An Exact duality Theory for Semide niteProgramming and its Complexity Implications

In this paper, an exact dual is derived for Semideenite Programming (SDP), for which strong duality properties hold without any regularity assumptions. Its main features are: i. The new dual is an explicit semideenite program with polynomially many variables and polynomial size coeecient bitlengths. ii. If the primal is feasible, then it is bounded if and only if the dual is feasible. iii. When the primal is feasible and bounded, then its optimum value equals that of the dual, i.e. there is no duality gap. Further, the dual attains this common optimum value. iv. It yields a precise theorem of the alternative for semideenite inequality systems, i.e. a characterization of the infeasibility of a semideenite inequality in terms of the feasibility of another polynomial size semideenite inequality. Note that the standard duality for linear programming satisses all of the above features, but no such explicit gap-free dual program of polynomial size was previously known for SDP, without Slater-like conditions being assumed. The dual is then applied to derive certain complexity results for SDP. The decision problem of Semideenite Feasibility (SDFP), i.e. that of determining if a given semideenite inequality system is feasible, is the central problem of interest. The complexity of SDFP is unknown, but we show the following: 1) In the Turing machine model, the membership or nonmembership of SDFP in NP and Co-NP is simultaneous; hence SDFP is not NP-Complete unless NP=Co-NP. 2) In the real number model of Blum, Shub and SmaleeBSS89], SDFP is in NP\Co-NP.