LP Cones

A cone of LP variables can specified separately. For example a randomly generated LP cone $A^T y \leq c$ with $3$ variables $y$ and $5$ inequality constraints can be specified in the following code.

> n=5; m=3;
> b = rand(m,1);
> At=rand(n,m);
> c=rand(n,1);
> AC{1,1} = 'LP';
> AC{1,2} = n;
> AC{1,3} = sparse([At c]);
> [STAT,y,X]=dsdp(b,AC);

Multiple cones of LP variables may be passed into the DSDP solver, but for efficiency reasons, it is best to group them all together. This cone may also be passed to the DSDP solver as a semidefinite cone, where the matrices $A_i$ and $C$ are diagonal. For efficiency reasons, however, it is best to identify them separately as belonging the the cone of 'LP' variables.

Although $y$ variables that are fixed to a constant can be preprocessed and removed from a model, it is often more convenient to leave them in the model. It is more efficient for to identify fixed variables to DSDP than to model these constraints as a pair of linear inequalities. The following example sets variables $1$ and $8$ to the values $2.4$ and $-6.1$, respectively.

> AC{j,1} = 'FIXED'; AC{j,2} = [ 1 8 ]; AC{j,3} = [ 2.4 -6.1 ];

The corresponding variables $x$ to these constraints may be positive or negative.