# 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 & Management Optimization, 2010, 6 (4) : 881-893. doi: 10.3934/jimo.2010.6.881
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