How can we rewrite this optimization problem in standard form?

Question

Consider the following problem $$ \begin{array}{crl} &\underset{y}{\max} & f(y)\\ s.t. & y_{1}A_{1}+y_{2}A_{2}+S_{1}&=C_{1},\\ & y_{3}A_{3}+y_{2}A_{4}+y_{5}A_{5}+S_{2}&=C_{2},\\ & y_{3}A_{6}+y_{4}A_{7}+S_{3}&=C_{3},\\ & S_{1},S_{2},S_{3}\succeq0, \end{array} $$ where $y_{i}$ are scalar variables and the matrices $A_{i}$ are symmetric real. How can we rewrite this optimzation problem in the following form $$ \begin{array}{crl} &\underset{y}{\max} & f(y)\\ s.t. & \sum_{i=1}^{m}y_{i}A_{i}+S&=C,\\ &S&\succeq0, \end{array} $$ where $x_{i}$ are scalar variables and the matrices $B_{i}$ are symmetric real. please? Thanks.

Answer 1

First rewrite the problem as $$ \begin{array}{crl} &\underset{y}{\max} & f(y)\\ s.t. & y_{1}A_{1}+y_{2}A_{2}+S_{1}&=C_{1},\\ & y_{3}A_{3}+y_{2}A_{4}+y_{4}A_{5}+S_{2}&=C_{2},\\ & y_{3}A_{6}+y_{4}A_{7}+S_{3}&=C_{3},\\ & S_{1},S_{2},S_{3}\succeq0, \end{array} $$ changing the variable $y_5$ by the variable $y_4$. The three equations that define equality constraints can be written in a single equation using block-partitioned matrices. $$ { \begin{array}{rl} \underset{y}{\max} & f(y)\\ y_1 \left[\begin{array}{ccc} A_1&0&0\\0&0&0\\0&0&0 \end{array}\right] + y_2 \left[\begin{array}{ccc} A_2&0&0\\0&A_4&0\\0&0&0 \end{array}\right] + y_3 \left[\begin{array}{ccc} 0&0&0\\0&A_3&0\\0&0&A_6 \end{array}\right] + y_4 \left[\begin{array}{ccc} 0&0&0\\0&A_5&0\\0&0&A_7 \end{array}\right] + \left[\begin{array}{ccc} S_1&0&0\\0&S_2&0\\0&0&S_3 \end{array}\right] &= \left[\begin{array}{ccc} C_1&0&0\\0&C_2&0\\0&0&C_3 \end{array}\right] \\ \left[\begin{array}{ccc} S_1&0&0\\0&S_2&0\\0&0&S_3 \end{array}\right] &\succeq 0 \end{array} } $$