Abstract.
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 program (SOCP) and its corresponding semidefinite program (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, in general no primal optimal solutions of the SOCP-related SDP can be an “arrow” matrix.
Similar content being viewed by others
References
Aldler, I., Alizadeh, F.: Primal-Dual Interior Point Algorithms for Convex Quadratically Constrained and Semidefinite Optimization Problems. RUTCOR Res. Report RRR, 46–95 (1995)
Chen, X.D., Sun, D., Sun, J.: Complementarity Functions and Numerical Experiments on some Smoothing Newton Methods for Second-Order-Cone Complementarity Problems. Comput. Optim. Appl. 25 (1–3), 39–56 (2003)
Halick’a, M., de Klerk, E., Roos, C.: On the Convergence of the Central Path in Semidefinite Optimization. SIAM J. Optim. 12 (4), 1090–1099 (2002)
Halick’a, M., de Klerk, E.: Private Communications, 2002
Lobo, M.S., Vandenberghe, L.: Stephen Boyd and Herv’e Lebret, Applications of Second-Order Cone Programming. Linear Alg. Appl. 284, 193–228 (1998)
Monteiro, R.D.C., Tsuchiya, T.: Polynomial Convergence of Primal-Dual Algorithms for the Second-Order Cone Program based on the MZ-family of directions. Math. Program. Ser. A 88, 61–83 (2000)
Nesterov, Y., Nemirovskii, A.: Interior Point Polynomial Algorithms in Convex Programming. Society for Industrial and Applied Mathematics, Philadelphia, 1984
Peng, J., Roos, C., Terlaky, T.: Primal-Dual Interior-point Methods for Second-Order Conic Optimization based on Self-Regular Proximities. SIAM J. Optim. 13 (1), 179–203 (2002)
Tsuchiya, T.: A Polynomial Primal-Dual Path-Following Algorithm for Second-Order Cone Programming. Research Memorandum No. 649, The Institute of Statistical Mathematics, Tokyo, Japan, October (Revised: December 1997)
Tsuchiya, T.: A Convergence Analysis of the Scaling-Invariant Primal-Dual Path-Following Algorithms for Second-Order Cone Programming. Optim. Meth. Soft. 11 & 12, 141–182 (1999)
Author information
Authors and Affiliations
Additional information
Mathematics Subject Classification (2000): 20E28, 20G40, 20C20
Rights and permissions
About this article
Cite this article
Sim, CK., Zhao, G. A note on treating a second order cone program as a special case of a semidefinite program. Math. Program. 102, 609–613 (2005). https://doi.org/10.1007/s10107-004-0546-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-004-0546-3