/*!
\page PDForm Standard Form

The %DSDP package uses a dual-scaling algorithm to solve conic
optimization problems of the form
\f[
  \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}
\f]
\f[
  \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}
\f]
where \f$K_j\f$ is a cone, \f$\langle \cdot , \cdot \rangle\f$ 
is the associated inner product,
and \f$b_i\f$, \f$y_i\f$ are scalars.
For semidefinite programming each cone \f$K_j\f$ is a set of 
symmetric positive definite matrices,
the data \f$A_{i,j}\f$ and \f$C_j\f$ are symmetric matrices of the same dimension, 
and the inner product
of two \f$n \times n\f$ matrices \f$C=(c_{k,l})\f$ and \f$X=(x_{k,l})\f$
is defined by
\f$\langle C , X \rangle := trace (C^T X) = \sum_{k,l}c_{k,l} x_{k,l} \f$.
In linear programming the cone \f$K_j\f$ is the nonnegative orthant \f$(\Re_{+}^n)\f$,
the data  \f$A_{i}\f$ and \f$C\f$ are vectors of the same dimension, 
and the inner product
\f$\langle C , X \rangle\f$ is the usual vector inner product.

The convergence of the algorithm assumes that both (P) and (D) have
an interior feasible region and the current solutions are elements of the interior.
To satisfy these assumptions,
DSDP bounds
the variables \f$y\f$ such that \f$l \leq y \leq u\f$ where \f$l,u \in \Re^m\f$.  
Furthermore,
DSDP bounds the trace of \f$X\f$ by a penalty parameter \f$\Gamma\f$.
By adding these auxiliary variables and parameters, DSDP solves following pair of problems:
\f[
  \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}
\f]
\f[
  \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}
\f]
where \f$I_j\f$ is the identity element of \f$K_j\f$, and \f$x^l, x^u \in \Re^m \f$ .
The variables \f$x^l\f$ and \f$x^u\f$ correspond to the Lagrangian multipliers of
the bounds  \f$l\f$ and \f$u\f$.  The difference of these variables
can be interpreted as the infeasibility in the solution of (P).
The positive variable \f$r\f$ augments the variable \f$S_{j}\f$ so that it
is a element of \f$K_{j}\f$.
The parameters \f$\Gamma\f$, \f$l\f$, and \f$u\f$ bound these 
auxiliary variables and penalize infeasibility in (D) and (P).
By default, the bounds and the penalty parameter \f$\Gamma\f$ are very large to force
\f$r\f$, \f$x^l\f$, and \f$x^u\f$ to be near zero at the solution.
Unbounded and infeasible solutions to either (P) or (D) are
determined through examination of the solutions to (PP) and (DD).
Note that the reformulation (PP) and (DD) is bounded and feasible,
and it can be expressed in the form of (P) and (D).

*/
