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In this work we present some exactness conditions for the Shor relaxation ofdiagonal QCQPs, which extend the conditions introduced in different recentpapers about the same topic. It is shown that the Shor relaxation is equivalentto two convex quadratic relaxations. Then, sufficient conditions for theexactness of the relaxations are derived from their KKT systems. It will beshown that, in some cases, by this derivation previous conditions in theliterature, which can be viewed as dual conditions, since they only involve theLagrange multipliers appearing in the KKT systems, can be extended toprimal-dual conditions, which also involve the primal variables appearing inthe KKT systems.

KKT-based primal-dual exactness conditions for the Shor relaxation