Results 1  10
of
32
Optimization of polynomials on compact semialgebraic sets
 SIAM J. Optim
"... Abstract. A basic closed semialgebraic subset S of R n is defined by simultaneous polynomial inequalities g1 ≥ 0,..., gm ≥ 0. We give a short introduction to Lasserre’s method for minimizing a polynomial f on a compact set S of this kind. It consists of successively solving tighter and tighter conve ..."
Abstract

Cited by 43 (5 self)
 Add to MetaCart
Abstract. A basic closed semialgebraic subset S of R n is defined by simultaneous polynomial inequalities g1 ≥ 0,..., gm ≥ 0. We give a short introduction to Lasserre’s method for minimizing a polynomial f on a compact set S of this kind. It consists of successively solving tighter and tighter convex relaxations of this problem which can be formulated as semidefinite programs. We give a new short proof for the convergence of the optimal values of these relaxations to the infimum f ∗ of f on S which is constructive and elementary. In the case where f possesses a unique minimizer x ∗ , we prove that every sequence of “nearly ” optimal solutions of the successive relaxations gives rise to a sequence of points in R n converging to x ∗. 1. Introduction to Lasserre’s method Throughout the paper, we suppose 1 ≤ n ∈ N and abbreviate (X1,..., Xn) by ¯X. We let R [ ¯ X] denote the ring of real polynomials in n indeterminates. Suppose we are given a so called basic closed semialgebraic set, i.e., a set S: = {x ∈ R n  g1(x) ≥ 0,..., gm(x) ≥ 0}
Semidefinite Representations for Finite Varieties
 MATHEMATICAL PROGRAMMING
, 2002
"... We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equalities and inequalities. When the polynomial equalities have a finite number of complex solutions and define a radical ideal we can reformulate this problem as a semidefinite programming prob ..."
Abstract

Cited by 37 (7 self)
 Add to MetaCart
We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equalities and inequalities. When the polynomial equalities have a finite number of complex solutions and define a radical ideal we can reformulate this problem as a semidefinite programming problem. This semidefinite program involves combinatorial moment matrices, which are matrices indexed by a basis of the quotient vector space R[x 1 , . . . , x n ]/I. Our arguments are elementary and extend known facts for the grid case including 0/1 and polynomial programming. They also relate to known algebraic tools for solving polynomial systems of equations with finitely many complex solutions. Semidefinite approximations can be constructed by considering truncated combinatorial moment matrices; rank conditions are given (in a grid case) that ensure that the approximation solves the original problem at optimality.
Positivity sums of squares and the multidimensional moment problem
 II, Adv. Geom
, 2005
"... Abstract. Let K be the basic closed semialgebraic set in Rn defined by some finite set of polynomials S and T, the preordering generated by S. For K compact, f apolynomialinnvariables nonnegative on K and real ɛ>0, we have that f + ɛ ∈ T. In particular, the KMoment Problem has a positive solution. ..."
Abstract

Cited by 27 (7 self)
 Add to MetaCart
Abstract. Let K be the basic closed semialgebraic set in Rn defined by some finite set of polynomials S and T, the preordering generated by S. For K compact, f apolynomialinnvariables nonnegative on K and real ɛ>0, we have that f + ɛ ∈ T. In particular, the KMoment Problem has a positive solution. In the present paper, we study the problem when K is not compact. For n = 1, we show that the KMoment Problem has a positive solution if and only if S is the natural description of K (see Section 1). For n ≥ 2, we show that the KMoment Problem fails if K contains a cone of dimension 2. On the other hand, we show that if K is a cylinder with compact base, then the following property holds: (‡) ∀f ∈ R[X],f ≥ 0onK ⇒∃q ∈ T such that ∀ real ɛ>0,f + ɛq ∈ T. This property is strictly weaker than the one given in Schmüdgen (1991), but in turn it implies a positive solution to the KMoment Problem. Using results of Marshall (2001), we provide many (noncompact) examples in hypersurfaces for which (‡) holds. Finally, we provide a list of 8 open problems.
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 23 (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.
On the complexity of Putinar’s Positivstellensatz
, 2005
"... Abstract. 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 ..."
Abstract

Cited by 19 (4 self)
 Add to MetaCart
Abstract. 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. 1.
approximations of nonnegative polynomials via simple high degree perturbations
 Math. Z
"... Abstract. We show that every real polynomial f nonnegative on [−1, 1] n can be approximated in the l1norm of coefficients, by a sequence of polynomials {fεr} that are sums of squares. This complements the existence of s.o.s. approximations in the denseness result of Berg, Christensen and Ressel, as ..."
Abstract

Cited by 16 (10 self)
 Add to MetaCart
Abstract. We show that every real polynomial f nonnegative on [−1, 1] n can be approximated in the l1norm of coefficients, by a sequence of polynomials {fεr} that are sums of squares. This complements the existence of s.o.s. approximations in the denseness result of Berg, Christensen and Ressel, as we provide a very simple and explicit approximation sequence. Then we show that if the moment problem holds for a basic closed semialgebraic set KS ⊂ R n with nonempty interior, then every polynomial nonnegative on KS can be approximated in a similar fashion by elements from the corresponding preordering. Finally, we show that the degree of the perturbation in the approximating sequence depends on ɛ as well as the degree and the size of coefficients of the nonnegative polynomial f, but not on the specific values of its coefficients. 1.
Distinguished representations of nonnegative polynomials
"... Abstract. Let g1,..., gr ∈ R[x1,..., xn] such that the set K = {g1 ≥ 0,..., gr ≥ 0} in R n is compact. We study the problem of representing polynomials f with fK ≥ 0 in the form f = s0 + s1g1 + · · · + srgr with sums of squares si, with particular emphasis on the case where f has zeros in K. Assu ..."
Abstract

Cited by 13 (1 self)
 Add to MetaCart
Abstract. Let g1,..., gr ∈ R[x1,..., xn] such that the set K = {g1 ≥ 0,..., gr ≥ 0} in R n is compact. We study the problem of representing polynomials f with fK ≥ 0 in the form f = s0 + s1g1 + · · · + srgr with sums of squares si, with particular emphasis on the case where f has zeros in K. Assuming that the quadratic module of all such sums is archimedean, we establish a localglobal condition for f to have such a representation, visàvis the zero set of f in K. This criterion is most useful when f has only finitely many zeros in K. We present a number of concrete situations where this result can be applied. As another application we solve an open problem from [8] on onedimensional quadratic modules.
R.: Nfold integer programming
 Disc. Optim
"... Abstract. Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. Th ..."
Abstract

Cited by 13 (5 self)
 Add to MetaCart
Abstract. Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms. We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considerations
Convex sets with semidefinite representation. Optimization Online
, 2006
"... Abstract. We provide a sufficient condition on a class of compact basic semialgebraic sets K ⊂ R n for their convex hull co(K) to have a semidefinite representation (SDr). This SDr is explicitly expressed in terms of the polynomials gj that define K. Examples are provided. We also provide an approxi ..."
Abstract

Cited by 13 (1 self)
 Add to MetaCart
Abstract. We provide a sufficient condition on a class of compact basic semialgebraic sets K ⊂ R n for their convex hull co(K) to have a semidefinite representation (SDr). This SDr is explicitly expressed in terms of the polynomials gj that define K. Examples are provided. We also provide an approximate SDr; that is, for every fixed ɛ> 0, there is a convex set Kɛ such that co(K) ⊆ Kɛ ⊆ co(K) + ɛB (where B is the unit ball of R n), and Kɛ has an explicit SDr in terms of the gj’s. For convex and compact basic semialgebraic sets K defined by concave polynomials, we provide a simpler explicit SDr when the nonnegative Lagrangian Lf associated with K and any linear f ∈ R[X] is a sum of squares. We also provide an approximate SDr specific to the convex case. 1.
Matrix SumofSquares Relaxations for Robust SemiDefinite Programs
 Math. Program
, 2006
"... We consider robust semidefinite programs which depend polynomially or rationally on some uncertain parameter that is only known to be contained in a set with a polynomial matrix inequality description. On the basis of matrix sumofsquares decompositions, we suggest a systematic procedure to con ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
We consider robust semidefinite programs which depend polynomially or rationally on some uncertain parameter that is only known to be contained in a set with a polynomial matrix inequality description. On the basis of matrix sumofsquares decompositions, we suggest a systematic procedure to construct a family of linear matrix inequality relaxations for computing upper bounds on the optimal value of the corresponding robust counterpart. With a novel matrixversion of Putinar’s sumofsquares representation for positive polynomials on compact semialgebraic sets, we prove asymptotic exactness of the relaxation family under a suitable constraint qualification. If the uncertainty region is a compact polytope, we provide a new duality proof for the validity of Putinar’s constraint qualification with an a priori degree bound on the polynomial certificates. Finally, we point out the consequences of our results for constructing relaxations based on the socalled fullblock Sprocedure, which allows to apply recently developed tests in order to computationally verify the exactness of possibly smallsized relaxations.