# a Rank Minimization Heuristic with Application to Minimum Order System Approximation

Rank Minimization Problem (RMP) has application in a variety of areas such as control, system identification, statistics and signal processing. Except in some special cases RMP is known to be computationaly hard. $\begin{array}{ l l } \mbox{minimize} & \mbox{Rank } X \\ \mbox{subject to: } & X \in C \end{array}$

If the matrix is symmetric and positive semidifinite, trace minimization is a very effective heuristic for rank minimization problem. The trace minimization results in a semidefinite problem which can be easily solved. $\begin{array}{ l l } \mbox{minimize} & \mbox{Tr } X \\ \mbox{subject to: } & X \in C \end{array}$

This paper focuses on the following problems:

1. Describing a generalization of the trace heuristic for genaral non-square matrices.
2. Showing that the new heuristic can be reduced to an SDP, and hence effictively solved.
3. Applying the mothod on the minimum order system approximation.

### The Heuristic

This heurisitic minimizes the sum of the singular values of the matrix, i.e, the nuclear norm.

$\begin{array}{ l l } \mbox{minimize} & ||X||_* \\ \mbox{subject to: } & X \in C \end{array} \mbox{where} ||X||_*=\sum_{i=1}^{\min{m,n} \sigma_i(X)$