An exact duality theory for semidefinite programming and its complexity implications (1997) [35 citations — 3 self]
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@ARTICLE{Ramana97anexact,author = {Motakuri Ramana},
title = {An exact duality theory for semidefinite programming and its complexity implications},
journal = {Mathematical Programming},
year = {1997},
volume = {77}
}
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Abstract:
In this paper, an exact dual is derived for Semidefinite Programming (SDP), for which strong duality properties hold without any regularity assumptions. Its main features are: ffl The new dual is an explicit semidefinite program with polynomially many variables and polynomial size coefficient bitlengths. ffl If the primal is feasible, then it is bounded if and only if the dual is feasible. ffl 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. ffl It yields a precise Farkas Lemma for semidefinite feasibility systems, i.e. a characterization of the infeasibility of a semidefinite inequality in terms of the feasibility of another polynomial size semidefinite inequality. Note that the standard duality for Linear Programming satisfies all of the above features, but no such explicit duality theory 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 Semidefinite Feasibility (SDFP), i.e. that of determining if a given semidefinite inequality system is feasible, is the central
