In the following we will give a con

In the following we will give a concise introduction to semidefinite programming and show its connection to the SOS
problem.
2.1. Semidefinite programming background
In this section we will give a brief introduction to semidefinite programming and its basic underlying ideas. We refer the
reader to [42,5] for a comprehensive treatment of the topic.
A semidefinite program is defined as the following convex optimization problem:
minimize hC; Xi
subject to hAi; Xi D bi; i D 1; : : : ;m;
X  0;
(P)
where X 2 Sn is the decision variable and the matrices C; Ai 2 Sn, and b 2 Rm are the problem data. The problem is convex
since its objective function and the feasible region defined by the constraints are convex. A geometric interpretation is the
minimization of a linear function over the intersection of the set of positive semidefinite matrices with an affine subspace.
Problem (P) is called strictly feasible if there exists some X  0 which satisfies the equality constraints in (P). The problem
above has an associated dual problem being
maximize hb; yi
subject to A.y/ VD C