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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 ..."
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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. 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
A convex polynomial that is not sosconvex
 Mathematical Programming
"... A multivariate polynomial p(x) = p(x1,...,xn) is sosconvex if its Hessian H(x) can be factored as H(x) = M T (x)M(x) with a possibly nonsquare polynomial matrix M(x). It is easy to see that sosconvexity is a sufficient condition for convexity of p(x). Moreover, the problem of deciding sosconvex ..."
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Cited by 7 (3 self)
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A multivariate polynomial p(x) = p(x1,...,xn) is sosconvex if its Hessian H(x) can be factored as H(x) = M T (x)M(x) with a possibly nonsquare polynomial matrix M(x). It is easy to see that sosconvexity is a sufficient condition for convexity of p(x). Moreover, the problem of deciding sosconvexity of a polynomial can be cast as the feasibility of a semidefinite program, which can be solved efficiently. Motivated by this computational tractability, it has been recently speculated whether sosconvexity is also a necessary condition for convexity of polynomials. In this paper, we give a negative answer to this question by presenting an explicit example of a trivariate homogeneous polynomial of degree eight that is convex but not sosconvex. Interestingly, our example is found with software using sum of squares programming techniques and the duality theory of semidefinite optimization. As a byproduct of our numerical procedure, we obtain a simple method for searching over a restricted family of nonnegative polynomials that are not sums of squares. 1
On Hilbert’s construction of positive polynomials
"... Abstract. In 1888, Hilbert described how to find real polynomials which take only nonnegative values but are not a sum of squares of polynomials. His construction was so restrictive that no explicit examples appeared until the late 1960s. We revisit and generalize Hilbert’s construction and present ..."
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Cited by 6 (1 self)
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Abstract. In 1888, Hilbert described how to find real polynomials which take only nonnegative values but are not a sum of squares of polynomials. His construction was so restrictive that no explicit examples appeared until the late 1960s. We revisit and generalize Hilbert’s construction and present many such polynomials. 1. History and Overview A real polynomial f(x1,...,xn) is psd or positive if f(a) ≥ 0 for all a ∈ R n; it is sos or a sum of squares if there exist real polynomials hj so that f = ∑ h 2 j. For forms, we follow the notation of [4] and use Pn,m to denote the cone of real psd forms of even degree m in n variables, Σn,m to denote its subcone of sos forms and let ∆n,m = Pn,m � Σn,m. The Fundamental Theorem of Algebra implies that ∆2,m = ∅; ∆n,2 = ∅ follows from the diagonalization of psd quadratic forms. The first suggestion that a psd form might not be sos was made by Minkowski in the oral defense of his 1885 doctoral dissertation: Minkowski proposed the thesis that not every psd form is sos. Hilbert was one of his official “opponents ” and remarked that Minkowski’s arguments had convinced him that this thesis should be true for ternary forms. (See [14], [15] and [24].) Three years later, in a single remarkable paper, Hilbert [11] resolved the question. He first showed that F ∈ P3,4 is a sum of three squares of quadratic forms; see [23] and [26] for recent expositions and [17, 18] for another approach. Hilbert then described a construction of forms in ∆3,6 and ∆4,4; after multiplying these by powers of linear forms if necessary, it follows that ∆n,m ̸ = ∅ if n ≥ 3 and m ≥ 6 or n ≥ 4 and m ≥ 4. The goal of this paper is to isolate the underlying mechanism of Hilbert’s construction, show that it applies to situations more general than those in [11], and use it to produce many new examples. In [11], Hilbert first worked with polynomials in two variables, which homogenize to ternary forms. Suppose f1(x, y) and f2(x, y) are two relatively prime real cubic polynomials with nine distinct real common zeros – {πi}, indexed arbitrarily – so that
POLYNOMIALS NONNEGATIVE ON A STRIP
"... Abstract. We prove that if f(x, y) is a polynomial with real coefficients which is nonnegative on the strip [0, 1] × R, then f(x, y) has a presentation of the form k∑ f(x, y) = gi(x, y) 2 ℓ∑ + hj(x, y) 2 x(1 − x), i=1 j=1 where the gi(x, y) and hj(x, y) are polynomials with real coefficients. 1. ..."
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Abstract. We prove that if f(x, y) is a polynomial with real coefficients which is nonnegative on the strip [0, 1] × R, then f(x, y) has a presentation of the form k∑ f(x, y) = gi(x, y) 2 ℓ∑ + hj(x, y) 2 x(1 − x), i=1 j=1 where the gi(x, y) and hj(x, y) are polynomials with real coefficients. 1.
Products of positive forms, linear matrix inequalities, and Hilbert 17th problem for ternary forms
, 2001
"... A form p on R n (homogeneous nvariate polynomial) is called positive semidenite (p.s.d.) if it is nonnegative on R n . In other words, the zero vector is a global minimizer of p in this case. The famous 17th conjecture of Hilbert [9] (later proven by Artin [1]) is that a form p is p.s.d. if and ..."
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Cited by 3 (1 self)
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A form p on R n (homogeneous nvariate polynomial) is called positive semidenite (p.s.d.) if it is nonnegative on R n . In other words, the zero vector is a global minimizer of p in this case. The famous 17th conjecture of Hilbert [9] (later proven by Artin [1]) is that a form p is p.s.d. if and only if it can be decomposed a sum of squares of rational functions. In this paper we give an algorithm to compute such a decomposition for ternary forms (n = 3). This algorithm involves the solution of a series of systems of linear matrix inequalities (LMI's). In particular, for a given p.s.d. ternary form p of degree 2m, we show that the abovementioned decomposition can be computed by solving at most m=4 systems of LMI's of dimensions polynomial in m. The underlying methodology is largely inspired by the original proof of Hilbert, who had been able to prove his conjecture for the case of ternary forms. 1
Lyapunov Based Analysis and Controller Synthesis for Polynomial Systems using SumofSquares Optimization
, 2003
"... Lyapunov Based Analysis and Controller Synthesis for Polynomial Systems using SumofSquares Optimization by Zachary William JarvisWloszek Doctor of Philosophy in EngineeringMechanical Engineering University of California, Berkeley Professor Andrew K. Packard, Chair This thesis considers a Lyapuno ..."
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Lyapunov Based Analysis and Controller Synthesis for Polynomial Systems using SumofSquares Optimization by Zachary William JarvisWloszek Doctor of Philosophy in EngineeringMechanical Engineering University of California, Berkeley Professor Andrew K. Packard, Chair This thesis considers a Lyapunov based approach to analysis and controller synthesis for systems whose dynamics are described by polynomials. We restrict the candidate Lyapunov functions as well as the controllers to be polynomials, so that the conditions in the Lyapunov theorem involve only polynomials. The Positivstellensatz delineates the exact manner to ascertain (ie. "certify") if the theorem's conditions hold. For computational reasons we further restrict the choice of certificates to those, which, with fixed Lyapunov functions and controllers, can be checked using sumofsquares optimization. Following these steps, we pose convex or coordinatewise convex (convex in one variable when the others are held fixed) iterative algorithms to search for Lyapunov functions and controllers. We provide a basic review of polynomials, the Positivstellensatz and the sumofsquares optimization results, which gives the necessary background to follow the subsequent developments that lead to our proposed algorithms. First, we consider global stability by constructing convex algorithms to search for Lyapunov functions that demonstrate semiglobal exponential stability. We then extend these algorithms in a coordinatewise convex form for both state and output feedback controller design. Additionally, we include a convex procedure to quantify a system's performance by bounding the induced norm from disturbances to outputs. Examples are included for illustration. Since we do not always desire global results, we provide two a...
Cylinders with compact crosssection and the strip conjecture
, 2008
"... The proof of the strip conjecture given just recently in [7] (see Theorem 3.1 below for a statement of the result) is best understood as a refinement of a proof of a general result for cylinders with compact crosssection (see Theorem 2.2 below). We elaborate on this statement. We give a detailed pr ..."
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Cited by 2 (1 self)
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The proof of the strip conjecture given just recently in [7] (see Theorem 3.1 below for a statement of the result) is best understood as a refinement of a proof of a general result for cylinders with compact crosssection (see Theorem 2.2 below). We elaborate on this statement. We give a detailed proof of Theorem 2.2 and we
ON THE ABSENCE OF UNIFORM DENOMINATORS IN HILBERT’S 17TH PROBLEM
, 2003
"... Abstract. Hilbert showed that for most (n, m) there exist psd forms p(x1,..., xn) of degree m which cannot be written as a sum of squares of forms. His 17th problem asked whether, in this case, there exists a form h so that h 2 p is a sum of squares of forms; that is, p is a sum of squares of ration ..."
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Cited by 1 (0 self)
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Abstract. Hilbert showed that for most (n, m) there exist psd forms p(x1,..., xn) of degree m which cannot be written as a sum of squares of forms. His 17th problem asked whether, in this case, there exists a form h so that h 2 p is a sum of squares of forms; that is, p is a sum of squares of rational functions with denominator h. We show that, for every such (n, m) there does not exist a single form h which serves in this way as a denominator for every psd p(x1,..., xn) of degree m. 1.
CLOSURE OF THE CONE OF SUMS OF 2dPOWERS IN CERTAIN WEIGHTED ℓ1SEMINORM TOPOLOGIES
"... Abstract. In [3] Berg, Christensen and Ressel prove that the closure of the cone of sums of squares ∑ R[X] 2 in the polynomial ring R[X]: = R[X1,..., Xn] in the topology induced by the ℓ1norm is equal to Pos([−1, 1] n), the cone consisting of all polynomials which are nonnegative on the hypercube ..."
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Abstract. In [3] Berg, Christensen and Ressel prove that the closure of the cone of sums of squares ∑ R[X] 2 in the polynomial ring R[X]: = R[X1,..., Xn] in the topology induced by the ℓ1norm is equal to Pos([−1, 1] n), the cone consisting of all polynomials which are nonnegative on the hypercube [−1, 1] n. The result is deduced as a corollary of a general result, also established in [3], which is valid for any commutative semigroup. In later work Berg and Maserick [5] and Berg, Christensen and Ressel [4] establish an even more general result, for a commutative semigroup with involution, for the closure of the cone of sums of squares of symmetric elements in the weighted ℓ1seminorm topology associated to an absolute value. In the present paper we give a new proof of these results which is based on Jacobi’s representation theorem [13]. At the same time, we use Jacobi’s representation theorem to extend these results from sums of squares to sums of 2dpowers, proving, in particular, that for any integer d ≥ 1, the closure of the cone of sums of 2dpowers ∑ R[X] 2d in R[X] in the topology induced by the ℓ1norm is equal to Pos([−1, 1] n).