On Semidefinite Representations of Second-order Conic Optimization Problems

Second-order conic optimization (SOCO) can be considered as a special case of semidefinite optimization (SDO). In the literature it has been advised that a SOCO problem can be embedded in an SDO problem using the arrow-head matrix transformation. However, a primal-dual solution pair cannot be mapped simultaneously using the arrow-head transformation as we might lose complementarity and duality in some cases. To address this issue, we investigate the relationship between SOCO problems, and their SDO counterpart. Through derivation of standard semidefinite representations of SOCO problems, we introduce admissible mappings. We show that the proposed mappings preserve both feasibility and optimality. Further, we discuss how the optimal partition of a SOCO problem maps to the optimal partition of its SDO counterpart.