Proving Inequalities and Solving Global Optimization Problems via Simplified CAD Projection

TL;DR

This paper introduces a simplified CAD projection operator, nproj, which reduces computational complexity in proving inequalities and solving global optimization problems involving polynomials.

Contribution

The paper proposes a new simplified projection operator, nproj, that reduces the projection scale and simplifies the CAD process for polynomial inequality problems.

Findings

The nproj operator has a projection scale no larger than Brown's.

The new algorithms are correct and more efficient for many problems.

Comparison shows improved effectiveness over existing tools.

Abstract

Let $\xx_n=(x_1,\ldots,x_n)$ and $f\in \R[\xx_n,k]$. The problem of finding all $k_0$ such that $f(\xx_n,k_0)\ge 0$ on $\mathbb{R}^n$ is considered in this paper, which obviously takes as a special case the problem of computing the global infimum or proving the semi-definiteness of a polynomial. For solving the problems, we propose a simplified Brown's CAD projection operator, \Nproj, of which the projection scale is always no larger than that of Brown's. For many problems, the scale is much smaller than that of Brown's. As a result, the lifting phase is also simplified. Some new algorithms based on \Nproj\ for solving those problems are designed and proved to be correct. Comparison to some existing tools on some examples is reported to illustrate the effectiveness of our new algorithms.