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 the supports for sums of squares (SOS) polynomials that lead to efficient SOS and semidefinite program (SDP) relaxations are obtained. Numerical results from various test problems are included to show the improved performance of the SOS and SDP relaxations.
To obtain a primal-dual pair of conic programming problems having zero duality gap, two methods have been proposed: the facial reduction algorithm due to Borwein and Wolkowicz [1,2] and the conic expansion method due to Luo, Sturm, and Zhang [5]. We establish a clear relationship between them. Our results show that although the two methods can be regarded as dual to each other, the facial reduction algorithm can produce a finer sequence of faces including the feasible region. We illustrate the facial reduction algorithm in LP, SOCP and an example of SDP. A simple proof of the convergence of the facial reduction algorithm for conic programming is also presented.
Abstract.Representation of a given nonnegative multivariate polynomial in terms of a sum of squares of polynomials has become an essential subject in recent developments of sums of squares optimization and SDP (semidefinite programming) relaxation of polynomial optimization problems. We discuss effective methods to obtain a simpler representation of a "sparse" polynomial as a sum of squares of sparse polynomials by eliminating redundancy.
Abstract.A sensor network localization problem can be formulated as a quadratic optimization problem (QOP). For quadratic optimization problems, semidefinite programming (SDP) relaxation by Lasserre with relaxation order 1 for general polynomial optimization problems (POPs) is known to be equivalent to the sparse SDP relaxation by Waki et al. with relaxation order 1, except the size and sparsity of the resulting SDP relaxation problems. We show that the sparse SDP relaxation applied to the QOP is at least as strong as the Biswas-Ye SDP relaxation for the sensor network localization problem. A sparse variant of the Biswas-Ye SDP relaxation, which is equivalent to the original Biswas-Ye SDP relaxation, is also derived. Numerical results are compared with the Biswas-Ye SDP relaxation and the edge-based SDP relaxation by Wang et al.. We show that the proposed sparse SDP relaxation is faster than the Biswas-Ye SDP relaxation. In fact, the computational efficiency in solving the resulting SDP problems increases as the number of anchors and/or the radio range grow. The proposed sparse SDP relaxation also provides more accurate solutions than the edge-based SDP relaxation when exact distances are given between sensors and anchors and there are only a small number of anchors.
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