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51
On the complexity of Putinar’s Positivstellensatz
, 2008
"... Let S = {x ∈ R n  g1(x) ≥ 0,..., gm(x) ≥ 0} be a basic closed semialgebraic set defined by real polynomials gi. Putinar’s Positivstellensatz says that, under a certain condition stronger than compactness of S, every real polynomial f positive on S posesses a representation f = ∑ m i=0 σigi wher ..."
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Cited by 39 (8 self)
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Let S = {x ∈ R n  g1(x) ≥ 0,..., gm(x) ≥ 0} be a basic closed semialgebraic set defined by real polynomials gi. Putinar’s Positivstellensatz says that, under a certain condition stronger than compactness of S, every real polynomial f positive on S posesses a representation f = ∑ m i=0 σigi where g0: = 1 and each σi is a sum of squares of polynomials. Such a representation is a certificate for the nonnegativity of f on S. We give a bound on the degrees of the terms σigi in this representation which depends on the description of S, the degree of f and a measure of how close f is to having a zero on S. As a consequence, we get information about the convergence rate of Lasserre’s procedure for optimization of a polynomial subject to polynomial constraints.
BiQuadratic Optimization over Unit Spheres and Semidefinite Programming Relaxations
, 2008
"... Abstract. This paper studies the socalled biquadratic optimization over unit spheres min x∈R n,y∈R m bijklxiyjxkyl ..."
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Cited by 32 (15 self)
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Abstract. This paper studies the socalled biquadratic optimization over unit spheres min x∈R n,y∈R m bijklxiyjxkyl
Sum of squares methods for sensor network localization
, 2006
"... We formulate the sensor network localization problem as finding the global minimizer of a quartic polynomial. Then sum of squares (SOS) relaxations can be applied to solve it. However, the general SOS relaxations are too expensive to implement for large problems. Exploiting the special features of t ..."
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Cited by 28 (3 self)
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We formulate the sensor network localization problem as finding the global minimizer of a quartic polynomial. Then sum of squares (SOS) relaxations can be applied to solve it. However, the general SOS relaxations are too expensive to implement for large problems. Exploiting the special features of this polynomial, we propose a new structured SOS relaxation, and discuss its various properties. When distances are given exactly, this SOS relaxation often returns true sensor locations. At each step of interior point methods solving this SOS relaxation, the complexity is O(n 3), where n is the number of sensors. When the distances have small perturbations, we show that the sensor locations given by this SOS relaxation are accurate within a constant factor of the perturbation error under some technical assumptions. The performance of this SOS relaxation is tested on some randomly generated problems.
Global optimization of polynomials using gradient tentacles and sums of squares
 SIAM Journal on Optimization
"... We consider the problem of computing the global infimum of a real polynomial f on R n. Every global minimizer of f lies on its gradient variety, i.e., the algebraic subset of R n where the gradient of f vanishes. If f attains a minimum on R n, it is therefore equivalent to look for the greatest low ..."
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Cited by 26 (0 self)
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We consider the problem of computing the global infimum of a real polynomial f on R n. Every global minimizer of f lies on its gradient variety, i.e., the algebraic subset of R n where the gradient of f vanishes. If f attains a minimum on R n, it is therefore equivalent to look for the greatest lower bound of f on its gradient variety. Nie, Demmel and Sturmfels proved recently a theorem about the existence of sums of squares certificates for such lower bounds. Based on these certificates, they find arbitrarily tight relaxations of the original problem that can be formulated as semidefinite programs and thus be solved efficiently. We deal here with the more general case when f is bounded from below but does not necessarily attain a minimum. In this case, the method of Nie, Demmel and Sturmfels might yield completely wrong results. In order to overcome this problem, we replace the gradient variety by larger semialgebraic subsets of R n which we call gradient tentacles. It now gets substantially harder to prove the existence of the necessary sums of squares certificates.
Sparse SOS relaxations for minimizing functions that are summations of small polynomials
 SIAM Journal On Optimization
, 2008
"... This paper discusses how to find the global minimum of functions that are summations of small polynomials (“small ” means involving a small number of variables). Some sparse sum of squares (SOS) techniques are proposed. We compare their computational complexity and lower bounds with prior SOS relaxa ..."
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Cited by 23 (4 self)
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This paper discusses how to find the global minimum of functions that are summations of small polynomials (“small ” means involving a small number of variables). Some sparse sum of squares (SOS) techniques are proposed. We compare their computational complexity and lower bounds with prior SOS relaxations. Under certain conditions, we also discuss how to extract the global minimizers from these sparse relaxations. The proposed methods are especially useful in solving sparse polynomial system and nonlinear least squares problems. Numerical experiments are presented, which show that the proposed methods significantly improve the computational performance of prior methods for solving these problems. Lastly, we present applications of this sparsity technique in solving polynomial systems derived from nonlinear differential equations and sensor network localization. Key words: Polynomials, sum of squares (SOS), sparsity, nonlinear least squares, polynomial system, nonlinear differential equations, sensor network localization 1
An exact Jacobian SDP relaxation for polynomial optimization
 Mathematical Programming, Series A
"... Given polynomials f(x), gi(x), hj(x), we study how to minimize f(x) on the set S = {x ∈ Rn: h1(x) = · · · = hm1(x) = 0, g1(x) ≥ 0,..., gm2(x) ≥ 0}. Let fmin be the minimum of f on S. Suppose S is nonsingular and fmin is achievable on S, which are true generically. This paper proposes a new t ..."
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Cited by 21 (7 self)
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Given polynomials f(x), gi(x), hj(x), we study how to minimize f(x) on the set S = {x ∈ Rn: h1(x) = · · · = hm1(x) = 0, g1(x) ≥ 0,..., gm2(x) ≥ 0}. Let fmin be the minimum of f on S. Suppose S is nonsingular and fmin is achievable on S, which are true generically. This paper proposes a new type semidefinite programming (SDP) relaxation which is the first one for solving this problem exactly. First, we construct new polynomials ϕ1,..., ϕr, by using the Jacobian of f, hi, gj, such that the above problem is equivalent to min x∈Rn f(x) s.t. hi(x) = 0, ϕj(x) = 0, 1 ≤ i ≤ m1, 1 ≤ j ≤ r, g1(x) ν1 · · · gm2(x)νm2 ≥ 0, ∀ν ∈ {0, 1}m2. Second, we prove that for all N big enough, the standard Nth order Lasserre’s SDP relaxation is exact for solving this equivalent problem, that is, its optimal value is equal to fmin. Some variations and examples are also shown.
Semidefinite approximations for global unconstrained polynomial optimization
 SIAM J. OPTIM
, 2005
"... We consider the problem of minimizing a polynomial function on R n, known to be hard even for degree 4 polynomials. Therefore approximation algorithms are of interest. Lasserre [15] and Parrilo [23] have proposed approximating the minimum of the original problem using a hierarchy of lower bounds ob ..."
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Cited by 21 (0 self)
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We consider the problem of minimizing a polynomial function on R n, known to be hard even for degree 4 polynomials. Therefore approximation algorithms are of interest. Lasserre [15] and Parrilo [23] have proposed approximating the minimum of the original problem using a hierarchy of lower bounds obtained via semidefinite programming relaxations. We propose here a method for computing tight upper bounds based on perturbing the original polynomial and using semidefinite programming. The method is applied to several examples.
Representations of positive polynomials on noncompact semialgebraic sets via KKT ideals
, 2006
"... This paper studies the representation of a positive polynomial f(x) on a noncompact semialgebraic set S = {x ∈ R n: g1(x) ≥ 0, · · · , gs(x) ≥ 0} modulo its KKT (KarushKuhnTucker) ideal. Under the assumption that the minimum value of f(x) on S is attained at some KKT point, we show that f(x) ..."
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Cited by 19 (4 self)
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This paper studies the representation of a positive polynomial f(x) on a noncompact semialgebraic set S = {x ∈ R n: g1(x) ≥ 0, · · · , gs(x) ≥ 0} modulo its KKT (KarushKuhnTucker) ideal. Under the assumption that the minimum value of f(x) on S is attained at some KKT point, we show that f(x) can be represented as sum of squares (SOS) of polynomials modulo the KKT ideal if f(x)> 0 on S; furthermore, when the KKT ideal is radical, we have that f(x) can be represented as sum of squares (SOS) of polynomials modulo the KKT ideal if f(x) ≥ 0 on S. This is a generalization of results in [18], which discuss the SOS representations of nonnegative polynomials over gradient ideals. Key words: Polynomials, semialgebraic set, sum of squares (SOS), KarushKuhnTucker (KKT) system, KKT ideal. 1
Certifying Convergence of Lasserre’s Hierarchy via Flat Truncation
, 2013
"... Abstract. Consider the optimization problem of minimizing a polynomial function subject to polynomial constraints. A typical approach for solving it globally is applying Lasserre’s hierarchy of semidefinite relaxations, based on either Putinar’s or Schmüdgen’s Positivstellensatz. A practical questi ..."
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Cited by 14 (7 self)
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Abstract. Consider the optimization problem of minimizing a polynomial function subject to polynomial constraints. A typical approach for solving it globally is applying Lasserre’s hierarchy of semidefinite relaxations, based on either Putinar’s or Schmüdgen’s Positivstellensatz. A practical question in applications is: how to certify its convergence and get minimizers? In this paper, we propose flat truncation as a certificate for this purpose. Assume the set of global minimizers is nonempty and finite. Our main results are: i) Putinar type Lasserre’s hierarchy has finite convergence if and only if flat truncation holds, under some generic assumptions; the same conclusion holds for the Schmüdgen type one under weaker assumptions. ii) Flat truncation is asymptotically satisfied for Putinar type Lasserre’s hierarchy if the archimedean condition holds; the same conclusion holds for the Schmüdgen type one if the feasible set is compact. iii) We show that flat truncation can be used as a certificate to check exactness of standard SOS relaxations and Jacobian SDP relaxations.
Discriminants and Nonnegative Polynomials
, 2010
"... For a semialgebraic set K in R n, let Pd(K) = {f ∈ R[x]≤d: f(u) ≥ 0 ∀ u ∈ K} be the cone of polynomials in x ∈ R n of degrees at most d that are nonnegative on K. This paper studies the geometry of its boundary ∂Pd(K). When K = R n and d is even, we show that its boundary ∂Pd(K) lies on the irredu ..."
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Cited by 14 (3 self)
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For a semialgebraic set K in R n, let Pd(K) = {f ∈ R[x]≤d: f(u) ≥ 0 ∀ u ∈ K} be the cone of polynomials in x ∈ R n of degrees at most d that are nonnegative on K. This paper studies the geometry of its boundary ∂Pd(K). When K = R n and d is even, we show that its boundary ∂Pd(K) lies on the irreducible hypersurface defined by the discriminant ∆(f) of f. When K = {x ∈ R n: g1(x) = · · · = gm(x) = 0} is a real algebraic variety, we show that ∂Pd(K) lies on the hypersurface defined by the discriminant ∆(f, g1,..., gm) of f, g1,...,gm. When K is a general semialgebraic set, we show that ∂Pd(K) lies on a union of hypersurfaces defined by the discriminantal equations. Explicit formulae for the degrees of these hypersurfaces and discriminants are given. We also prove that typically Pd(K) does not have a barrier of type − log ϕ(f) when ϕ(f) is required to be a polynomial, but such a barrier exits if ϕ(f) is allowed to be semialgebraic. Some illustrating examples are shown.