LMIs in Control/Global Minimum of Polynomial via SOS Method

[UNDER CONSTRUCTION] - CME

The global minimum of a certain polynomial functions $f(x)$ can be found using Sum-of-Squares (SOS) methods. This is a useful starting point to the more-useful but less straightforward issue of local minimums.

Required Data

A polynomial function $f(x)$ whose minimum is desired.

The Problem

To find the minimum value of $f(x)$, we suppose that there exists a scalar $p$ such that

$f(x)-p\geq 0$

This is equivalent to determining if $(f(x)-p)$ has a SOS representation, since a SOS polynomial is never negative. This becomes an optimization problem by attempting to find the largest possible $p$ such that a representation exists. In other words, the optimization problem becomes

$\min _{p}-p$ subject to $f(x)-p\geq 0$

Example

Code example in SOStools probably worthwhile!

Reference

SOS tools manual is a good one

Related LMIs

SOS Basics

Local Minimum is a solid follow-up