Title: Treating Second Order Cone Programs as a Special Case of Semidefinite Programs

It is well known that a vector is in a second order cone if and only if its "arrow" matrix is positive semidefinite. But much less well-known is about the relation between a second order cone problem (SOCP) and its corresponding semidefinite problem (SDP). The correspondence between the dual problem of SOCP and SDP is quite direct and the correspondence between the primal problems is much more complicated. Given a SDP primal optimal solution which is not necessarily "arrow-shaped", we can construct a SOCP primal optimal solution. The mapping from the primal optimal solution of SDP to the primal optimal solution of SOCP can be shown to be unique. Conversely, given a SOCP primal optimal solution, we can construct a SDP primal optimal solution which is not an "arrow" matrix. Indeed, we will show that in general no primal optimal solutions of the SOCP-related SDP can be an "arrow" matrix.