# American Institute of Mathematical Sciences

October  2016, 12(4): 1187-1197. doi: 10.3934/jimo.2016.12.1187

## Semidefinite programming via image space analysis

 1 College of Economics and Business Administration, Chongqing University, Chongqing 400044, China

Received  January 2015 Revised  April 2015 Published  January 2016

In this paper, we investigate semidefinite programming by using the image space analysis and present some equivalence between the (regular) linear separation and the saddle points of the Lagrangian functions related to semidefinite programming. Some necessary and sufficient optimality conditions for semidefinite programming are also given under some suitable assumptions. As an application, we obtain some equivalent characterizations for necessary and sufficient optimality conditions for linear semidefinite programming under Slater assumption.
Citation: Shouhong Yang. Semidefinite programming via image space analysis. Journal of Industrial and Management Optimization, 2016, 12 (4) : 1187-1197. doi: 10.3934/jimo.2016.12.1187
##### References:
 [1] P. H. Dien, G. Mastroeni, M. Pappalardo and P. H. Quang, Regularity conditions for constrained extremum problems via image space, J. Optim. Theory Appl., 80 (1994), 19-37. doi: 10.1007/BF02196591. [2] F. Giannessi, Theorems of the alternative and optimality conditions, J. Optim. Theory Appl., 42 (1984), 331-365. doi: 10.1007/BF00935321. [3] F. Giannessi, Constrained Optimization and Image Space Analysis, Springer, New York, 2005. [4] F. Giannessi and G. Mastroeni, Separation of sets and Wolfe duality, J. Global Optim., 42 (2008), 401-412. doi: 10.1007/s10898-008-9301-2. [5] C. Helmberg, Semidefinite programming, European J. Oper. Res., 137 (2002), 461-482. doi: 10.1016/S0377-2217(01)00143-6. [6] J. Li and N. J. Huang, Image space analysis for vector variational inequalities with matrix inequality constraints and applications, J. Optim. Theory Appl., 145 (2010), 459-477. doi: 10.1007/s10957-010-9691-4. [7] J. Li and N. J. Huang, Image space analysis for variational inequalities with cone constraints and applications to traffic equilibria, Sci. China Math., 55 (2012), 851-868. doi: 10.1007/s11425-011-4287-5. [8] D. T. Luc, Theory of Vector Optimization, Springer Verlag, Berlin, 1989. [9] Y. Nesterov and A. Nemirovskii, Interior-point Polynomial Algorithms in Convex Programming, SIAM Studies in Applied Mathematics, Philadelphia, 1994. doi: 10.1137/1.9781611970791. [10] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1970. [11] A. Shapiro and K. Scheinberg, Duality and optimality conditions, in Handbook of Semidefinite Programming: Theory, Algorithms and Applications (eds. H. Wolkowicz, R. Saigal and L. Vandenberghe), Kluwer Acad. Publ., 27, Boston, MA, 2000, 67-110. doi: 10.1007/978-1-4615-4381-7_4. [12] L. Vandenberghe and S. Boyd, Semidefinite programming, SIAM Rev., 38 (1996), 49-95. doi: 10.1137/1038003. [13] G. Wanka, R. I. Boţ and S. M. Grad, Multiobjective duality for convex semidefinite programming problems, Z. Anal. Anwendungen, 22 (2003), 711-728. doi: 10.4171/ZAA/1169. [14] S. K. Zhu and S. J. Li, Unified duality theory for constrained extremum problems. Part I: Image space analysis, J. Optim. Theory Appl., 161 (2014), 738-762. doi: 10.1007/s10957-013-0468-4. [15] J. Zowe and M. Kočvara, Semidefinite programming, in Modern Optimization and its Applications in Engineering (eds. A. Ben-Tal and A. Nemirovski), Haifa (Israel), Technion, 2000.

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##### References:
 [1] P. H. Dien, G. Mastroeni, M. Pappalardo and P. H. Quang, Regularity conditions for constrained extremum problems via image space, J. Optim. Theory Appl., 80 (1994), 19-37. doi: 10.1007/BF02196591. [2] F. Giannessi, Theorems of the alternative and optimality conditions, J. Optim. Theory Appl., 42 (1984), 331-365. doi: 10.1007/BF00935321. [3] F. Giannessi, Constrained Optimization and Image Space Analysis, Springer, New York, 2005. [4] F. Giannessi and G. Mastroeni, Separation of sets and Wolfe duality, J. Global Optim., 42 (2008), 401-412. doi: 10.1007/s10898-008-9301-2. [5] C. Helmberg, Semidefinite programming, European J. Oper. Res., 137 (2002), 461-482. doi: 10.1016/S0377-2217(01)00143-6. [6] J. Li and N. J. Huang, Image space analysis for vector variational inequalities with matrix inequality constraints and applications, J. Optim. Theory Appl., 145 (2010), 459-477. doi: 10.1007/s10957-010-9691-4. [7] J. Li and N. J. Huang, Image space analysis for variational inequalities with cone constraints and applications to traffic equilibria, Sci. China Math., 55 (2012), 851-868. doi: 10.1007/s11425-011-4287-5. [8] D. T. Luc, Theory of Vector Optimization, Springer Verlag, Berlin, 1989. [9] Y. Nesterov and A. Nemirovskii, Interior-point Polynomial Algorithms in Convex Programming, SIAM Studies in Applied Mathematics, Philadelphia, 1994. doi: 10.1137/1.9781611970791. [10] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1970. [11] A. Shapiro and K. Scheinberg, Duality and optimality conditions, in Handbook of Semidefinite Programming: Theory, Algorithms and Applications (eds. H. Wolkowicz, R. Saigal and L. Vandenberghe), Kluwer Acad. Publ., 27, Boston, MA, 2000, 67-110. doi: 10.1007/978-1-4615-4381-7_4. [12] L. Vandenberghe and S. Boyd, Semidefinite programming, SIAM Rev., 38 (1996), 49-95. doi: 10.1137/1038003. [13] G. Wanka, R. I. Boţ and S. M. Grad, Multiobjective duality for convex semidefinite programming problems, Z. Anal. Anwendungen, 22 (2003), 711-728. doi: 10.4171/ZAA/1169. [14] S. K. Zhu and S. J. Li, Unified duality theory for constrained extremum problems. Part I: Image space analysis, J. Optim. Theory Appl., 161 (2014), 738-762. doi: 10.1007/s10957-013-0468-4. [15] J. Zowe and M. Kočvara, Semidefinite programming, in Modern Optimization and its Applications in Engineering (eds. A. Ben-Tal and A. Nemirovski), Haifa (Israel), Technion, 2000.
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