Section: New Results

Global optimization

Let $f_1,\cdots ,f_p$ be in $\mathbb{Q}\left[𝐗\right]$, where $𝐗={(X_1,\cdots ,X_n)}^t$, that generate a radical ideal and let $V$ be their complex zero-set. Assume that $V$ is smooth and equidimensional. Given $f\in \mathbb{Q}\left[X\right]$ bounded below, consider the optimization problem of computing $f^{☆}={inf}_{x\in V\cap {ℝ}^n}f\left(x\right)$. For $𝐀\in G{L}_n\left(ℂ\right)$, we denote by $f^{𝐀}$ the polynomial $f\left(𝐀𝐗\right)$ and by $V^{𝐀}$ the complex zero-set of $f_1^{𝐀},...,f_p^{𝐀}$. In [9] , we construct families of polynomials ${𝖬}_0^{𝐀},...,{𝖬}_d^{𝐀}$ in $\mathbb{Q}\left[𝐗\right]$: each ${𝖬}_i^{𝐀}$ is related to the section of a linear subspace with the critical locus of a linear projection. We prove that there exists a non-empty Zariski-open set $O\subset G{L}_n\left(ℂ\right)$ such that for all $𝐀\in O\cap G{L}_n\left(\mathbb{Q}\right)$, $f\left(x\right)$ is non-negative for all $x\in V\cap {ℝ}^n$ if, and only if, $f^{𝐀}$ can be expressed as a sum of squares of polynomials on the truncated variety generated by the ideal $\langle {𝖬}_i^{𝐀}\rangle$, for $0\le i\le d$. Hence, we can obtain algebraic certificates for lower bounds on $f^{☆}$ using semidefinite programs. Some numerical experiments are given. We also discuss how to decrease the number of polynomials in ${𝖬}_i^{𝐀}$.