Lower Bound for a Polynomial on a product of hyperellipsoids using geometric programming
Mehdi Ghasemi, Murray Marshall
TL;DR
This work advances polynomial optimization by providing a GP-based method to compute lower bounds for a polynomial $f$ on basic semialgebraic sets defined by $g_j$ of a hyperellipsoid-like form. It specializes the framework to $g_j = 1- frac{1}{N_i^d} x_i^d$ over partitions, enabling a scalable, sparsity-aware algorithm that can outperform SDP in large-scale or sparse problems. The main contribution is showing that the supremum of GP-computable bounds, $s(f,f g)$, yields valid lower bounds on $f$ over $K_{f g}$, with explicit expressions and special cases including a hypercube bound when $m=n$ and $I_j=igl\u002e jigr floor$. A Sage implementation is provided, and the method is contrasted with SDP approaches in terms of speed and applicability to large problems, offering practical value for fast, approximate certifiability in polynomial optimization on structured domains.
Abstract
Let $f$ be a polynomial in $n$ variables $x_1,\dots,x_n$ with real coefficients. In [Ghasemi-Marshal], Ghasemi and Marshall give an algorithm, based on geometric programming, which computes a lower bound for $f$ on $\mathbb{R}^n$. In [Ghasemi-Lasserre-Marshall] Ghasemi, Lasserre and Marshall show how the algorithm in [Ghasemi-Marshal] can be modified to compute a lower bound for $f$ on the hyperellipsoid $\sum_{i=1}^n x_i^d \le M.$ Here $d$ is a fixed even integer, $d \ge \max\{ 2, °(f)\}$ and $M$ is a fixed positive real number. Suppose now that $g_j := 1-\sum_{i\in I_j} (\frac{x_i}{N_i})^d$, $j=1,\dots,m$, where $d$ is a fixed even integer $d \ge \max\{ 2, °(f)\}$, $N_i$ is a fixed positive real number, $i=1,\dots,n$ and $I_1,\dots, I_m$ is a fixed partition of $\{ 1,\dots,n\}$. The present paper gives an algorithm based on geometric programming for computing a lower bound for $f$ on the subset of $\mathbb{R}^n$ defined by the inequalities $g_j\ge 0$, $j=1,\dots,m$. The algorithm is implemented in a SAGE program developed by the first author. The bound obtained is typically not as sharp as the bound obtained using semidefinite programming, but it has the advantage that it is computable rapidly, even in cases where the bound obtained by semidefinite programming is not computable. When $m=1$ and $N_i = \root d \of{M}$, $i=1,\dots,n$ the algorithm produces the lower bound obtained in [Ghasemi-Lasserre-Marshall]. When $m=n$ and $I_j = \{ j \}$, $j=1,\dots,n$ the algorithm produces a lower bound for $f$ on the hypercube $\prod_{i=1}^n [-N_i,N_i]$, which in certain cases can be computed by a simple formula.