Proving inequalities and solving global optimization problems via simplified CAD projection. (January 2016)

• Record Type: Journal Article
• Authors: Han, Jingjun; Jin, Zhi; Xia, Bican
• Is Part Of: Journal of symbolic computation. Volume 72(2016)
• Page Start: 206
• Page End: 230
• Publication Date: 2016-01
• Subjects: CAD projection -- Global optimization -- Semi-definiteness -- Polynomials; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2015.02.007
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: Let x n = ( x 1, …, x n ) and f ∈ R [ x n, k ] . The problem of finding all k 0 such that f ( x n, k 0 ) ≥ 0 on 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–McCallum's CAD projection operator, Nproj, of which the projection scale is always no larger than that of Brown–McCallum's. For many problems, the projection scale is much smaller than that of Brown–McCallum's. As a result, the lifting phase is also simplified. Some new algorithms based onNproj 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.