M. Putinar, "Positive polynomials on compact semi-algebraic sets", Indiana University Mathematics Journal, 42 (1993) 969--984. 17  Home/Search   Document Not in Database   Summary   Related Articles   Check

...., where u # i (x) is defined as # (u # i ) # x and analogously for u ## j (x) It is proved in [11] that program (7) is, in fact, the semidefinite dual of (5) thus, weak semidefinite duality implies that p # t p # . 10) A result in real algebraic geometry (Schmudgen [21] Putinar [20], Jacobi and Prestel [7] asserts that, when F is compact (and satisfies some additional technical condition) then every positive polynomial on F has a decomposition (6) This result implies the asymptotic convergence of the lower bounds # # t , p # t to p # as t goes to infinity [11] In the ....

M. Putinar. Positive polynomials on compact semi-algebraic sets. Indiana University Mathematics Journal, 42:969--984, 1993.

.... aim at representing the polynomial g 0 (x) p # , nonnegative on the (semi algebraic) feasible set K of P, as a linear combination of the g i s with weights being polynomials that are sums of squares, as in Putinar s representation of polynomials, strictly positive on a semi algebraic set [19]. In brief, the primal LMI relaxations of P are relaxations of the problem (equivalent to P) p # = min g 0 d (K) 1 , where the minimum is taken over all the probability measures on the feasible set K of P, whereas the dual relaxations # i solve # i , q k # # i = ....

M. Putinar. Positive polynomials on compact semi-algebraic sets. Indiana University Mathematics Journal, Vol. 42, pp. 969--984, 1993.

.... aim at representing the polynomial g 0 (x) p # , nonnegative on the (semi algebraic) feasible set K of P, as a linear combination of the g i s with weights being polynomials that are sums of squares, as in Putinar s representation of polynomials, strictly positive on a semi algebraic set [Put93]. In brief, the primal LMI relaxations of P are relaxations of the problem (equivalent to P) p # = min g 0 d (K) 1 , where the minimum is taken over all the probability measures on the feasible set K of P, whereas the dual relaxations # i solve max # i , q k # g ....

M. Putinar. Positive polynomials on compact semi-algebraic sets. Indiana University Mathematics Journal, Vol. 42, pp. 969--984, 1993.

.... matrix subject to LMI constraints: subject to X (2) where is here an a#ne subset of the semidefinite cone (a LMI) In this case also, minimizing the nuclear norm #X# # of X will produce excellent approximate solutions (see [FHB00] In this paper, using results by [Cas84] Sho87] [Put93], CLR95] Nes00] Las01] PS01] and [Las02] we show that the MinCard(x) and MinRank(X) problems in (1) and (2) are equivalent to large semidefinite programs (see [NN94] To be precise, based on a reformulation a la [Sho87] of problems (1) and (2) we use the technique in [Las02] to produce a ....

....and the sum of squares representation of positive polynomials. We also summarize the application of these representations to semialgebraic problems. In section 3, we show that both the MinCard(x) and the MinRank(X) problems are equivalent to large scale semidefinite programs. Based on the work by [Put93], Nes00] and [Las02] we explicitly construct in section 4 a sequence of semidefinite programs solving problems (1) and (2) We also show how the problem of finding optimal convex lower bounds on the objective function can be represented in a similar way. Finally, in section 5, we discuss the ....

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M. Putinar, Positive polynomials on compact semi-algebraic sets, Indiana University Mathematics Journal 42 (1993), no. 3, 969--984.

.... the computation of sum of squares decompositions, a technique pioneered by Shor [Sho87] see also [Nes00] Distinguished representations: Let # be the set of polynomials that can be written as sums of squares, i.e. # : s # R[x] s = i q i (x) q i R[x] Recent work by Putinar [Put93], Jacobi Prestel [JP01] and others (see [PD01] for an up to date detailed treatment of most available results) establish that if the polynomial f(x) is strictly positive over a set as in (1) that is compact, then under some mild additional assumptions, there exists an a#ne representation: ....

M. Putinar. Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J., 42(3):969--984, 1993.

....m 2n 1 # # # also be positive semidefinite. In multiple dimensions, it is generally unknown which are the exact necessary and su#cient conditions for M# to be a valid moment sequence, when we are working over a general domain. For a wide class of domains, however, Schmudgen [10] and Putinar [8] find such conditions. We review this work briefly, and use it to derive the necessary and su#cient conditions for M# to be a moment sequence. An Operator Approach Given a closed subset# of R , a sequence of numbers M# defines a valid moment sequence if there exists a measure such ....

.... The Motzkin polynomial in R P (x, y, z) x x z z is an example that shows that in higher dimensions, the sum of squares decomposition of a nonnegative polynomial is not in general possible (see Parrilo [7] or Reznick [9] for details) However, Schmudgen [10] and Putinar [8] give a representation of all polynomials that are nonnegative over a compact finitely generated semialgebraic set K, as defined in the theorem below. This leads to necessary and su#cient conditions for a moment sequence to be valid on K. Theorem 2 (Putinar [8] Suppose K : f i ....

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Putinar, M., "Positive Polynomials on Compact Semi-algebraic Sets," Indiana University Math. J., Vol 42, (3) 1993.

....sequences ( i#I g i ) y for all I [1, m] Thus what Schmudgen shows is that both sets (F ) and # (g 1 , g m ) have the same sets of nonnegative linear functionals. From this follows that every polynomial p which is positive on F belongs to # (g 1 , g m ) Putinar [Pu93] shows the following stronger result. Theorem 15. Let F be a compact semi algebraic set as in (43) Assume that there exists a polynomial u . g m# for which the set 0 is compact. If p is a polynomial positive on F , then p . As we see below, this result plays a ....

M. Putinar. Positive polynomials on compact semi-algebraic sets. Indiana University Mathematics Journal, 42:969--984, 1993.

....( Q i2I g i ) y for all I [1; m] Thus what Schmudgen shows is that both sets P (F ) and Sigma 2 (g 1 ; g m ) have the same sets of nonnegative linear functionals. From this follows that every polynomial p which is positive on F belongs to Sigma 2 (g 1 ; g m ) Putinar [Pu93] shows the following stronger result. Theorem 15. Let F be a compact semi algebraic set as in (43) Assume that there exists a polynomial u 2 Sigma 2 g 1 Sigma 2 : g m Sigma 2 for which the set fx 2 R n j u(x) 0g is compact. If p is a polynomial positive on F , then p 2 ....

M. Putinar. Positive polynomials on compact semi-algebraic sets. Indiana University Mathematics Journal, 42:969--984, 1993.

....of moments and recent results on the representation of polynomials that are strictly positive on a compact semi algebraic set. For results on the theory of moments and the representation of positive polynomials, the reader is referred to Curto and Fialkow [2] Berg [1] Schmudgen [11] Putinar [10], Jacobi [5] Jacobi and Prestel [6] It turns out that this theory is a natural and appropriate tool for global optimization since g 0 #x# # p # is precisely a (non strictly) positive polynomial, a feature that distinguishes p # from the other (local) minima. Moreover, the LMI relaxations ....

....# (13) for a finite family of polynomials #q j #x##, j # 1####r 0 ,and #q kj #x##, j # 1####r k , k # 1####m. In fact, a necessary and sufficient condition for the representation (13) to exist is that there exists a polynomial u#x# of the form (13) such that #u#x# # 0# is compact (see Putinar [10] and Jacobi [5] For instance, the representation (13) holds whenever #g k #x# # 0# is compact for some k, or when all the g k #x# are linear and # is compact. In particular, it holds for every 0 1 program. Indeed, write the integral constraints as x 2 i # x i # 0andx i # x 2 i # 0, for ....

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M. PUTINAR, Positive polynomials on compact semialgebraic sets, Ind. Univ. Math. J. 42 (1993), pp. 969--984.

....the real holomorphy ring of a field. In [2] Becker and Schwartz give a short elementary proof in the commutative case. In his solution of the moment problem in [8] Schmudgen gives a representation of polynomials strictly positive on a bounded basic closed semi algebraic set in R n . Putinar [7] gives a criterion for linear representations to exist. Jacobi and Prestel [5] show how Schmudgen s representation can be improved and determine exactly when the linear representations considered by Putinar are possible. Schmudgen and Putinar use methods from functional analysis. In [9] Wormann ....

M. Putinar, Positive polynomials on compact semi-algebraic sets, Ind. Univ. Math. J. 42 (1993), 969--984.

....a standard tool in real algebraic geometry; see [2] 5] 7] In his solution of the K moment problem in [11] Schmudgen proves a surprisingly strong version of the Positivstellensatz in the compact case. Schmudgen s result has since been extended and improved in various ways; see [1] 4] 6] 9] [10]. In the present paper we describe an extension in another direction, to the non compact case. Let V be an algebraic set in R n . The coordinate ring R[V ] of V is the ring of all polynomial functions f : V R. R[V ] is generated as an R algebra by x 1 ; x n where x i : V R denotes the ....

M. Putinar, Positive polynomials on compact semi-algebraic sets, Ind. Univ. Math. J. 42 (1993), 969--984.

.... traces back to work of Shor [16] 17] and is further developed by Parrilo [10] and by Parrilo and Sturmfels [11] and by Lasserre [7] 8] In [7] 8] Lasserre describes an extension of the method to minimizing a polynomial on an arbitrary basic closed semialgebraic set and uses a result due to Putinar [13] to prove that the method produces the exact minimum in the compact case. In the general case it produces a lower bound for the minimum. The ideas involved come from three branches of mathematics: algebraic geometry (positive polynomials) functional analysis (the moment problem) and optimization. ....

....L p (g) g(p) Suppose p K S . Then f(p) L p (f) Thus f # . The inequality f is obvious. If f M S then, #L # 0. Since L is linear and L(1) 1 this implies L(f) 0, i.e. L(f) # for any such #. This proves f . The following result is due to Putinar [13]. Jacobi gives another proof in [4] based on an extension of the Kadison Dubois Theorem. 1.2 Theorem [13] Suppose K S is compact and r M S for some real number r. Then, for any f R[x] f 0 on K S M S . If K S is compact then K S is completely inside some big ball centered at the ....

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M. Putinar, Positive polynomials on compact semialgebraic sets, Ind. Univ. Math. J. 43/3 (1993), 969--984.

.... we shall describe the structure of all polynomial functions that are positive on a semi algebraic set given by a simultaneous system of polynomial inequalities (including, after homogenization, the points at infinity) Similar results were obtained, in the case of compact semi algebraic sets, in [10]. In fact we work with homogeneous polynomials and sets given by homogeneous inequalities. This restriction is imposed by the proof and by the results of Section 3. However, for continuity with the previous section, we state all results for nonhomogeneous polynomials. 1102 MIHAI PUTINAR AND ....

M. Putinar, Positive polynomials on compact semi-algebraic sets, Indiana Univ. Math. J. 42 (1993), 969--984.

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M. Putinar, "Positive polynomials on compact semi-algebraic sets", Indiana University Mathematics Journal, 42 (1993) 969--984. 17

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M. Putinar, "Positive polynomials on compact semi-algebraic sets", Indiana University Mathematics Journal, 42 (1993) 969--984.

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M. Putinar, "Positive polynomials on compact semi-algebraic sets", Indiana University Mathematics Journal, 42 (1993) 969--984.

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M. Putinar. Positive polynomials on compact semi-algebraic sets. Indiana University Mathematics Journal, Vol. 42, pp. 969--984, 1993.

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M. Putinar, Positive polynomials on compact semi-algebraic sets, Ind. Univ. Math. J., 42 (1993), pp. 969--984.

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M. Putinar. Positive polynomials on compact semi-algebraic sets. Indiana University Mathematics Journal, 42:969--984, 1993.

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M. Putinar. Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J. 42 (1993), 969--984.

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M. Putinar, "Positive polynomials on compact semi-algebraic sets", Indiana University Mathematics Journal, 42 (1993) 969--984.

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M. Putinar, "Positive polynomials on compact semi-algebraic sets", Indiana University Mathematics Journal, 42 (1993) 969--984. 17

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M. Putinar. Positive polynomials on compact semi-algebraic sets. Ind. Univ. Math. J., 42:969--984, 1993.

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M. Putinar, "Positive polynomials on compact semi-algebraic sets", Indiana University Mathematics Journal, 42 (1993) 969--984.

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M. Putinar. Positive polynomials on compact semialgebraic sets. Indiana University Mathematical Journal, 42:969--984, 1993.

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M. Putinar. Positive polynomials on compact semi-algebraic sets. Ind. Univ. Math. J., 42:969--984, 1993.

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M. Putinar. Positive polynomials on compact semialgebraic sets. Indiana. Univ. Math. J., 42:969--984, 1993.

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M. Putinar, "Positive polynomials on compact semi-algebraic sets", Indiana University Mathematics Journal, 42 (1993) 969--984.

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M. Putinar, "Positive polynomials on compact semi-algebraic sets", Indiana University Mathematics Journal, 42 (1993) 969--984.

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M. Putinar, Positive polynomials on compact semi-algebraic sets, Ind. Univ. Math. J. 42/3 (1993), 969--984.

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