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