SemiDefinite What?

This page contains definitions for SemiDefinite things like matrices, programs, etc.

== SemiDefinite Matrices ==

A positive SemiDefinite matrix is a HermitianMatrix all of whose eigenvalues are nonnegative. Thus any symmetric matrix that has a 0 on the diagonal is a SemiDefinite matrix.

SemiDefinite Programming

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Suppose you have a loop in a program where the values of the variable values at the start of the loop are denoted $x_0 \in \mathbb{R}^n$ and the values at the end of the loop are denoted $x$ where $n$ is the number of variables. Then $x_k$ is the variable values after the $k^{th}$ loop.

Let 
\[
     M(x)=M_0 + \sum\limits^m_{k=1} x_k.M_k 
\]
with symmetric matrices $(M_k = M_k^{\top}$, and positive semidefiniteness (can you say that?) defined as:
\[
     M(x) \succeq 0 \equiv \forall X \in \mathbb{R}^N : XM(x)X^{\top} \ge 0
\]
Somehow I believe that $M$ is an encoding of the constraints in the loop. It doesn't say that in Cousot-05-vmaci, but that is what I think it means. Then the SemiDefinite programming optimization problem is to find a solution to the constraints:
\[
      \left\{\begin{array}{l}
         \exists x \in \mathbb{R}^m : M(x) \succeq 0 \\
         Minimizing~~c^{\top} x
      \end{array}\right.
\]
where $c \in \mathbb{R}^m is a real given vector and $M$ is called the {\em linear matrix inequality}