2021/11/05 by Sunyoung Kim, Kim, Sunyoung, Masakazu Kojima +1
Computer Science · Engineering · Mathematics · #90C20 #90C22 #90C25 #90C26 #90C27 #Advanced Optimization Algorithms Research #FOS: Mathematics #Optimization and Control (math.OC) #Optimization and Mathematical Programming #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.2111.03251
openalex publication_date 2021/11/05 · openalex created_date 2021/11/22 · openalex updated_date 2026/07/28
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 view of strong duality of COPs for the known results. For our framework, we employ a COP minimizing a linear function in a vector variable x subject to a single hyperplane constraint x ∈ H and two cone constraints x ∈ K1, x ∈ K2. It can be identically reformulated as a simpler COP with the single hyperplane constraint x ∈ H and the single cone constraint x ∈ K1 ∩ K2. 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 K1 and K2 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.