A note on quadratic constraints with indicator variables: Convex hull description and perspective relaxation

In this paper, we study the mixed-integer nonlinear set given by a separable quadratic constraint on continuous variables, where each continuous variable is controlled by an additional indicator. This set occurs pervasively in optimization problems with uncertainty and in machine learning. We show t...

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Bibliographic Details
Published inOperations research letters Vol. 52; p. 107059
Main Authors Gómez, Andrés, Xie, Weijun
Format Journal Article
LanguageEnglish
Published Elsevier B.V 01.01.2024
Subjects
Convexification
Mixed-integer optimization
Perspective relaxation
Quadratic optimization
Quadratic optimization
Convexification
Perspective relaxation
Mixed-integer optimization
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Summary:In this paper, we study the mixed-integer nonlinear set given by a separable quadratic constraint on continuous variables, where each continuous variable is controlled by an additional indicator. This set occurs pervasively in optimization problems with uncertainty and in machine learning. We show that optimization over this set is NP-hard. Despite this negative result, we discover links between the convex hull of the set under study, and a family of polyhedral sets studied in the literature. Moreover, we show that although perspective relaxation in the literature for this set fails to match the structure of its convex hull, it is guaranteed to be a close approximation.
ISSN:0167-6377
1872-7468
DOI:10.1016/j.orl.2023.107059