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.

We consider the problem of learning dissimilarities between points via formulations which preserve a specified ordering between points rather than the numerical values of the dissimilarities. Dissimilarity ranking (dranking) learns from instances like “A is more similar to B than C is to D ” or “The distance between E and F is larger than that between G and H”. Three formulations of d-ranking problems are presented and new algorithms are presented for two of them, one by semidefinite programming (SDP) and one by quadratic programming (QP). Among the novel capabilities of these approaches are outof-sample prediction and scalability to large problems. 1.

. Symmetricity of an optimal solution of SemiDefinite Program (SDP) with certain symmetricity is discussed based on symmetry property of the central path that is traced by a primaldual interior-point method. Introducing some operators for rearranging elements of matrices and vectors, three types of symmetric SDPs are defined by using those operators. The symmetricity of the solution on the central path is proved for each of symmetric SDPs. Therefore, it is theoretically guaranteed that a symmetric optimal solution is always obtained by using a primal-dual interiorpoint method even if there are other asymmetric optimal solutions. As an application of this result, we consider topology optimization problems of symmetric trusses that belong to one of the three types of symmetric SDPs, and we shall show that the symmetric optimal solution can be found regardless of the choice of member numbering and coordinate systems. Numerical experiments by using several algorithms for SDP illustrate rap...

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

Abstract. A basic framework for exploiting sparsity via positive semidefinite matrix completion is presented for an optimization problem with linear and nonlinear matrix inequalities. The sparsity, characterized with a chordal graph structure, can be detected in the variable matrix or in a linear or nonlinear matrix-inequality constraint of the problem. We classify the sparsity in two types, the domain-space sparsity (d-space sparsity) for the symmetric matrix variable in the objective and/or constraint functions of the problem, which is required to be positive semidefinite, and the range-space sparsity (r-space sparsity) for a linear or nonlinear matrix-inequality constraint of the problem. Four conversion methods are proposed in this framework: two for exploiting the d-space sparsity and the other two for exploiting the r-space sparsity. When applied to a polynomial semidefinite program (SDP), these conversion methods enhance the structured sparsity of the problem called the correlative sparsity. As a result, the resulting polynomial SDP can be solved more effectively by applying the sparse SDP relaxation. Preliminary numerical results on the conversion methods for quadratic semidefinite programs indicate their potential for improving the efficiency of solving various

This paper is a continuation of the work in [11] and [2] on the problem of estimating by a linear estimator, N unobservable input vectors, undergoing the same linear transformation, from noise-corrupted observable output vectors. Whereas in the aforementioned papers, only the matrix representing the linear transformation was assumed uncertain, here we are concerned with the case in which the second order statistics of the noise vectors (i.e., their covariance matrices) are also subjected to uncertainty. We seek a robust mean-squared error estimator immuned against both sources of uncertainty. We show that the optimal robust mean-squared error estimator has a special form represented by an elementary block circulant matrix, and moreover when the uncertainty sets are ellipsoidal-like, the problem of finding the optimal estimator matrix can be reduced to solving an explicit semidefinite programming problem, whose size is independent of N.

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