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Convex hull of two quadratic or a conic quadratic and a quadratic inequality. (English) Zbl 1393.90074
This paper contributes to extending the development of strong valid inequalities to mixed integer conic quadratic programming. The analyis is based on results on the convex hull of open sets defined by two strict non-homogenuous quadratic inequalities in the reference [U. Yildiran, IMA J. Math. Control Inf. 26, No. 4, 417–450 (2009; Zbl 1187.90227)]. The authors prove three main results:
how to extend the aggregation technique of [loc. cit.] to yield valid conic quadratic inequalities for the convex hull of open sets defined by two strict quadratic inequalities or by a strict conic quadratic inequality and a strict quadratic inequality;
that under an additional containment assumption, these inequalities charcterize the convex hull exactly for sets defined by a strict conic quadratic and a strict quadratic inequality;
that under certain topological assumptions the results can be transferred to characterize the closed convex hull of sets defined with non-strict conic and quadratic inequalities.
The authors provide illustrative examples and compare their results to closed convex hull charcaterizations in [S. Burer and F. Kılınç-Karzan, Math. Program. 162, No. 1–2 (A), 393–429 (2017; Zbl 1358.90095)].

MSC:
90C11 Mixed integer programming
90C26 Nonconvex programming, global optimization
90C20 Quadratic programming
Software:
SCIP
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References:
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