An effective decision method for semidefinite polynomials

Zeng, Guangxing; Xiao-ning, Zeng

doi:10.1016/s0747-7171(03)00073-7

Cited by 8 publications

(1 citation statement)

References 9 publications

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“…The entire sequence of events takes around one second. Numerous similar instances of classic elementary inequalities can be proved efficiently, including almost all the examples in [12] and the following:…”

Section: Examplesmentioning

confidence: 99%

Verifying Nonlinear Real Formulas Via Sums of Squares

Harrison

Lecture Notes in Computer Science

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Abstract. Techniques based on sums of squares appear promising as a general approach to the universal theory of reals with addition and multiplication, i.e. verifying Boolean combinations of equations and inequalities. A particularly attractive feature is that suitable 'sum of squares' certificates can be found by sophisticated numerical methods such as semidefinite programming, yet the actual verification of the resulting proof is straightforward even in a highly foundational theorem prover. We will describe our experience with an implementation in HOL Light, noting some successes as well as difficulties. We also describe a new approach to the univariate case that can handle some otherwise difficult examples. Verifying nonlinear formulas over the realsOver the real numbers, there are algorithms that can in principle perform quantifier elimination from arbitrary first-order formulas built up using addition, multiplication and the usual equality and inequality predicates. A classic example of such a quantifier elimination equivalence is the criterion for a quadratic equation to have a real root:The first quantifier elimination algorithm for this theory was developed by Tarski [32], 1 who actually demonstrated completeness and quantifier elimination just for the theory of real-closed fields, which can be characterized as ordered fields where all nonnegative elements have square roots (∀x. 0 ≤ x ⇒ ∃y. x = y 2 ) and all non-trivial polynomials of odd degree have a root. There are several interesting models of these axioms besides the reals (e.g. the algebraic reals, the computable reals, the hyperreals) yet Tarski's result shows that these different models satisfy exactly the same properties in the first-order language under consideration.However, Tarski's procedure is complicated and inefficient. Many alternative decision methods were subsequently proposed; two that are significantly simpler were given by Seidenberg [30] and Cohen [8], while the CAD algorithm [9], apparently the first ever to be implemented, is significantly more efficient, though relatively complicated. Cohen's ideas were recast by Hörmander [17] into a relatively simple algorithm. However, even CAD has poor worst-case complexity (doubly exponential), and the CohenHörmander algorithm is generally still slower. Thus, there has been limited progress on 1 Tarski actually discovered the procedure in 1930, but it remained unpublished for many years afterwards.

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Section: Examplesmentioning

confidence: 99%

Verifying Nonlinear Real Formulas Via Sums of Squares

Harrison

Lecture Notes in Computer Science

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Global minimization of multivariate polynomials using nonstandard methods

Zeng

Xiao

2011

J Glob Optim

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The purpose of this paper is to present two algorithms for global minimization of multivariate polynomials. For a multivariate real polynomial f , we provide an effective algorithm for deciding whether or not the infimum of f is finite. In the case of f having a finite infimum, the infimum of f can be accurately coded as (h; a, b), where h is a real polynomial in one variable, a and b is two rational numbers with a < b, and (h, a, b) stands for the only real root of h in the open interval ]a, b[. Moreover, another algorithm is provided to decide whether or not the infimum of f is attained when the infimum of f is finite. Our methods are called "nonstandard", because an infinitesimal element is introduced in our arguments.

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Semi-algebraically connected components of minimum points of a polynomial function

Xiao

Zeng

2013

J Syst Sci Complex

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An effective decision method for semidefinite polynomials

Cited by 8 publications

References 9 publications

Verifying Nonlinear Real Formulas Via Sums of Squares

Verifying Nonlinear Real Formulas Via Sums of Squares

Global minimization of multivariate polynomials using nonstandard methods

Semi-algebraically connected components of minimum points of a polynomial function

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