Abstract
In this paper, we first demonstrate that positive semidefiniteness of a large well-structured sparse symmetric matrix can be represented via positive semidefiniteness of a bunch of smaller matrices linked, in a linear fashion, to the matrix. We derive also the “dual counterpart” of the outlined representation, which expresses the possibility of positive semidefinite completion of a well-structured partially defined symmetric matrix in terms of positive semidefiniteness of a specific bunch of fully defined submatrices of the matrix. Using the representations, we then reformulate well-structured large-scale semidefinite problems into smooth convex–concave saddle point problems, which can be solved by a Prox-method developed in [6] with efficiency \(\mathcal {O}(\epsilon^{-1})\). Implementations and some numerical results for large-scale Lovász capacity and MAXCUT problems are finally presented.
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Lu, Z., Nemirovski, A. & Monteiro, R.D.C. Large-scale semidefinite programming via a saddle point Mirror-Prox algorithm. Math. Program. 109, 211–237 (2007). https://doi.org/10.1007/s10107-006-0031-2
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DOI: https://doi.org/10.1007/s10107-006-0031-2