Exploiting Sparsity in Semide nite Programming via Matrix Completion I : General Framework ?

A critical disadvantage of primal-dual interior-point methods against dual interior-point methods for large scale SDPs (semidenite programs) has been that the primal positive semidenite variable matrix becomes fully dense in general even when all data matrices are sparse. Based on some fundamental results about positive semidenite matrix completion, this article proposes a general method of exploiting the aggregate sparsity pattern over all data matrices to overcome this disadvantage. Our method is used in two ways. One is a conversion of a sparse SDP having a large scale positive semidenite variable matrix into an SDP having multiple but smaller size positive semidenite variable matrices to which we can eectively apply any interior-point method for SDPs employing a standard block-diagonal matrix data structure. The other way is an incorporation of our method into primal-dual interior-point methods which we can apply directly to a given SDP. In Part II of this article, we will investigate an implementation of such a primal-dual interior-point method based on positive denite matrix completion, and report some numerical results.