Consider the following problem
\begin{eqnarray*} \underset{y}{\max} & f(y)\\ s.t. & y_{1}A_{1}+y_{2}A_{2}+y_{3}A_{3}+S_{1}=C_{1},\\ & y_{4}A_{4}+y_{5}A_{5}+y_{6}A_{6}+y_{7}A_{7}+S_{2}=C_{2},\\ & y_{4}A_{8}+y_{5}A_{9}+y_{6}A_{10}+y_{7}A_{11}+S_{3}=C_{3},\\ & y_{4}A_{12}+y_{5}A_{13}+y_{6}A_{14}+y_{7}A_{15}+y_{2}A_{16}+y_{3}A_{17}+S_{4}=C_{4},\\ & S_{1},S_{2},S_{3},S_{4}\succeq0, \end{eqnarray*} where $y_{i}$ are scalar variables. How can we rewrite this optimzation problem in the following form \begin{eqnarray*} \underset{x}{\max} & f(x)\\ s.t. & \sum_{i=1}^{i=m}x_{i}B_{i}+T=D,\\ & T\succeq0, \end{eqnarray*} where $x_{i}$ are scalar variables please? Thanks.
We have $$ { \begin{array}{rl} \underset{y}{\max} & f(y)\\ y_1 \left[\begin{array}{cccc} A_1&0&0&0\\0&0&0&0\\0&0&0&0\\0&0&0&0 \end{array}\right] + y_2 \left[\begin{array}{cccc} A_2&0&0&0\\0&0&0&0\\0&0&0&0\\0&0&0&A_{16} \end{array}\right] + y_3 \left[\begin{array}{cccc} A_3&0&0&0\\0&0&0&0\\0&0&0&0\\0&0&0&A_{17} \end{array}\right] +& \\ +y_4 \left[\begin{array}{cccc} 0&0&0&0\\0&A_4&0&0\\0&0&A_8&0\\0&0&0&A_{12} \end{array}\right] +y_5 \left[\begin{array}{cccc} 0&0&0&0\\0&A_5&0&0\\0&0&A_{9}&0\\0&0&0&A_{13} \end{array}\right] +y_6 \left[\begin{array}{cccc} 0&0&0&0\\0&A_6&0&0\\0&0&A_{10}&0\\0&0&0&A_{14} \end{array}\right] + &\\ +y_7 \left[\begin{array}{cccc} 0&0&0&0\\0&A_7&0&0\\0&0&A_{11}&0\\0&0&0&A_{15} \end{array}\right] + \left[\begin{array}{ccc} S_1&0&0&0\\0&S_2&0&0\\0&0&S_3&0\\0&0&0&S_4 \end{array}\right] &= \left[\begin{array}{cccc} C_1&0&0&0\\0&C_2&0&0\\0&0&C_3&0\\0&0&0&C_4 \end{array}\right] \\ \left[\begin{array}{ccc} S_1&0&0&0\\0&S_2&0&0\\0&0&S_3&0\\0&0&0&S_4 \end{array}\right] &\succeq 0 \end{array} } $$