4 - The cone of polynomials nonnegative on K

Summary

Introduction

Recall from Chapter 2 that the convex cone C_d ⊂ ℝ[x]_2d of nonnegative polynomials of degree at most 2d (a nonnegative polynomial has necessarily even degree) is much harder to characterize than its subcone ∑[x]_d of sums of squares (SOS). Indeed, while we have seen that the latter has a simple semidefinite representation in a higher dimensional space, so far there is no such simple representation for the former. In addition, when d is fixed Blekherman (2006) has shown that after proper normalization, the “gap” between C_d and Σ[x]_d increases unboundedly with the number of variables.

Similarly, for a subset K ⊂ ℝⁿ, the (finite-dimensional) convex cone C_d (K) of polynomials of degree at most d and nonnegative on K does not have a simple and tractable representation. This is why when f ∈ ℝ[x]_d the optimization problem f* = inf { f (x) : x ∈ K }, which is equivalent to f * = sup { λ : f (x) − λ ≥ 0, ∀x ∈ K }, is very difficult to solve in general even though it is also the finite-dimensional convex optimization problem sup { λ : f − λ ∈ C_d (K) }. Finite-dimensional convex optimization problems are not always tractable!

However, we next show that the results of Chapter 2 and Chapter 3 provide us with tractable inner and outer approximations of C_d (K), respectively. Those approximations will be very useful to approximate as closely as desired the global optimum f* of the optimization problem P in (1.1).