Strong (Lagrangian) duality of general conic optimization problems (COPs) has long been studied and its profound and complicated results appear in different forms in a wide range of literatures. As a result, characterizing the known and unknown results can sometimes be difficult. The aim of this article is to provide a unified and geometric overview of strong duality of COPs for the known results, and to explain the essential ideas of the duality results with the simplest geometry. For our framework, we employ a COP minimizing a linear function in a vector variable (Formula presented.) subject to a single hyperplane constraint (Formula presented.) and two cone constraints (Formula presented.), (Formula presented.). It can be identically reformulated as a simpler COP with the single hyperplane constraint (Formula presented.) and the single cone constraint (Formula presented.). This simple COP and its dual as well as their duality relation can be represented geometrically, and they have no duality gap without any constraint qualification. The dual of the original target COP is equivalent to the dual of the reformulated COP if the Minkowski sum of the duals of the two cones (Formula presented.) and (Formula presented.) is closed or if the dual of the reformulated COP satisfies a certain Slater condition. Thus, these two conditions make it possible to transfer all duality results, including the existence and/or boundedness of optimal solutions, on the reformulated COP to the ones on the original target COP, and further to the ones on a standard primal-dual pair of COPs with symmetry.
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- closedness of the Minkowski sum of two cones
- geometry of strong duality
- simple conic optimization problems
- the Slater condition