Strong duality of a conic optimization problem with a single hyperplane and two cone constraints

Sunyoung KIm (skimewha.ac.kr)
Masakazu Kojima (kojimais.titech.ac.jp)

Abstract: Our target conic optimization problem (COP) is minimizing a linear function in a vector variable $x$ subject to a single hyperplane constraint $x \in H$ and two cone constraints $x \in K_1$, $x \in K_2$. This COP is geometrically representable and extensive enough to study strong (Lagrangian) duality of more general COPs with cone and multiple linear inequality constraints. It can also be identically reformulated as a simpler COP with the single hyperplane constraint $x \in H$ and the single cone constraint $x \in K_1 \cap K_2$, which has 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 $K_1$ and $K_2$ is closed or if the dual of the reformulated COP satisfies a 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.

Keywords: Duality, conic optimization problems, simple conic optimization problems, generalizing the Slater condition, closedness of the Minkowski sum of two cones

Category 1: Linear, Cone and Semidefinite Programming (Other )

Category 2: Convex and Nonsmooth Optimization (Convex Optimization )

Category 3: Global Optimization (Theory )

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Entry Submitted: 11/04/2021
Entry Accepted: 11/05/2021
Entry Last Modified: 11/15/2021

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