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CERTIFIED RELAXATION FOR POLYNOMIAL OPTIMIZATION ON SEMIALGEBRAIC SETS
, 2013
"... In this paper, we describe a relaxation method to compute the minimal critical value of a real polynomial function on a semialgebraic set S and the ideal defining the points at which the minimal critical value is reached. We show that any relaxation hierarchy which is the projection of the KarushK ..."
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In this paper, we describe a relaxation method to compute the minimal critical value of a real polynomial function on a semialgebraic set S and the ideal defining the points at which the minimal critical value is reached. We show that any relaxation hierarchy which is the projection of the KarushKuhnTucker relaxation stops in a finite number of steps and the ideal defining the minimizers is generated by the kernel of the associated moment matrix in that degree. Assuming the minimizer ideal is zerodimensional, we give a new criterion to detect when the minimum is reached and we prove that this criterion is satisfied for a sufficiently high degree. This exploits new representation of positive polynomials as elements of the preordering modulo the KKT ideal, which only involves polynomials in the initial set of variables.
REPRESENTATIONS OF NONNEGATIVE POLYNOMIALS HAVING FINITELY MANY ZEROS
, 2004
"... Let K be a basic closed semialgebraic set in Rn defined by polynomial inequalities g1 ≥ 0,...,gs ≥ 0, where g1,...,gs ∈ R[x1,...,xn], and let f ∈ R[x1,...,xn]. In [12], Schmüdgen proves that, if K is compact and f is strictly positive on K, thenfbelongs to the quadratic preordering generated by g1,. ..."
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Let K be a basic closed semialgebraic set in Rn defined by polynomial inequalities g1 ≥ 0,...,gs ≥ 0, where g1,...,gs ∈ R[x1,...,xn], and let f ∈ R[x1,...,xn]. In [12], Schmüdgen proves that, if K is compact and f is strictly positive on K, thenfbelongs to the quadratic preordering generated by g1,...,gs. DenotebyMthe (smaller) quadratic module generated by g1,...,gs. Results of Putinar [8] and Jacobi [1] show that, if M is archimedean, then f>0onK implies f ∈ M. The question of exactly when M is archimedean is studied in detail in [2]. In [11], extending earlier results in the preordering case in [10], Scheiderer gives sufficient conditions for f ≥ 0onK to imply f ∈ M. In [11, Cor. 2.6], as an application of his methods, Scheiderer extends the PutinarJacobi result to include the case where f ≥ 0onK and, at each zero of f in K, the partial derivatives of f vanish and the hessian of f is positive definite. The PutinarJacobi result serves as the theoretical underpinning for an optimization algorithm based on semidefinite programming due to Lasserre; see [4] or [5]. According to the PutinarJacobi result, if M is archimedean, the minimum value of any polynomial f on K is equal to sup{c ∈ R  f − c ∈ M}. This latter number can be approximated by Lasserre’s algorithm. One is naturally interested in knowing when f − c ∈ M holds when c is the exact minimum of f on K, e.g., see [4, Th. 2.1 and Remark 2.2]. Although [11, Cor. 2.6] sheds light on this question, its usefulness is limited by the unrealistic constraints on the boundary zeros. In Section 1 we review basic terminology and results and, at the same time, we use the Basic Lemma in [3] to give a short proof of the main result in [11]. In Section 2 we prove that the constraints on the boundary zeros in [11, Cor. 2.6] can be replaced by constraints which are much less restrictive and much more natural; see Theorem 2.3. In the Appendix, we examine the Basic Lemma in [3], and we compare this result to Lemma 2.6 in [10], which is the key result in the approach taken by Scheiderer in [10] and [11]. 1. the main result in [11] Let A be a commutative ring with 1. For simplicity, assume 1
ERROR ESTIMATES IN THE OPTIMIZATION OF DEGREE TWO POLYNOMIALS ON A DISCRETE HYPERCUBE
"... Abstract. The paper considers the distribution of values Q(x), x ∈{−1, 1} n,whereQis a quadratic form in n variables with real coefficients. Error estimates are established for approximations of the maximum and minimum value of Q on {−1, 1} n which can be obtained by semidefinite programming. Bounds ..."
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Abstract. The paper considers the distribution of values Q(x), x ∈{−1, 1} n,whereQis a quadratic form in n variables with real coefficients. Error estimates are established for approximations of the maximum and minimum value of Q on {−1, 1} n which can be obtained by semidefinite programming. Bounds are given involving the sum of the absolute values of the offdiagonal entries. Other bounds are given which are useful in the case of extreme skewness. Used in conjunction with earlier bounds of Nesterov in [5], these new bounds lead to improvements on the bound given by the trace. The trigonometric description of the maximum and minimum given in [5], which is based on the rounding argument introduced by Goemans and Williamson in [1], is a major tool in obtaining these bounds. 1. introduction Let Q be a polynomial in n variables with real coefficients, S ⊆ Rn a basic closed semialgebraic set. Lasserre’s algorithm [3] produces an increasing sequence of lower bounds for Q on S computable via semidefinite programming which, in case S is compact, converges to the exact minimum of Q on S. In [4] a refinement of Lasserre’s algorithm is described which takes into account the fact that S may have dimension less than n. This involves
REPRESENTATION OF NONNEGATIVE POLYNOMIALS VIA THE KKT IDEALS
"... Abstract. This paper studies the representation of a nonnegative polynomial f on a noncompact semialgebraic set K modulo its KKT (KarushKuhnTucker) ideal. Under the assumption that f satisfies the boundary Hessian conditions (BHC) at each zero of f in K; we show that f can be represented as a s ..."
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Abstract. This paper studies the representation of a nonnegative polynomial f on a noncompact semialgebraic set K modulo its KKT (KarushKuhnTucker) ideal. Under the assumption that f satisfies the boundary Hessian conditions (BHC) at each zero of f in K; we show that f can be represented as a sum of squares (SOS) of real polynomials modulo its KKT ideal if f ≥ 0 on K. 1. introduction We see that a polynomial in one variable f(x) ∈ R[x] and f(x) ≥ 0, for all x ∈ R, then f(x) = ∑m i=1 g2 i (x), where gi(x) ∈ R[x], i.e., f is a sum of squares in R[x] (SOS for short). However, in the multivariables case, this is false. A counterexample was given by Motzkin in 1967; if f(x, y) = 1 + x4y2 + x2y4 − 3x2y2, then f(x, y) ≥ 0, for all x, y ∈ R, but f is not a SOS in R[x, y]. To overcome this, we will consider the polynomials that are positive on K, where K is a semialgebraic set in Rn. For example, Schmüdgen’s famous theorem [17] says that
Minimizing Polynomials Over Semialgebraic Sets
, 2005
"... This paper concerns a method for finding the minimum of a polynomial on a semialgebraic set, i.e., a set in R m defined by finitely many polynomial equations and inequalities, using the KarushKuhnTucker (KKT) system and sum of squares (SOS) relaxations. This generalizes results in the recent paper ..."
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This paper concerns a method for finding the minimum of a polynomial on a semialgebraic set, i.e., a set in R m defined by finitely many polynomial equations and inequalities, using the KarushKuhnTucker (KKT) system and sum of squares (SOS) relaxations. This generalizes results in the recent paper [15], which considers minimizing polynomials on algebraic sets, i.e., sets in R m defined by finitely many polynomial equations. Most of the theorems and conclusions in [15] generalize to semialgebraic sets, even in the case where the semialgebraic set is not compact. We discuss the method in some special cases, namely, when the semialgebraic set is contained in the nonnegative orthant R n + or in box constraints [a, b]n. These constraints make the computations more efficient.
Exact relaxation for polynomial optimization on semialgebraic sets
, 2014
"... In this paper, we study the problem of computing the infimum of a real polynomial function f on a closed basic semialgebraic set S and the points where this infimum is reached, if they exist. We show that when the infimum is reached, a SemiDefinite Program hierarchy constructed from the KarushKu ..."
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In this paper, we study the problem of computing the infimum of a real polynomial function f on a closed basic semialgebraic set S and the points where this infimum is reached, if they exist. We show that when the infimum is reached, a SemiDefinite Program hierarchy constructed from the KarushKuhnTucker ideal is always exact and that the vanishing ideal of the KKT minimizer points is generated by the kernel of the associated moment matrix in that degree, even if this ideal is not zerodimensional. We also show that this relaxation allows to detect when there is no KKT minimizer. Analysing the properties of the Fritz John variety, we show how to find all the minimizers of f. We prove that the exactness of the relaxation depends only on the real points which satisfy these constraints. This exploits representations of positive polynomials as elements of the preordering modulo the KKT ideal, which only involves polynomials in the initial set of variables. The approach provides a uniform treatment of different optimization problems considered previously. Applications to global optimization, optimization on semialgebraic sets defined by regular sets of constraints, optimization on finite semialgebraic sets and real radical computation are given.
REAL IDEAL AND THE DUALITY OF SEMIDEFINITE PROGRAMMING FOR POLYNOMIAL OPTIMIZATION
, 901
"... Abstract. We study the ideal generated by polynomials vanishing on a semialgebraic set and propose an algorithm to calculate the generators, which is based on some techniques of the cylindrical algebraic decomposition. By applying these, polynomial optimization problems with polynomial equality cons ..."
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Abstract. We study the ideal generated by polynomials vanishing on a semialgebraic set and propose an algorithm to calculate the generators, which is based on some techniques of the cylindrical algebraic decomposition. By applying these, polynomial optimization problems with polynomial equality constraints can be modified equivalently so that the associated semidefinite programming relaxation problems have no duality gap. Elementary proofs for some criteria on reality of ideals are also given. 1.