## Semidefinite approximations for global unconstrained polynomial optimization

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Venue: | SIAM J. Optim |

Citations: | 15 - 0 self |

### BibTeX

@ARTICLE{Jibetean_semidefiniteapproximations,

author = {Dorina Jibetean and Monique Laurent},

title = {Semidefinite approximations for global unconstrained polynomial optimization},

journal = {SIAM J. Optim},

year = {}

}

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### Abstract

Abstract. 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. We consider the problem:

### Citations

10897 |
Computers and Intractability: A Guide to the Theory of NP- Completeness
- Garey, Johnson
- 1979
(Show Context)
Citation Context ...jx 2 i x2 j ≥ 0 for all x ∈ Rn , i.e., if p∗ = 0. Alternatively, problem (1) contains the problem of deciding whether an integer sequence a1,...,an can be partitioned, which is known to be NPcomplete =-=[7]-=-, with a1,...,an being partitionable if there exists x ∈{±1} n such that aT x = 0, i.e., if the infimum of the polynomial p(x) :=(aTx) 2 + �n i=1 (x2i − 1)2 is equal to 0. 1.1. Some known approaches t... |

346 |
Using Algebraic Geometry
- Cox, Little, et al.
- 2005
(Show Context)
Citation Context ...sed for determining the real solutions to this system of polynomial equations; e.g., using Gröbner bases and the eigenvalue method, using resultants and discriminants, or homotopy methods (see, e.g., =-=[3]-=-; see [25] for a discussion and comparison). However, there are several difficulties with such an approach. It is computationally expensive (e.g., computing a Gröbner basis may be computationally very... |

205 | Ideals, Varieties and Algorithms - Cox, Little, et al. |

98 | Gloptipoly: global optimization over polynomials with matlab and sedumi
- Henrion, J
(Show Context)
Citation Context ...isk (like 6∗ ), this means that we have rescaled the problem for SeDuMi (setting pars.scaling = [1 10]). (This is advised when the expected solutions have large entries; see the manual for GloptiPoly =-=[10]-=-. Without rescaling, the solution returned by GloptiPoly is approximatively 1, which is the value of p at the point (0,0) of Vλ, and thus not the true minimum.) Recall that |S3| = 10, |S4| = 15, |S5| ... |

47 | The truncated complex K-moment problem
- Curto, Fialkow
(Show Context)
Citation Context ...shows that Mk(y) � 0. If the support of μ is contained in {x | h(x) ≥ 0}, one can verify that pT Mk−d(hy)p = � p(x) 2h(x)dμ(x) ≥ 0 for all p ∈ RSk−d , which shows that Mk−d(hy) � 0. Curto and Fialkow =-=[4, 5]-=- prove some results showing that, under some rank condition, the necessary conditions from the above lemma are also sufficient for the existence of a representing measure. A key notion is that of “fla... |

47 | Detecting global optimality and extracting solutions in gloptipoly
- Henrion, Lasserre
- 2005
(Show Context)
Citation Context ...en the ideal Igrad generated by the polynomials ∂p (i =1,...,n) is radical (see [6]). (By case ∂xi (II) there is finite convergence of p∗ k to p∗ when Igrad is zero-dimensional.) Henrion and Lasserre =-=[11]-=- gave the following stopping criterion: If the optimum solution y to (4) satisfies the rank condition (6) rank Mk(y) = rank Mk−d(y), where d := max(d1,...,dℓ), then p ∗ k = p∗ . See section 2.2 for de... |

42 |
Solution of the truncated complex moment problem for flat data
- Curto, Fialkow
- 1996
(Show Context)
Citation Context ...shows that Mk(y) � 0. If the support of μ is contained in {x | h(x) ≥ 0}, one can verify that pT Mk−d(hy)p = � p(x) 2h(x)dμ(x) ≥ 0 for all p ∈ RSk−d , which shows that Mk−d(hy) � 0. Curto and Fialkow =-=[4, 5]-=- prove some results showing that, under some rank condition, the necessary conditions from the above lemma are also sufficient for the existence of a representing measure. A key notion is that of “fla... |

28 | Minimizing polynomials via sum of squares over the gradient ideal”, Mathematical programming
- Nie, Demmel, et al.
- 2006
(Show Context)
Citation Context ...n | ∂p (x) =0(i =1,...,n)}. Then there is asymptotic ∂xi convergence of ρ ∗ k and p∗ k to p∗ , and finite convergence when the ideal Igrad generated by the polynomials ∂p (i =1,...,n) is radical (see =-=[6]-=-). (By case ∂xi (II) there is finite convergence of p∗ k to p∗ when Igrad is zero-dimensional.) Henrion and Lasserre [11] gave the following stopping criterion: If the optimum solution y to (4) satisf... |

19 | Global minimization of a multivariate polynomial using matrix methods
- Hanzon, Jibetean
- 2010
(Show Context)
Citation Context ...BAL UNCONSTRAINED POLYNOMIAL MINIMIZATION 491 this approach applies only if the polynomial p attains its minimum. We will come back to this type of approach later in this section. Hanzon and Jibetean =-=[9]-=- (see also Jibetean [12]) proposed going around these difficulties by considering a perturbation � n� � (2) pλ(x):=p(x)+λ of the original polynomial p for small λ>0. Set i=1 p ∗ λ := inf pλ(x). x∈Rn x... |

11 |
Algebraic Optimization with Applications to System Theory
- Jibetean
- 2003
(Show Context)
Citation Context ...OMIAL MINIMIZATION 491 this approach applies only if the polynomial p attains its minimum. We will come back to this type of approach later in this section. Hanzon and Jibetean [9] (see also Jibetean =-=[12]-=-) proposed going around these difficulties by considering a perturbation � n� � (2) pλ(x):=p(x)+λ of the original polynomial p for small λ>0. Set i=1 p ∗ λ := inf pλ(x). x∈Rn x 2m+2 i Thus, p∗ ≤ p∗ λ ... |

9 | P.: On the equivalence of algebraic approaches to the minimization of forms on the simplex
- Klerk, Laurent, et al.
- 2005
(Show Context)
Citation Context .... Hence, ρ∗ k = p∗ L,k = p∗B = 0, implying that p can be written as p = u+(1− � i x2i )v, where u, v are sums of squares. As p is homogeneous, this implies easily that p must be a sum of squares (see =-=[14]-=-), yielding a contradiction.) Let us illustrate this in our current example. Table 7c shows the values p∗ L,k obtained for the moment relaxations (4) for the minimum p∗B of p over the unit ball. Recal... |

7 |
L.: Computing global minima to polynomial optimization problems using gröbner bases
- Hägglöf, Lindberg, et al.
- 1995
(Show Context)
Citation Context ... p(x) :=(aTx) 2 + �n i=1 (x2i − 1)2 is equal to 0. 1.1. Some known approaches to polynomial unconstrained minimization. An approach followed by some authors (e.g., by Hägglöf, Lindberg, and Stevenson =-=[8]-=-) is to look at the first order conditions ∂p/∂xi =0(i =1,...,n). Various algebraic techniques can be used for determining the real solutions to this system of polynomial equations; e.g., using Gröbne... |

2 |
A test for copositive matrices
- Kaplan
- 2000
(Show Context)
Citation Context ...opositive when q is nonnegative. Testing matrix copositivity is a co-NP-complete problem [21]. Although some necessary and sufficient conditions for the copositivity of a matrix are known (see, e.g., =-=[13]-=-), their algorithmic application is computationally too expensive. An alternative consists therefore of using numerical algorithms for testing (non)copositivity. Parrilo [22, 23] introduced the follow... |