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This paper develops a new storage-optimal algorithm that provably solvesgeneric semidefinite programs (SDPs) in standard form. This method isparticularly effective for weakly constrained SDPs. The key idea is toformulate an approximate complementarity principle: Given an approximatesolution to the dual SDP, the primal SDP has an approximate solution whoserange is contained in the eigenspace with small eigenvalues of the dual slackmatrix. For weakly constrained SDPs, this eigenspace has very low dimension, sothis observation significantly reduces the search space for the primalsolution.  This result suggests an algorithmic strategy that can be implemented withminimal storage:  (1) Solve the dual SDP approximately;  (2) compress the primal SDP to the eigenspace with small eigenvalues of thedual slack matrix;  (3) solve the compressed primal SDP.  The paper also provides numerical experiments showing that this approach issuccessful for a range of interesting large-scale SDPs.

An Optimal-Storage Approach to Semidefinite Programming using Approximate Complementarity