Theorems of the Alternative for Inequality Systems of Real Polynomials

Abstract

In this paper, we establish theorems of the alternative for inequality systems of real polynomials. For the real quadratic inequality system, we present two new results on the matrix decomposition, by which we establish two theorems of the alternative for the inequality system of three quadratic polynomials under an assumption that at least one of the involved forms be negative semidefinite. We also extend a theorem of the alternative to the case with a regular cone. For the inequality system of higher degree real polynomials, defined by even order tensors, a theorem of the alternative for the inequality system of two higher degree polynomials is established under suitable assumptions. As a byproduct, we give an equivalence result between two statements involving two higher degree polynomials. Based on this result, we investigate the optimality condition of a class of polynomial optimization problems under suitable assumptions.

Unifying Farkas lemma and S-lemma: new theory and applications in nonquadratic nonconvex optimization

“…Since the objective function is decreasing with respect to z, the inequality holds as an equality at the optimal solution. Therefore, x an optimal solution to (BE) (15) if and only if…”

“…Jeyakumar et al [14] extended the domain of the system from R n to any linear manifold. Hu and Huang [15] generalized S-lemma to even order tenors. Xia et al [12] established necessary and sufficient conditions for S-lemma with equality.…”

A theorem of the alternative with an arbitrary number of inequalities and quadratic programming

“…However, some of them predate that fundamental result, as the Gordan theorem [14], which dates back to 1873. Notwithstanding its simplicity, Gordan's theorem and some of its extensions are essential tools and active research topics in mathematical programming (some illustrative examples may be found in [28,18,7,12,21,20,19,6,11,17,39,25]) and have wide-ranging applications in many different fields (see, for instance, [13,26,4,31,29]).…”

“…Yakubovič from the Dines theorem [40], or in its generalization [20,Corollary 3.7]. Instead, sometimes one looks for a convex set containing the joint range set and applies the Separation theorem: see for instance [17,Theorem 2.4]. We will show in the next section that the use of the theorem of Gordan, Theorem 2.3, avoids this kind of technique and leads to a more direct approach for the study of quadratic programs, which in particular allows us to deduce quite general results.…”