# feasp

Compute solution to given system of LMIs

## Syntax

## Description

`[`

computes a solution, if any exists, of a system of LMIs and returns a vector
`tmin`

,`xfeas`

] = feasp(`lmisys`

)`xfeas`

of particular values of the decision variables for which all
LMIs in the system are satisfied.

Given an LMI system,

$${N}^{T}LxN\le {M}^{T}R(x)M,$$

`feasp`

computes `xfeas`

by solving the auxiliary
convex program:

Minimize t subject to *N ^{T}L*(

*x*)

*N*–

*M*(

^{T}R*x*)

*M*≤

*tI*.

The global minimum of this program is the scalar value `tmin`

. The
LMI constraints are feasible if `tmin ≤ 0`

, and strictly feasible if
`tmin < 0`

.

## Examples

## Input Arguments

## Output Arguments

## Tips

When the least-squares problem solved at each iteration becomes ill conditioned, the

`feasp`

solver switches from Cholesky-based to QR-based linear algebra (see Memory Problems for details). Since the QR mode typically requires much more memory, MATLAB^{®}may run out of memory and display the following message.??? Error using ==> feaslv Out of memory. Type HELP MEMORY for your options.

If you see this message, increase your swap space. If no additional swap space is available, set

`options(4) = 1`

. Doing so prevents switching to QR and causes`feasp`

to terminate when Cholesky fails due to numerical instabilities.

## Algorithms

The feasibility solver of `feasp`

is based on Nesterov and Nemirovski's
Projective Method described in [1] and [2].

## References

[1] Nesterov, Y., and A. Nemirovski,
*Interior Point Polynomial Methods in Convex Programming: Theory and
Applications*, SIAM, Philadelphia, 1994.

[2] Nemirovski, A., and P. Gahinet,
“The Projective Method for Solving Linear Matrix Inequalities,” *Proc. Amer. Contr.
Conf.*, 1994, Baltimore, Maryland, p. 840–844.

## Version History

**Introduced before R2006a**