Positivity of trigonometric polynomials

The paper introduces a modification of the well-known sum-of-squares relaxation scheme for semi-algebraic programming by Shor based on replacing the ordinary polynomials by their trigonometric counterparts. It is shown that the new scheme has certain theoretical advantages over the classical one: in particular, a trigonometric polynomial is positive if and only if it can be represented as a sum of squares of a finite number of trigonometric polynomials. A dual version of the SOS relaxation is also introduced and discussed. An example of a quantized finite horizon optimal control application with state constraints, typical for model predictive control, is discussed.