\form#0:\[ \begin{array}{llllll} \mbox{Maximize} & & y_1 & + & y_2 \\ \mbox{Subject to} & & 4 y_1 & + & 2 y_2 & \leq 6 \\ & & 3 y_1 & + & 7 y_2 & \leq 10 \\ & & & & - y_2 & \leq 12 \\ \end{array} \]
\form#1:$A^T$
\form#2:$A_{i,j}$
\form#3:$n \times n$
\form#4:$ n^2 $
\form#5:\[ \begin{array}{lllllllllll} [ a_{1,1} & \ldots & a_{1,n} & a_{2,1} & \ldots & a_{2,n} & \ldots & a_{n,1} & \ldots & a_{n,n} ] \\ \end{array} \]
\form#6:$a_{i,j}, i<j $
\form#7:$\frac{n(n+1)}{2}$
\form#8:\[ \begin{array}{llllllll} [ a_{1,1} & a_{2,1} & a_{2,2} & a_{3,1} & a_{3,2} & a_{3,3} & \ldots & a_{n,n} ] \\ \end{array} \]
\form#9:$ a_{1,1} $
\form#10:$a_{1,1}$
\form#11:$a_{2,1}$
\form#12:$a_{3,2}$
\form#13:$A_{i,j}= alpha * v * v^T $
\form#14:$ V V^T= X $
\form#15:$ v^T A_{i,j} v $
\form#16:$(x^u - x^l)_i$
\form#17:$\frac{\bar{z} - b^Ty}{\rho}$
\form#18:$\sqrt{-\nabla \psi^T(y^k,\bar{z}^k ) \Delta y}$
\form#19:$ \mbox{maximize} \ \ {\displaystyle \sum_{i=1}^m b_i \ y_i } $
\form#20:$ [ -1 \ y_1 \ \ldots \ y_m \ r ] $
\form#21:$ b - mu * A(S^{-1}) $
\form#22:\[ \begin{array}{llllllllll} (P) \ \ \ & \mbox{minimize} & {\displaystyle \sum_{j=1}^{n_b} \langle C_j, X_j \rangle } &\mbox{subject to}& {\displaystyle \sum_{j=1}^{n_b} \langle A_{i,j}, X_{j} \rangle } = b_i ,& i=1,\ldots, m, & & X_j \in K_j,\\ \end{array} \]
\form#23:\[ \begin{array}{lllllllll} (D) \ \ \ & \mbox{maximize} & {\displaystyle \sum_{i=1}^m b_i \ y_i } &\mbox{subject to}&{\displaystyle \sum_{i=1}^m A_{i,j}y_i + S_{j} } = C_{j}, & j=1, \ldots, n_b, & S_j \in K_j, \\ \end{array} \]
\form#24:$K_j$
\form#25:$\langle \cdot , \cdot \rangle$
\form#26:$b_i$
\form#27:$y_i$
\form#28:$C_j$
\form#29:$C=(c_{k,l})$
\form#30:$X=(x_{k,l})$
\form#31:$\langle C , X \rangle := trace (C^T X) = \sum_{k,l}c_{k,l} x_{k,l} $
\form#32:$(\Re_{+}^n)$
\form#33:$A_{i}$
\form#34:$C$
\form#35:$\langle C , X \rangle$
\form#36:$y$
\form#37:$l \leq y \leq u$
\form#38:$l,u \in \Re^m$
\form#39:$X$
\form#40:$\Gamma$
\form#41:\[ \begin{array}{llccccll} \vspace{0.25cm} (PP) & \mbox{minimize} & {\displaystyle \sum_{j=1}^{n_b} \langle C_j, X_j \rangle } & + & u^T x^u - l^T x^l \\ &\mbox{subject to}& {\displaystyle \sum_{j=1}^{n_b} \langle A_{i,j}, X_{j} \rangle} & + & x^u_i - x^l_i & = & b_i ,& i=1,\ldots, m, \\ & & {\displaystyle \sum_{j=1}^{n_b} \langle I_j , X_{j} \rangle } & & & \leq & \Gamma, \\ & & X_j \in K_j, & & x^u, x^l & \geq & 0, \\ \end{array} \]
\form#42:\[ \begin{array}{llrcll} \vspace{0.25cm} (DD) & \mbox{maximize} & {\displaystyle \sum_{i=1}^m b_i \ y_i - \Gamma r} \\ & \mbox{subject to}&{\displaystyle C_{j} - \sum_{i=1}^m A_{i,j}y_i + r I_j } & = & S_{j} \in K_j, & j=1, \ldots, n_b, \\ & & l_i \leq y_i \leq u_i, & & & i=1,\ldots, m, \\ & & r \geq 0, \\ \end{array} \]
\form#43:$I_j$
\form#44:$x^l, x^u \in \Re^m $
\form#45:$x^l$
\form#46:$x^u$
\form#47:$l$
\form#48:$u$
\form#49:$r$
\form#50:$S_{j}$
\form#51:$K_{j}$
\form#52:\[ \begin{array}{llllllllll} (P) \ \ \ & \mbox{minimize} & {\displaystyle \sum_{j=0}^{n_b-1} C_j \bullet X_j } &\mbox{subject to}& {\displaystyle \sum_{j=0}^{n_b-1} A_{i,j} \bullet X_{j} = b_i } ,& i=1,\ldots, m, & & X_j \succeq 0, \\ \end{array} \]
\form#53:\[ \begin{array}{lllllllll} (D) \ \ \ & \mbox{maximize} & {\displaystyle \sum_{i=1}^m b_i \ y_i } &\mbox{subject to}&{\displaystyle \sum_{i=1}^m A_{i,j}y_i + S_{j} } = C_{j}, & j=0, \ldots, n_b-1, & S_j \succeq 0 \\ \end{array} \]
\form#54:$ C \bullet X := trace (C^T X) = \sum_{k,l}c_{k,l} x_{k,l} $
\form#55:$ A_{0,j}$
\form#56:$ n \times n$
\form#57:$ n(n+1)/2 $
\form#58:\[ \begin{array}{llllllll} [ a_{1,1} & a_{2,1} & a_{2,2} & a_{3,1} & a_{3,2} & a_{3,3} & \ldots & a_{n,n} ] \\ \end{array}. \]
\form#59:\[ \begin{array}{lllllllll} \mbox{Maximize} & b^T y & & \mbox{such that} & A^T y & \leq c \\ \mbox{Minimize} & c^T x & & \mbox{such that} & A x & = b, & x \geq 0 \\ \end{array} \]
\form#60:$ s:= c - A^Ty $
\form#61:$ x,s \in \Re^n$