# American Institute of Mathematical Sciences

October  2010, 6(4): 881-893. doi: 10.3934/jimo.2010.6.881

## Duality formulations in semidefinite programming

 1 Department of Mathematics and Computer Science, Northern Michigan University, Marquette, MI 49855, United States 2 Department of Mathematics, Nantong Vacational College, Nantong 226007, China, China

Received  October 2009 Revised  June 2010 Published  September 2010

In this paper, duals for standard semidefinite programming problems from both the primal and dual sides are studied. Explicit expressions of the minimal cones and their dual cones are obtained under closeness assumptions of certain sets. As a result, duality formulations resulting from regularizations for both primal and dual problems can be expressed explicitly in terms of equality and inequality constraints involving three vector and matrix variables under such assumptions. It is proved in this paper that these newly developed duals can be cast as the Extended Lagrange-Slater Dual (ELSD) and the Extended Lagrange-Slater Dual of the Dual (ELSDD) with one reduction step. Therefore, the duals formulated in this paper guarantee strong duality, i.e., a zero duality gap and dual attainment.
Citation: Qinghong Zhang, Gang Chen, Ting Zhang. Duality formulations in semidefinite programming. Journal of Industrial and Management Optimization, 2010, 6 (4) : 881-893. doi: 10.3934/jimo.2010.6.881
##### References:
 [1] G. Barker and D. Carlson, Cones of diagonally dominant matrices, Pacific J. Math., 57 (1975), 15-32. [2] J. Bonnans and A. Shapiro, "Perturbation Analysis of Optimization Problems," Springer, New York, 2000. [3] J. Borwein and H. Wolkowicz, Facial reduction for a cone-convex programming problem, J. Austral. Math. Soc. Ser. A, 30 (1980/1981), 369-380. [4] J. Borwein and H. Wolkowicz, Regularizing the abstract convex program, J. Math. Anal. Appl., 83 (1981), 495-530. doi: 10.1016/0022-247X(81)90138-4. [5] E. de Klerk, C. Roos and T. Terlaky, Infeasible start semidefinite programming algorithms via self-dual embeddings, Fields Inst. Commun., 18 (1998), 215-236. [6] K. Kortanek and Q. Zhang, Perfect duality in semi-infinite and semidefinite programming, Math. Program., Ser. A, 91 (2001), 127-144. [7] Z. Luo, J. Sturm and S. Zhang, "Duality Results for Conic Convex Programming," Technical Report, Econometric Institute Report No. 9719/A, Econometric Institute, Erasumus University, Rotterdam, The Netherlands, April 1997. [8] M. Ramana, An exact duality theory for semidefinite programming and its complexity implications, Math. Program., Ser. B, 77 (1997), 129-162. [9] M. Ramana, L. Tunçel and H. Wolkowicz, Strong duality for semidefinite programming, SIAM J. Optim., 7 (1997), 641-662. [10] R. Rockafellar, "Convex Analysis," Princeton University Press, Princeton, New Jersey, 1970. [11] J. Sturm, "Primal-Dual Interior Point Approach to Semidefinite Programming," Ph.D thesis, Erasmus University, 1997. [12] M. Todd, Semidefinite optimization, Acta. Numer., 10 (2001), 515-560. doi: 10.1017/S0962492901000071. [13] L. Vandenberghe and S. Boyd, Semidefinite programming, SIAM Review, 38 (1996), 49-95. doi: 10.1137/1038003. [14] H. Wolkowicz, Some applications of optimization in matrix theory, Linear Algebra Appl., 40 (1981), 101-118. doi: 10.1016/0024-3795(81)90143-9. [15] Q. Zhang, Embedding methods for semidefinite programming, submitted for publication.

show all references

##### References:
 [1] G. Barker and D. Carlson, Cones of diagonally dominant matrices, Pacific J. Math., 57 (1975), 15-32. [2] J. Bonnans and A. Shapiro, "Perturbation Analysis of Optimization Problems," Springer, New York, 2000. [3] J. Borwein and H. Wolkowicz, Facial reduction for a cone-convex programming problem, J. Austral. Math. Soc. Ser. A, 30 (1980/1981), 369-380. [4] J. Borwein and H. Wolkowicz, Regularizing the abstract convex program, J. Math. Anal. Appl., 83 (1981), 495-530. doi: 10.1016/0022-247X(81)90138-4. [5] E. de Klerk, C. Roos and T. Terlaky, Infeasible start semidefinite programming algorithms via self-dual embeddings, Fields Inst. Commun., 18 (1998), 215-236. [6] K. Kortanek and Q. Zhang, Perfect duality in semi-infinite and semidefinite programming, Math. Program., Ser. A, 91 (2001), 127-144. [7] Z. Luo, J. Sturm and S. Zhang, "Duality Results for Conic Convex Programming," Technical Report, Econometric Institute Report No. 9719/A, Econometric Institute, Erasumus University, Rotterdam, The Netherlands, April 1997. [8] M. Ramana, An exact duality theory for semidefinite programming and its complexity implications, Math. Program., Ser. B, 77 (1997), 129-162. [9] M. Ramana, L. Tunçel and H. Wolkowicz, Strong duality for semidefinite programming, SIAM J. Optim., 7 (1997), 641-662. [10] R. Rockafellar, "Convex Analysis," Princeton University Press, Princeton, New Jersey, 1970. [11] J. Sturm, "Primal-Dual Interior Point Approach to Semidefinite Programming," Ph.D thesis, Erasmus University, 1997. [12] M. Todd, Semidefinite optimization, Acta. Numer., 10 (2001), 515-560. doi: 10.1017/S0962492901000071. [13] L. Vandenberghe and S. Boyd, Semidefinite programming, SIAM Review, 38 (1996), 49-95. doi: 10.1137/1038003. [14] H. Wolkowicz, Some applications of optimization in matrix theory, Linear Algebra Appl., 40 (1981), 101-118. doi: 10.1016/0024-3795(81)90143-9. [15] Q. Zhang, Embedding methods for semidefinite programming, submitted for publication.
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