Polynomials and multilinear mappings in topological vector-spaces

Bochnak, Jacek; Sičiak, Józef

doi:10.4064/sm-39-1-59-76

Cited by 103 publications

(88 citation statements)

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“…The above inequality was shown by Lojasiewicz (see [6]), and an alternative proof can be found in Nesterov and Nemirovski [43]. Yet, it also follows from Corollary 4.2 by setting A = G T G.…”

Section: If F ∈ ℜ M D Is a D-th Order Super-symmetric Tensor With Fmentioning

confidence: 66%

Maximum Block Improvement and Polynomial Optimization

Chen¹,

He²,

Li³

et al. 2012

SIAM J. Optim.

160

177

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Abstract. In this paper we propose an efficient method for solving the spherically constrained homogeneous polynomial optimization problem. The new approach has the following three main ingredients. First, we establish a block coordinate descent type search method for nonlinear optimization, with the novelty being that we only accept a block update that achieves the maximum improvement, hence the name of our new search method: Maximum Block Improvement (MBI). Convergence of the sequence produced by the MBI method to a stationary point is proven. Second, we establish that maximizing a homogeneous polynomial over a sphere is equivalent to its tensor relaxation problem, thus we can maximize a homogeneous polynomial function over a sphere by its tensor relaxation via the MBI approach. Third, we propose a scheme to reach a KKT point of the polynomial optimization, provided that a stationary solution for the relaxed tensor problem is available. Numerical experiments have shown that our new method works very efficiently: for a majority of the test instances that we have experimented with, the method finds the global optimal solution at a low computational cost.Key words. block coordinate descent, polynomial optimization problem, tensor form AMS subject classifications. 90C26, 90C30, 15A69, 49M271. Introduction. The optimization models whose objective and constraints are polynomial functions have recently attracted much research attention. This is in part due to an increased demand on the application side (cf. the sample applications in numerical linear algebra [50,26,28], material sciences [57], quantum physics [9,18], and signal processing [16,4,53]), and in part due to its own strong theoretical appeal. Indeed, polynomial optimization is a challenging task; at the same time it is rich enough to be fruitful. For instance, even the simplest instances of polynomial optimization, such as maximizing a cubic polynomial over a sphere, is NP-hard (Nesterov [42]). However, the problem is so elementary that it can be even attempted in an undergraduate calculus class. For readers interested in polynomial optimization with simple constraints, we refer to De Klerk [12] for a survey on the computational complexity of optimizing various classes of polynomial functions over a simplex, hypercube or sphere. In particular, De Klerk et al.[13] designed a polynomial-time approximation scheme (PTAS) for minimizing polynomials of fixed degree over the simplex.So far, a few results have been obtained for approximation algorithms with guaranteed worst-case performance ratios for higher degree polynomial optimization problems. Luo and Zhang [39] derived a polynomial-time approximation algorithm to optimize a multivariate quartic polynomial over a region defined by quadratic inequalities.

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Section: If F ∈ ℜ M D Is a D-th Order Super-symmetric Tensor With Fmentioning

confidence: 66%

Maximum Block Improvement and Polynomial Optimization

Chen¹,

He²,

Li³

et al. 2012

SIAM J. Optim.

160

177

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“…(2) In general, the Lie algebra encountered in the Milnor-Moore Theorem is not the full Lie algebra of infinitesimal characters (a fact which is somewhat opaque in [10]). This is due to the fact that one needs to take the restricted dual, i.e.…”

Section: B) Hence As Locally Convex Spaces L(g(h B)) = T 1 a (G(hmentioning

confidence: 99%

Character groups of Hopf algebras as infinite-dimensional Lie groups

Bogfjellmo¹,

Dahmen²,

Schmeding³

2016

Ann. inst. Fourier

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In this article character groups of Hopf algebras are studied from the perspective of infinite-dimensional Lie theory. For a graded and connected Hopf algebra we construct an infinite-dimensional Lie group structure on the character group with values in a locally convex algebra. This structure turns the character group into a Baker-Campbell-Hausdorff-Lie group which is regular in the sense of Milnor. Furthermore, we show that certain subgroups associated to Hopf ideals become closed Lie subgroups of the character group.If the Hopf algebra is not graded, its character group will in general not be a Lie group. However, we show that for any Hopf algebra the character group with values in a weakly complete algebra is a pro-Lie group in the sense of Hofmann and Morris.

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“…While it is not hard to see that every C 1 -homomorphism between smooth Lie groups modelled on real locally convex spaces is automatically smooth [13], such a result is not available here. 4 It is therefore necessary to prove by hand that every one-parameter group Z p → G is C k…”

Section: L3mentioning

confidence: 99%