Sum of squares methods for minimizing polynomial forms over spheres and hypersurfaces

Abstract

This paper studies the problem of minimizing a homogeneous polynomial (form) f(x) over the unit sphere $\mathbb{S}^{n - 1} = \left\{ {x \in \mathbb{R}^n :\left\| x \right\|_2 = 1} \right\}$. The problem is NP-hard when f(x) has degree 3 or higher. Denote by f _min (resp. f _max) the minimum (resp. maximum) value of f(x) on $\mathbb{S}^{n - 1}$. First, when f(x) is an even form of degree 2d, we study the standard sum of squares (SOS) relaxation for finding a lower bound of the minimum f _min:

$$\max \gamma s.t. f\left( x \right) - \gamma \cdot \left\| x \right\|_2^{2d} is SOS.$$

Let f _sos be the above optimal value. Then we show that for all n ⩾ 2d,

$$1 \leqslant \frac{{f_{\max } - f_{sos} }} {{f_{\max } - f_{\min } }} \leqslant C(d)\sqrt {\left( {_{2d}^n } \right)} .$$

Here, the constant C(d) is independent of n. Second, when f(x) is a multi-form and $\mathbb{S}^{n - 1}$ becomes a multi-unit sphere, we generalize the above SOS relaxation and prove a similar bound. Third, when f(x) is sparse, we prove an improved bound depending on its sparsity pattern; when f(x) is odd, we formulate the problem equivalently as minimizing a certain even form, and prove a similar bound. Last, for minimizing f(x) over a hypersurface H(g) = {x ∈ ℝⁿ: g(x) = 1} defined by a positive definite form g(x), we generalize the above SOS relaxation and prove a similar bound.