On Cones of Nonnegative Quadratic Functions

Sturm, J.F.; Zhang, Shuzhong

doi:10.1287/moor.28.2.246.14485

Cited by 270 publications

(235 citation statements)

References 19 publications

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“…A specific extraction procedure was also described in [19,Section 5] in the case of quadratic optimization problems with one (possibly non-convex) quadratic constraint, or one linear constraint jointly with one concave quadratic constraint.…”

Section: Resultsmentioning

confidence: 99%

Detecting Global Optimality and Extracting Solutions in GloptiPoly

Henrion¹,

Lasserre²

2005

Lecture Notes in Control and Information Sciences

254

296

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GloptiPoly is a Matlab/SeDuMi add-on to build and solve convex linear matrix inequality (LMI) relaxations of non-convex optimization problems with multivariate polynomial objective function and constraints, based on the theory of moments. In contrast with the dual sum-of-squares decompositions of positive polynomials, the theory of moments allows to detect global optimality of an LMI relaxation and extract globally optimal solutions. In this report, we describe and illustrate the numerical linear algebra algorithm implemented in GloptiPoly for detecting global optimality and extracting solutions. We also mention some related heuristics that could be useful to reduce the number of variables in the LMI relaxations.

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Section: Resultsmentioning

confidence: 99%

Detecting Global Optimality and Extracting Solutions in GloptiPoly

Henrion¹,

Lasserre²

2005

Lecture Notes in Control and Information Sciences

254

296

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“…Moreover, Algorithm MBI is simple to implement in this case, as optimizing one block while fixing all other blocks is a trivial problem to solve. In fact, simultaneously optimizing over two vectors of variables, while fixing other vectors, are also easy to implement; see [59,64]. In particular, if d is even, then we may partition the blocks as…”

Section: Spherically Constrained Homogeneous Polynomial Optimizationmentioning

confidence: 99%

Maximum Block Improvement and Polynomial Optimization

Chen¹,

He²,

Li³

et al. 2012

SIAM J. Optim.

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159

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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“…Later it was shown by Barvinok in [3] that the bound is essentially tight. Along a different line, Sturm and Zhang [19] proposed a matrix rank-1 decomposition scheme, leading to an algorithmic approach to finding rank-1 solutions for SDP, provided that the number of constraints in the SDP problem is small. The matrix decomposition scheme of Sturm and Zhang was extended to the complex matrix case by Huang and Zhang [11].…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, Huang and Zhang [11] also proposed a polynomial-time procedure, in the complex case, for finding a low rank solution with a bound on the rank similar to that in Barvinok [2] and Pataki [14]. In the case where the number of constraints in the SDP is small, the rank-1 decomposition method of Sturm and Zhang [19] (and Huang and Zhang [11] for the complex SDP) can be considered as an alternative polynomial-time procedure to find the low rank matrix solutions (in this case rank-1), other than the procedure suggested in Barvinok [2] and Pataki [14].…”

Section: Introductionmentioning

confidence: 99%