An Optimal-Storage Approach to Semidefinite Programming using Approximate Complementarity
This addresses computational efficiency for researchers and practitioners dealing with large-scale optimization problems, representing an incremental improvement in algorithmic storage.
The paper tackles the problem of solving large-scale semidefinite programs (SDPs) by developing a storage-optimal algorithm that reduces the search space for primal solutions, particularly for weakly constrained SDPs, with numerical experiments demonstrating success across various cases.
This paper develops a new storage-optimal algorithm that provably solves generic semidefinite programs (SDPs) in standard form. This method is particularly effective for weakly constrained SDPs. The key idea is to formulate an approximate complementarity principle: Given an approximate solution to the dual SDP, the primal SDP has an approximate solution whose range is contained in the eigenspace with small eigenvalues of the dual slack matrix. For weakly constrained SDPs, this eigenspace has very low dimension, so this observation significantly reduces the search space for the primal solution. This result suggests an algorithmic strategy that can be implemented with minimal storage: (1) Solve the dual SDP approximately; (2) compress the primal SDP to the eigenspace with small eigenvalues of the dual slack matrix; (3) solve the compressed primal SDP. The paper also provides numerical experiments showing that this approach is successful for a range of interesting large-scale SDPs.