}\eta_{22} ^{2}\\ &+662004\varphi_{5}\varphi_{7}\eta_{11}^{4}\eta_{22}^{2}+6899760\varphi_{6} \varphi_{7}\eta_{11}^{3}\eta_{22}^{3}+3449880\varphi_{7}^{2}\eta_{11}^{4}\eta_ {22}^{2}\\ &+4957575\varphi_{5}\
u_{2}\eta_{11}^{3}V_{11}+5511520\varphi_{6}\
u_{2}\eta_{ 11}^{2}\eta_{22}V_{11}+6200460\varphi_{7}\
u_{2}\eta_{11}\eta_{22}^{2}V_{11} \\ &+4952920\
u_{2}^{2}V_{11}^{2}),\end{array}  \tag{24}\]  
\[