Results 1  10
of
57
Verifying nonlinear real formulas via sums of squares
 Theorem Proving in Higher Order Logics, TPHOLs 2007, volume 4732 of Lect. Notes in Comp. Sci
, 2007
"... 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 ..."
Abstract

Cited by 34 (3 self)
 Add to MetaCart
(Show Context)
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. 1 Verifying nonlinear formulas over the reals Over the real numbers, there are algorithms that can in principle perform quantifier elimination from arbitrary firstorder 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: ∀a b c. (∃x. ax 2 + bx + c = 0) ⇔ a = 0 ∧ (b = 0 ⇒ c = 0) ∨ a � = 0 ∧ b 2 ≥ 4ac
An Explicit Construction of Distinguished Representations of Polynomials Nonnegative Over Finite Sets
, 2002
"... We present a simple constructive proof of the existence of distinguished sum of squares representations for polynomials nonnegative over finite sets described by polynomial equalities and inequalities. A degree bound is directly obtained, as the cardinality of the support of the summands equals t ..."
Abstract

Cited by 24 (4 self)
 Add to MetaCart
We present a simple constructive proof of the existence of distinguished sum of squares representations for polynomials nonnegative over finite sets described by polynomial equalities and inequalities. A degree bound is directly obtained, as the cardinality of the support of the summands equals the number of points in the variety. Only basic results from commutative algebra are used in the construction.
Positive polynomials in scalar and matrix variables, the spectral theorem, and optimization
 , in vol. Structured Matrices and Dilations. A Volume Dedicated to the Memory of Tiberiu Constantinescu
"... We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decompositions, with emphasis on the interaction between classical moment problems, function theory of one or several complex variables and modern operator theory. The second par ..."
Abstract

Cited by 23 (8 self)
 Add to MetaCart
(Show Context)
We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decompositions, with emphasis on the interaction between classical moment problems, function theory of one or several complex variables and modern operator theory. The second part of the survey focuses on recently discovered connections between real algebraic geometry and optimization as well as polynomials in matrix variables and some control theory problems. These new applications have prompted a series of recent studies devoted to the structure of positivity and convexity in a free ∗algebra, the appropriate setting for analyzing inequalities on polynomials having matrix variables. We sketch some of these developments, add to them and comment on the rapidly growing literature.
On the Equivalence of Algebraic Approaches to the Minimization of Forms on the Simplex
, 2003
"... We consider the problem of minimizing a form on the standard simplex [equivalently, the problem of minimizing an even form on the unit sphere]. Converging hierarchies of approximations for this problem can be constructed, that are based, respectively, on results by SchmudgenPutinar and by Polya ..."
Abstract

Cited by 16 (9 self)
 Add to MetaCart
We consider the problem of minimizing a form on the standard simplex [equivalently, the problem of minimizing an even form on the unit sphere]. Converging hierarchies of approximations for this problem can be constructed, that are based, respectively, on results by SchmudgenPutinar and by Polya about representations of positive polynomials in terms of sums of squares. We show that the two approaches yield, in fact, the same approximations. The same type of argument also permits to establish some representation results a la Polya for positive polynomials on semialgebraic cones.
Closures of Quadratic Modules
 2 CLOSED SEPARATION THEOREMS 19
"... Abstract. We consider the problem of determining the closure M of a quadratic module M in a commutative Ralgebra with respect to the finest locally convex topology. This is of interest in deciding when the moment problem is solvable [26] [27] and in analyzing algorithms for polynomial optimization ..."
Abstract

Cited by 14 (8 self)
 Add to MetaCart
(Show Context)
Abstract. We consider the problem of determining the closure M of a quadratic module M in a commutative Ralgebra with respect to the finest locally convex topology. This is of interest in deciding when the moment problem is solvable [26] [27] and in analyzing algorithms for polynomial optimization involving semidefinite programming [12]. The closure of a semiordering is also considered, and it is shown that the space YM consisting of all semiorderings lying over M plays an important role in understanding the closure of M. The result of Schmüdgen for preorderings in [27] is strengthened and extended to quadratic modules. The extended result is used to construct an example of a nonarchimedean quadratic module describing a compact semialgebraic set that has the strong moment property. The same result is used to obtain a recursive description of M which is valid in many cases. In Section 1 we consider the general relationship between the closure C and the sequential closure C ‡ of a subset C of a real vector space V in the finest locally convex topology. We are mainly interested in the case where C is a cone in V. We consider cones with nonempty interior and cones satisfying C ∪ −C = V. In Section 2 we begin our investigation of the closure M of a quadratic module M of a commutative Ralgebra A; the focus is on finitely generated quadratic modules in finitely generated algebras. The closure of a semiordering Q of A is also considered, and it is shown that the space YM consisting of all semiorderings of A lying over M plays an important role in understanding the closure of M; see Propositions 2.2, 2.3 and 2.4. The result of Schmüdgen for preorderings in [27] is strengthened and extended to quadratic modules; see Theorem 2.8. In Section 3 we consider the case of quadratic modules that describe compact semialgebraic sets. We use Theorem 2.8 to deduce various results; see Theorems 3.1 and 3.4; and also to construct an example where KM is compact, M satisfies the strong moment property (SMP), but M is not archimedean; see Example 3.7.
A tutorial on sum of squares techniques for system analysis
 In Proceedings of the American control conference, ASCC
, 2005
"... Abstract — This tutorial is about new system analysis techniques that were developed in the past few years based on the sum of squares decomposition. We will present stability and robust stability analysis tools for different classes of systems: systems described by nonlinear ordinary differential e ..."
Abstract

Cited by 14 (1 self)
 Add to MetaCart
(Show Context)
Abstract — This tutorial is about new system analysis techniques that were developed in the past few years based on the sum of squares decomposition. We will present stability and robust stability analysis tools for different classes of systems: systems described by nonlinear ordinary differential equations or differential algebraic equations, hybrid systems with nonlinear subsystems and/or nonlinear switching surfaces, and timedelay systems described by nonlinear functional differential equations. We will also discuss how different analysis questions such as model validation and safety verification can be answered for uncertain nonlinear and hybrid systems. I.
An exact duality theory for semidefinite programming based on sums of squares. ArXiv eprints
, 2012
"... Abstract. Farkas ’ lemma is a fundamental result from linear programming providing linear certificates for infeasibility of systems of linear inequalities. In semidefinite programming, such linear certificates only exist for strongly infeasible linear matrix inequalities. We provide nonlinear algebr ..."
Abstract

Cited by 12 (2 self)
 Add to MetaCart
Abstract. Farkas ’ lemma is a fundamental result from linear programming providing linear certificates for infeasibility of systems of linear inequalities. In semidefinite programming, such linear certificates only exist for strongly infeasible linear matrix inequalities. We provide nonlinear algebraic certificates for all infeasible linear matrix inequalities in the spirit of real algebraic geometry: A linear matrix inequality A(x) ≽ 0 is infeasible if and only if −1 lies in the quadratic module associated to A. We also present a new exact duality theory for semidefinite programming, motivated by the real radical and sums of squares certificates from real algebraic geometry. 1.
PURE STATES, POSITIVE MATRIX POLYNOMIALS AND SUMS OF HERMITIAN SQUARES
, 907
"... Abstract. Let M be an archimedean quadratic module of real t×t matrix polynomials in n variables, and let S ⊆ R n be the set of all points where each element of M is positive semidefinite. Our key finding is a natural bijection between the set of pure states of M and S × P t−1 (R). This leads us to ..."
Abstract

Cited by 11 (2 self)
 Add to MetaCart
Abstract. Let M be an archimedean quadratic module of real t×t matrix polynomials in n variables, and let S ⊆ R n be the set of all points where each element of M is positive semidefinite. Our key finding is a natural bijection between the set of pure states of M and S × P t−1 (R). This leads us to conceptual proofs of positivity certificates for matrix polynomials, including the recent seminal result of Hol and Scherer: If a symmetric matrix polynomial is positive definite on S, then it belongs to M. We also discuss what happens for nonsymmetric matrix polynomials or in the absence of the archimedean assumption, and review some of the related classical results. The methods employed are both algebraic and functional analytic. 1.
The convex Positivstellensatz in a free algebra
 Adv. Math
"... Abstract. The main result of this paper establishes the perfect noncommutative Nichtnegativstellensatz on a convex semialgebraic set: suppose L is a monic linear pencil in g variables and let DL be its positivity domain DL: = ⋃ {} n×n g ..."
Abstract

Cited by 11 (8 self)
 Add to MetaCart
Abstract. The main result of this paper establishes the perfect noncommutative Nichtnegativstellensatz on a convex semialgebraic set: suppose L is a monic linear pencil in g variables and let DL be its positivity domain DL: = ⋃ {} n×n g
LMI approximations for cones of positive semidefinite forms
 Fachbereich Mathematik, Universität Konstanz, 78457
"... An interesting recent trend in optimization is the application of semidefinite programming techniques to new classes of optimization problems. In particular, this trend has been successful in showing that under suitable circumstances, polynomial optimization problems can be approximated via a sequen ..."
Abstract

Cited by 11 (4 self)
 Add to MetaCart
(Show Context)
An interesting recent trend in optimization is the application of semidefinite programming techniques to new classes of optimization problems. In particular, this trend has been successful in showing that under suitable circumstances, polynomial optimization problems can be approximated via a sequence of semidefinite programs. Similar ideas apply to conic optimization over the cone of copositive matrices, and to certain optimization problems involving random variables with some known moment information. We bring together several of these approximation results by studying the approximability of cones of positive semidefinite forms (homogeneous polynomials). Our approach enables us to extend the existing methodology to new approximation schemes. In particular, we derive a novel approximation to the cone of copositive forms, that is, the cone of forms that are positive semidefinite over the nonnegative orthant. The format of our construction can be extended to forms that are positive semidefinite over more general conic domains. We also construct polyhedral approximations to cones of positive semidefinite forms over a polyhedral domain. This opens the possibility of using linear programming technology in optimization problems over these cones.