## Sums of Squares and Semidefinite Programming Relaxations for Polynomial Optimization Problems with Structured Sparsity (2006)

Venue: | SIAM Journal on Optimization |

Citations: | 77 - 25 self |

### BibTeX

@ARTICLE{Waki06sumsof,

author = {Hayato Waki and Sunyoung Kim and Masakazu Kojima and Masakazu Muramatsu},

title = {Sums of Squares and Semidefinite Programming Relaxations for Polynomial Optimization Problems with Structured Sparsity},

journal = {SIAM Journal on Optimization},

year = {2006},

volume = {17},

pages = {218--242}

}

### OpenURL

### Abstract

Abstract. Unconstrained and inequality constrained sparse polynomial optimization problems (POPs) are considered. A correlative sparsity pattern graph is defined to find a certain sparse structure in the objective and constraint polynomials of a POP. Based on this graph, sets of supports for sums of squares (SOS) polynomials that lead to efficient SOS and semidefinite programming (SDP) relaxations are obtained. Numerical results from various test problems are included to show the improved performance of the SOS and SDP relaxations. Key words.

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Citation Context ...em is to find a cut that has the maximum number of edges. We formulate the maxcut problem as an equality constrained POP: maximize � ⎫ (1 − xixj)/2 ⎬ {i,j}∈E (40) ⎭ subject to x 2 i = 1 (i ∈ V ). See =-=[10]-=- or [27] for details of this formulation. In [25] and [26], it is shown that a further reduction of the variables is possible in the SOS relaxation if we deal with the integer equality constraints dir... |

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