Strong duality in minimizing a quadratic form subject to two homogeneous quadratic inequalities over the unit sphere

In this paper, we study the strong duality for an optimization problem to minimize a homogeneous quadratic function subject to two homogeneous quadratic constraints over the unit sphere, called Problem (P) in this paper. When a feasible (P) fails to have a Slater point, we show that (P) always adopt...

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Veröffentlicht in:Journal of global optimization Jg. 76; H. 1; S. 121 - 135
Hauptverfasser: Nguyen, Van-Bong, Nguyen, Thi Ngan, Sheu, Ruey-Lin
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York Springer US 01.01.2020
Springer
Springer Nature B.V
Schlagworte:
Computer Science
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Optimization
Quadratic equations
Quadratic forms
Real Functions
90C20
CDT problem
49M20
Joint numerical range
90C46
90C22
90C26
Quadratically constrained quadratic programming
Slater condition
S-lemma
ISSN:0925-5001, 1573-2916
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Zusammenfassung:In this paper, we study the strong duality for an optimization problem to minimize a homogeneous quadratic function subject to two homogeneous quadratic constraints over the unit sphere, called Problem (P) in this paper. When a feasible (P) fails to have a Slater point, we show that (P) always adopts the strong duality. When (P) has a Slater point, we propose a set of conditions, called “ Property J ”, on an SDP relaxation of (P) and its conical dual. We show that (P) has the strong duality if and only if there exists at least one optimal solution to the SDP relaxation of (P) which fails Property J. Our techniques are based on various extensions of S-lemma as well as the matrix rank-one decomposition procedure introduced by Ai and Zhang. Many nontrivial examples are constructed to help understand the mechanism.
Bibliographie:ObjectType-Article-1
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ISSN:0925-5001
1573-2916
DOI:10.1007/s10898-019-00835-5