\[f\left( x,y \right)=2 x^{2} + x y + 51 x + 3 y^{2} - 17 y\]

Метод градиентнгого спуска

Градиент

\[\nabla f\left({X_0}\right)=\left( {\begin{array}{*{20}{c}} 4 x + y + 51\\x + 6 y - 17 \end{array}} \right)\]

Начнём с точки \(X_0=\left( 0;0 \right)^T\)

Итерация 0

\[f\left({X_0}\right)=2 \cdot 0^{2} + 0 \cdot 0 + 51 \cdot 0 + 3 \cdot 0^{2} - 17 \cdot 0=0\]

\[\nabla f\left({X_0}\right)=\left( {\begin{array}{*{20}{c}} 4 \cdot 0 + 0 + 51\\0 + 6 \cdot 0 - 17 \end{array}} \right)=\left( {\begin{array}{*{20}{c}} 51\\-17 \end{array}} \right)\]

\[\left|\left|\nabla f\left({X_0}\right)\right|\right|=\sqrt {51^{2}+\left( -17 \right)^{2}}=53.7587\]

Итерация 1

\[X_{1}=X_{0}-t_{0} \nabla f\left({X_{0}}\right)=\left( {\begin{array}{*{20}{c}} 0\\0 \end{array}} \right)-0.1\left( {\begin{array}{*{20}{c}} 51\\-17 \end{array}} \right)=\left( {\begin{array}{*{20}{c}} -5.1\\1.7 \end{array}} \right)\]

Пусть \(t_0=0.1.\)

\[f\left({X_{1}}\right)=2 \cdot \left(-5.1\right)^{2} - 5.1 \cdot 1.7 + 51 \cdot \left(-5.1\right) + 3 \cdot 1.7^{2} - 17 \cdot 1.7=-236.98\]

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

\[\nabla f\left({X_{1}}\right)=\left( {\begin{array}{*{20}{c}} 4 \cdot \left(-5.1\right) + 1.7 + 51\\-5.1 + 6 \cdot 1.7 - 17 \end{array}} \right)=\left( {\begin{array}{*{20}{c}} 32.3\\-11.9 \end{array}} \right)\]

\[\left|\left|\nabla f\left({X_{1}}\right)\right|\right|=\sqrt {32.3^{2}+\left( -11.9 \right)^{2}}=34.4224\]

Итерация 2

\[X_{2}=X_{1}-t_{1} \nabla f\left({X_{1}}\right)=\left( {\begin{array}{*{20}{c}} -5.1\\1.7 \end{array}} \right)-0.1\left( {\begin{array}{*{20}{c}} 32.3\\-11.9 \end{array}} \right)=\left( {\begin{array}{*{20}{c}} -8.33\\2.89 \end{array}} \right)\]

\[f\left({X_{2}}\right)=2 \cdot \left(-8.33\right)^{2} - 8.33 \cdot 2.89 + 51 \cdot \left(-8.33\right) + 3 \cdot 2.89^{2} - 17 \cdot 2.89=-334.1996\]

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

\[\nabla f\left({X_{2}}\right)=\left( {\begin{array}{*{20}{c}} 4 \cdot \left(-8.33\right) + 2.89 + 51\\-8.33 + 6 \cdot 2.89 - 17 \end{array}} \right)=\left( {\begin{array}{*{20}{c}} 20.57\\-7.99 \end{array}} \right)\]

\[\left|\left|\nabla f\left({X_{2}}\right)\right|\right|=\sqrt {20.57^{2}+\left( -7.99 \right)^{2}}=22.0673\]

$f\left( x,y \right)=2 x^{2} + x y + 51 x + 3 y^{2} - 17 y$ \newline \textbf{Метод градиентнгого спуска} \newline Градиент \newline $\nabla f\left({X_0}\right)=\left( {\begin{array}{*{20}{c}} 4 x + y + 51\\x + 6 y - 17 \end{array}} \right)$ \newline Начнём с точки $X_0=\left( 0;0 \right)^T$ \newline \underline{Итерация 0} \newline $f\left({X_0}\right)=2 \cdot 0^{2} + 0 \cdot 0 + 51 \cdot 0 + 3 \cdot 0^{2} - 17 \cdot 0=0$ \newline $\nabla f\left({X_0}\right)=\left( {\begin{array}{*{20}{c}} 4 \cdot 0 + 0 + 51\\0 + 6 \cdot 0 - 17 \end{array}} \right)=\left( {\begin{array}{*{20}{c}} 51\\-17 \end{array}} \right)$ \newline $\left|\left|\nabla f\left({X_0}\right)\right|\right|=\sqrt {51^{2}+\left( -17 \right)^{2}}=53.7587$ \newline \underline{Итерация 1} \newline $X_{1}=X_{0}-t_{0} \nabla f\left({X_{0}}\right)=\left( {\begin{array}{*{20}{c}} 0\\0 \end{array}} \right)-0.1\left( {\begin{array}{*{20}{c}} 51\\-17 \end{array}} \right)=\left( {\begin{array}{*{20}{c}} -5.1\\1.7 \end{array}} \right)$ \newline Пусть $t_0=0.1.$ \newline $f\left({X_{1}}\right)=2 \cdot \left(-5.1\right)^{2} - 5.1 \cdot 1.7 + 51 \cdot \left(-5.1\right) + 3 \cdot 1.7^{2} - 17 \cdot 1.7=-236.98$ \newline $f\left({X_{1}}\right) < f\left({X_{0}}\right)$ \newline $\nabla f\left({X_{1}}\right)=\left( {\begin{array}{*{20}{c}} 4 \cdot \left(-5.1\right) + 1.7 + 51\\-5.1 + 6 \cdot 1.7 - 17 \end{array}} \right)=\left( {\begin{array}{*{20}{c}} 32.3\\-11.9 \end{array}} \right)$ \newline $\left|\left|\nabla f\left({X_{1}}\right)\right|\right|=\sqrt {32.3^{2}+\left( -11.9 \right)^{2}}=34.4224$ \newline \underline{Итерация 2} \newline $X_{2}=X_{1}-t_{1} \nabla f\left({X_{1}}\right)=\left( {\begin{array}{*{20}{c}} -5.1\\1.7 \end{array}} \right)-0.1\left( {\begin{array}{*{20}{c}} 32.3\\-11.9 \end{array}} \right)=\left( {\begin{array}{*{20}{c}} -8.33\\2.89 \end{array}} \right)$ \newline $f\left({X_{2}}\right)=2 \cdot \left(-8.33\right)^{2} - 8.33 \cdot 2.89 + 51 \cdot \left(-8.33\right) + 3 \cdot 2.89^{2} - 17 \cdot 2.89=-334.1996$ \newline $f\left({X_{2}}\right) < f\left({X_{1}}\right)$ \newline $\nabla f\left({X_{2}}\right)=\left( {\begin{array}{*{20}{c}} 4 \cdot \left(-8.33\right) + 2.89 + 51\\-8.33 + 6 \cdot 2.89 - 17 \end{array}} \right)=\left( {\begin{array}{*{20}{c}} 20.57\\-7.99 \end{array}} \right)$ \newline $\left|\left|\nabla f\left({X_{2}}\right)\right|\right|=\sqrt {20.57^{2}+\left( -7.99 \right)^{2}}=22.0673$