The Generalized Trust-Region Sub-Problem with Additional Linear Inequality Constraints—Two Convex Quadratic Relaxations and Strong Duality

In this paper, we study the problem of minimizing a general quadratic function subject to a quadratic inequality constraint with a fixed number of additional linear inequality constraints. Under a regularity condition, we first introduce two convex quadratic relaxations (CQRs), under two different c...

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Published inSymmetry (Basel) Vol. 12; no. 8; p. 1369
Main Authors Almaadeed, Temadher A., Taati, Akram, Salahi, Maziar, Hamdi, Abdelouahed
Format Journal Article
LanguageEnglish
Published Basel MDPI AG 01.08.2020
Subjects
Inequality
Linear functions
Optimization
Quadratic equations
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ISSN2073-8994
2073-8994
DOI10.3390/sym12081369

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Summary:In this paper, we study the problem of minimizing a general quadratic function subject to a quadratic inequality constraint with a fixed number of additional linear inequality constraints. Under a regularity condition, we first introduce two convex quadratic relaxations (CQRs), under two different conditions, that are minimizing a linear objective function over two convex quadratic constraints with additional linear inequality constraints. Then, we discuss cases where the CQRs return the optimal solution of the problem, revealing new conditions under which the underlying problem admits strong Lagrangian duality and enjoys exact semidefinite optimization relaxation. Finally, under the given sufficient conditions, we present necessary and sufficient conditions for global optimality of the problem and obtain a form of S-lemma for a system of two quadratic and a fixed number of linear inequalities.
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ISSN:2073-8994
2073-8994
DOI:10.3390/sym12081369