\[f\left( x_{1},x_{2} \right)=x_{1}^{2} + 2 x_{2}^{2} + e^{x_{1} + x_{2}}\]

Метод градиентнгого спуска

Градиент

\[\nabla f\left({X_0}\right)=\left( {\begin{array}{*{20}{c}} 2 x_{1} + e^{x_{1} + x_{2}}\\4 x_{2} + e^{x_{1} + x_{2}} \end{array}} \right)\]

Начнём с точки \(X_0=\left( 1;1 \right)^T\)

Итерация 0

\[f\left({X_0}\right)=1^{2} + 2 \cdot 1^{2} + e^{1 + 1}=10.389\]

\[\nabla f\left({X_0}\right)=\left( {\begin{array}{*{20}{c}} 2 \cdot 1 + e^{1 + 1}\\4 \cdot 1 + e^{1 + 1} \end{array}} \right)=\left( {\begin{array}{*{20}{c}} 9.3891\\11.3891 \end{array}} \right)\]

\[\left|\left|\nabla f\left({X_0}\right)\right|\right|=\sqrt {9.3891^{2}+11.3891^{2}}=14.7602\]

Итерация 1

\[X_{1}=X_{0}-t_{0} \nabla f\left({X_{0}}\right)=\left( {\begin{array}{*{20}{c}} 1\\1 \end{array}} \right)-0.1\left( {\begin{array}{*{20}{c}} 9.3891\\11.3891 \end{array}} \right)=\left( {\begin{array}{*{20}{c}} 0.0611\\-0.1389 \end{array}} \right)\]

Пусть \(t_0=0.1.\)

\[f\left({X_{1}}\right)=0.0611^{2} + 2 \cdot \left(-0.1389\right)^{2} + e^{0.0611 - 0.1389}=0.9674\]

\[f\left({X_{1}}\right) < f\left({X_{0}}\right)\]

\[\nabla f\left({X_{1}}\right)=\left( {\begin{array}{*{20}{c}} 2 \cdot \left(0.8 - 0.1 e^{2}\right) + e^{0.8 - 0.1 e^{2} - -0.6 + 0.1 e^{2}}\\4 \cdot \left(0.6 - 0.1 e^{2}\right) + e^{0.8 - 0.1 e^{2} - -0.6 + 0.1 e^{2}} \end{array}} \right)=\left( {\begin{array}{*{20}{c}} 1.0473\\0.3695 \end{array}} \right)\]

\[\left|\left|\nabla f\left({X_{1}}\right)\right|\right|=\sqrt {1.0473^{2}+0.3695^{2}}=1.1106\]

Итерация 2

\[X_{2}=X_{1}-t_{1} \nabla f\left({X_{1}}\right)=\left( {\begin{array}{*{20}{c}} 0.8 - 0.1 e^{2}\\0.6 - 0.1 e^{2} \end{array}} \right)-0.1\left( {\begin{array}{*{20}{c}} 1.0473\\0.3695 \end{array}} \right)=\left( {\begin{array}{*{20}{c}} -0.0436\\-0.1759 \end{array}} \right)\]

\[f\left({X_{2}}\right)=\left(-0.0436\right)^{2} + 2 \cdot \left(-0.1759\right)^{2} + e^{-0.0436 - 0.1759}=0.8666\]

\[f\left({X_{2}}\right) < f\left({X_{1}}\right)\]

\[\nabla f\left({X_{2}}\right)=\left( {\begin{array}{*{20}{c}} 2 \cdot \left(- 0.08 e^{2} - \frac{0.4055}{e^{0.2 e^{2}}} + 0.64\right) + e^{- 0.08 e^{2} - \frac{0.4055}{e^{0.2 e^{2}}} + 0.64 - -0.36 + \frac{0.4055}{e^{0.2 e^{2}}} + 0.06 e^{2}}\\4 \cdot \left(- 0.06 e^{2} - \frac{0.4055}{e^{0.2 e^{2}}} + 0.36\right) + e^{- 0.08 e^{2} - \frac{0.4055}{e^{0.2 e^{2}}} + 0.64 - -0.36 + \frac{0.4055}{e^{0.2 e^{2}}} + 0.06 e^{2}} \end{array}} \right)=\left( {\begin{array}{*{20}{c}} 0.7156\\0.0995 \end{array}} \right)\]

\[\left|\left|\nabla f\left({X_{2}}\right)\right|\right|=\sqrt {0.7156^{2}+0.0995^{2}}=0.7225\]

$f\left( x_{1},x_{2} \right)=x_{1}^{2} + 2 x_{2}^{2} + e^{x_{1} + x_{2}}$ \newline \textbf{Метод градиентнгого спуска} \newline Градиент \newline $\nabla f\left({X_0}\right)=\left( {\begin{array}{*{20}{c}} 2 x_{1} + e^{x_{1} + x_{2}}\\4 x_{2} + e^{x_{1} + x_{2}} \end{array}} \right)$ \newline Начнём с точки $X_0=\left( 1;1 \right)^T$ \newline \underline{Итерация 0} \newline $f\left({X_0}\right)=1^{2} + 2 \cdot 1^{2} + e^{1 + 1}=10.389$ \newline $\nabla f\left({X_0}\right)=\left( {\begin{array}{*{20}{c}} 2 \cdot 1 + e^{1 + 1}\\4 \cdot 1 + e^{1 + 1} \end{array}} \right)=\left( {\begin{array}{*{20}{c}} 9.3891\\11.3891 \end{array}} \right)$ \newline $\left|\left|\nabla f\left({X_0}\right)\right|\right|=\sqrt {9.3891^{2}+11.3891^{2}}=14.7602$ \newline \underline{Итерация 1} \newline $X_{1}=X_{0}-t_{0} \nabla f\left({X_{0}}\right)=\left( {\begin{array}{*{20}{c}} 1\\1 \end{array}} \right)-0.1\left( {\begin{array}{*{20}{c}} 9.3891\\11.3891 \end{array}} \right)=\left( {\begin{array}{*{20}{c}} 0.0611\\-0.1389 \end{array}} \right)$ \newline Пусть $t_0=0.1.$ \newline $f\left({X_{1}}\right)=0.0611^{2} + 2 \cdot \left(-0.1389\right)^{2} + e^{0.0611 - 0.1389}=0.9674$ \newline $f\left({X_{1}}\right) < f\left({X_{0}}\right)$ \newline $\nabla f\left({X_{1}}\right)=\left( {\begin{array}{*{20}{c}} 2 \cdot \left(0.8 - 0.1 e^{2}\right) + e^{0.8 - 0.1 e^{2} - -0.6 + 0.1 e^{2}}\\4 \cdot \left(0.6 - 0.1 e^{2}\right) + e^{0.8 - 0.1 e^{2} - -0.6 + 0.1 e^{2}} \end{array}} \right)=\left( {\begin{array}{*{20}{c}} 1.0473\\0.3695 \end{array}} \right)$ \newline $\left|\left|\nabla f\left({X_{1}}\right)\right|\right|=\sqrt {1.0473^{2}+0.3695^{2}}=1.1106$ \newline \underline{Итерация 2} \newline $X_{2}=X_{1}-t_{1} \nabla f\left({X_{1}}\right)=\left( {\begin{array}{*{20}{c}} 0.8 - 0.1 e^{2}\\0.6 - 0.1 e^{2} \end{array}} \right)-0.1\left( {\begin{array}{*{20}{c}} 1.0473\\0.3695 \end{array}} \right)=\left( {\begin{array}{*{20}{c}} -0.0436\\-0.1759 \end{array}} \right)$ \newline $f\left({X_{2}}\right)=\left(-0.0436\right)^{2} + 2 \cdot \left(-0.1759\right)^{2} + e^{-0.0436 - 0.1759}=0.8666$ \newline $f\left({X_{2}}\right) < f\left({X_{1}}\right)$ \newline $\nabla f\left({X_{2}}\right)=\left( {\begin{array}{*{20}{c}} 2 \cdot \left(- 0.08 e^{2} - \frac{0.4055}{e^{0.2 e^{2}}} + 0.64\right) + e^{- 0.08 e^{2} - \frac{0.4055}{e^{0.2 e^{2}}} + 0.64 - -0.36 + \frac{0.4055}{e^{0.2 e^{2}}} + 0.06 e^{2}}\\4 \cdot \left(- 0.06 e^{2} - \frac{0.4055}{e^{0.2 e^{2}}} + 0.36\right) + e^{- 0.08 e^{2} - \frac{0.4055}{e^{0.2 e^{2}}} + 0.64 - -0.36 + \frac{0.4055}{e^{0.2 e^{2}}} + 0.06 e^{2}} \end{array}} \right)=\left( {\begin{array}{*{20}{c}} 0.7156\\0.0995 \end{array}} \right)$ \newline $\left|\left|\nabla f\left({X_{2}}\right)\right|\right|=\sqrt {0.7156^{2}+0.0995^{2}}=0.7225$