Sum Of Squares Representation Research Articles

Weighted Sum-of-Squares Lower Bounds for Univariate Polynomials Imply VP≠VNP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext{VP} \ eq \ ext{VNP}$$\\end{document}

For a polynomial f,a weighted sum-of-squares representation (SOS) has the form f=∑i∈[s]cifi2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f = \\sum_{i\\in [s]} c_i f_i^2$$\\end{document}, where the weightsci\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c_i$$\\end{document} are field elements. The size of the representation is the number of monomials that appear across the fi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f_i$$\\end{document}'s.Its minimum across all such decompositions is called the support-sum S(f) of f.For a univariate polynomial f of degree d of full support,a lower bound for the support-sum is S(f)≥d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S(f) \\ge \\sqrt d$$\\end{document}.We show that the existence of an explicit univariate polynomial fwith support-sum just slightly larger than the lower bound, that is, S(f)≥d0.5+ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S(f) \\ge d^{0.5+\\varepsilon}$$\\end{document}, for some ε>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon > 0$$\\end{document},implies that VP≠\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ e$$\\end{document}VNP,the major open problem in algebraic complexity.In fact, our proof works for some subconstant functions ε(d)>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon(d) > 0$$\\end{document} as well.We also consider the sum-of-cubes representation (SOC) of polynomials. We show that an explicit hard polynomialimplies both blackbox-PIT is in P, and VP≠\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ eq$$\\end{document}VNP.

Sums of Squares Polynomial Program Reformulations for Adjustable Robust Linear Optimization Problems with Separable Polynomial Decision Rules

We show that adjustable robust linear programs with affinely adjustable box data uncertainties under separable polynomial decision rules admit exact sums of squares (SOS) polynomial reformulations. These problems share the same optimal values and admit a one-to-one correspondence between the optimal solutions. A sum of squares representation of non-negativity of a separable non-convex polynomial over a box plays a key role in the reformulation. This reformulation allows us to find adjustable robust solutions of uncertain linear programs under box data uncertainty by numerically solving their associated equivalent SOS polynomial optimization problem using semi-definite linear programming. We illustrate how the quality of the adjustable robust solution of a robust optimization problem with polynomial decision rules improves as the degree of the polynomial increases. Our results demonstrate that the adjustable robust solutions approach the actual optimal solution as the degree of the polynomial increases from one to fifteen.

Sum-of-squares chordal decomposition of polynomial matrix inequalities

We prove decomposition theorems for sparse positive (semi)definite polynomial matrices that can be viewed as sparsity-exploiting versions of the Hilbert–Artin, Reznick, Putinar, and Putinar–Vasilescu Positivstellensätze. First, we establish that a polynomial matrix P(x) with chordal sparsity is positive semidefinite for all xin mathbb {R}^n if and only if there exists a sum-of-squares (SOS) polynomial sigma (x) such that sigma P is a sum of sparse SOS matrices. Second, we show that setting sigma (x)=(x_1^2 + cdots + x_n^2)^nu for some integer nu suffices if P is homogeneous and positive definite globally. Third, we prove that if P is positive definite on a compact semialgebraic set mathcal {K}={x:g_1(x)ge 0,ldots ,g_m(x)ge 0} satisfying the Archimedean condition, then P(x) = S_0(x) + g_1(x)S_1(x) + cdots + g_m(x)S_m(x) for matrices S_i(x) that are sums of sparse SOS matrices. Finally, if mathcal {K} is not compact or does not satisfy the Archimedean condition, we obtain a similar decomposition for (x_1^2 + cdots + x_n^2)^nu P(x) with some integer nu ge 0 when P and g_1,ldots ,g_m are homogeneous of even degree. Using these results, we find sparse SOS representation theorems for polynomials that are quadratic and correlatively sparse in a subset of variables, and we construct new convergent hierarchies of sparsity-exploiting SOS reformulations for convex optimization problems with large and sparse polynomial matrix inequalities. Numerical examples demonstrate that these hierarchies can have a significantly lower computational complexity than traditional ones.

On sum of squares certificates of non-negativity on a strip

In [6] , Murray Marshall proved that every f ∈ R [ X , Y ] non-negative on the strip [ 0 , 1 ] × R can be written as f = σ 0 + σ 1 X ( 1 − X ) with σ 0 , σ 1 sums of squares in R [ X , Y ] . In this work, we present a few results concerning this representation in particular cases. First, under the assumption deg Y ⁡ f ≤ 2 , by characterizing the extreme rays of a suitable cone, we obtain a degree bound for each term. Then, we consider the case of f positive on [ 0 , 1 ] × R and non-vanishing at infinity , and we show again a degree bound for each term, coming from a constructive method to obtain the sum of squares representation. Finally, we show that this constructive method also works in the case of f having only a finite number of zeros, all of them lying on the boundary of the strip, and such that ∂ f ∂ X does not vanish at any of them.

On Sum of Squares Representation of Convex Forms and Generalized Cauchy--Schwarz Inequalities

A convex form of degree larger than one is always nonnegative since it vanishes together with its gradient at the origin. In 2007, Parrilo asked if convex forms are always sums of squares. A few ye...

Polynomial Norms

In this paper, we study polynomial norms, i.e. norms that are the $d^{\text{th}}$ root of a degree-$d$ homogeneous polynomial $f$. We first show that a necessary and sufficient condition for $f^{1/d}$ to be a norm is for $f$ to be strictly convex, or equivalently, convex and positive definite. Though not all norms come from $d^{\text{th}}$ roots of polynomials, we prove that any norm can be approximated arbitrarily well by a polynomial norm. We then investigate the computational problem of testing whether a form gives a polynomial norm. We show that this problem is strongly NP-hard already when the degree of the form is 4, but can always be answered by testing feasibility of a semidefinite program (of possibly large size). We further study the problem of optimizing over the set of polynomial norms using semidefinite programming. To do this, we introduce the notion of r-sos-convexity and extend a result of Reznick on sum of squares representation of positive definite forms to positive definite biforms. We conclude with some applications of polynomial norms to statistics and dynamical systems.

Design of Lyapunov functions for a class of homogeneous systems: Generalized forms approach

SummaryIn this paper, we provide a method to design Lyapunov functions (LFs) for a class of homogeneous systems described by functions that we call generalized forms (GFs). Homogeneous polynomial systems and several high‐order sliding modes are included in the class. The LF candidate is chosen from the same class of functions and it is parameterized in its coefficients. Since the derivative of the LF candidate along the system's trajectories is also a GF, the problem is reduced to verify positive definiteness of two GFs. We establish a procedure to represent a GF with a finite set of polynomials. Thus, the problem is changed to determine positive definiteness of a set of polynomials. Such a problem can be solved by means of Pólya's theorem or the sum of squares representation.

Design of Continuous Twisting Algorithm

For the double integrator with matched Lipschitz disturbances we propose a continuous homogeneous controller providing finite-time stability of the origin. The disturbance is compensated exactly in finite time using a discontinuous function through an integral action. Since the controller is dynamic, the closed loop is a third order system that achieves a third order sliding mode in the steady state. The stability and robustness properties of the controller are proven using a smooth and homogeneous strict Lyapunov function (LF). In a first stage, the gains of the controller and the LF are designed using a method based on Pólya’s Theorem. In a second stage the controller’s gains are adjusted through a sum of squares representation of the LF.

The entropic discriminant

The entropic discriminant is a non-negative polynomial associated to a matrix. It arises in contexts ranging from statistics and linear programming to singularity theory and algebraic geometry. It describes the complex branch locus of the polar map of a real hyperplane arrangement, and it vanishes when the equations defining the analytic center of a linear program have a complex double root. We study the geometry of the entropic discriminant, and we express its degree in terms of the characteristic polynomial of the underlying matroid. Singularities of reciprocal linear spaces play a key role. In the corank-one case, the entropic discriminant admits a sum of squares representation derived from the discriminant of a characteristic polynomial of a symmetric matrix.

The convex Positivstellensatz in a free algebra

Given a monic linear pencil L in g variables, let PL=(PL(n))n∈N where PL(n):={X∈Sng∣L(X)⪰0}, and Sng is the set of g-tuples of symmetric n×n matrices. Because L is a monic linear pencil, each PL(n) is convex with interior, and conversely it is known that convex bounded noncommutative semialgebraic sets with interior are all of the form PL. The main result of this paper establishes a perfect noncommutative Nichtnegativstellensatz on a convex semialgebraic set. Namely, a noncommutative matrix-valued polynomial p is positive semidefinite on PL if and only if it has a weighted sum of squares representation with optimal degree bounds: p=s∗s+∑jfinitefj∗Lfj, where s,fj are matrices of noncommutative polynomials of degree no greater than deg(p)2. This noncommutative result contrasts sharply with the commutative setting, where there is no control on the degrees of s,fj and assuming only p nonnegative, as opposed to p strictly positive, yields a clean Positivstellensatz so seldom that such cases are noteworthy.

Positivity, Complete Positivity, and SOS Representation

The gap between positive multivariate polynomials and the sum-of-squares (SOS) representation has resurfaced as a problem of importance and interest in control theory. A representation theorem for a bihermitian form is proved. A linear map and its dual are associated with this unique form. Complete positivity and not just positivity of the linear maps is necessary as well as sufficient for SOS representation of the form.

A positivstellensatz for non-commutative polynomials

A non-commutative polynomial which is positive on a bounded semi-algebraic set of operators has a weighted sum of squares representation. This Positivstellensatz parallels similar results in the commutative case. A broader issue is, to what extent does real semi-algebraic geometry extend to non-commutative polynomials? Our “strict" Positivstellensatz is positive news, on the opposite extreme from strict positivity would be a Real Nullstellensatz. We give an example which shows that there is no non-commutative Real Nullstellensatz along certain lines. However, we include a successful type of non-commutative Nullstellensatz proved by George Bergman.

Spectral Factorizations and Sums of Squares Representations via Semidefinite Programming

In this paper we find a characterization for when a multivariable trigonometric polynomial can be written as a sum of squares. In addition, the truncated moment problem is addressed. A numerical algorithm for finding a sum of squares representation is presented as well. In the one-variable case, the algorithm finds a spectral factorization. The latter may also be used to find inner-outer factorizations.

An algorithm for sums of squares of real polynomials

We present an algorithm to determine if a real polynomial is a sum of squares (of polynomials), and to find an explicit representation if it is a sum of squares. This algorithm uses the fact that a sum of squares representation of a real polynomial corresponds to a real, symmetric, positive semi-definite matrix whose entries satisfy certain linear equations.