Strong duality of a conic optimization problem with a single hyperplane and two cone constraints
- 주제(키워드) Duality , simple conic optimization problems , geometry of strong duality , the Slater condition , closedness of the Minkowski sum of two cones
- 주제(기타) Operations Research & Management Science; Mathematics, Applied
- 설명문(일반) [Kim, Sunyoung] Ewha W Univ, Dept Math, Seoul, South Korea; [Kojima, Masakazu] Chuo Univ, Dept Ind & Syst Engn, Tokyo, Japan
- 관리정보기술 faculty
- 등재 SCIE, SCOPUS
- OA유형 Green Submitted
- 발행기관 TAYLOR & FRANCIS LTD
- 발행년도 2023
- 총서유형 Journal
- URI http://www.dcollection.net/handler/ewha/000000211539
- 본문언어 영어
- Published As https://doi.org/10.1080/02331934.2023.2251987
초록/요약
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 x subject to a single hyperplane constraint x. H and two cone constraints x epsilon K-1, x epsilon K-2. It can be identically reformulated as a simpler COP with the single hyperplane constraint x. H and the single cone constraint x epsilon K-1 boolean AND K-2. 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 targetCOPis equivalent to the dual of the reformulated COP if the Minkowski sum of the duals of the two cones K-1 andK(2) is closed or if the dual of the reformulatedCOPsatisfies 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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