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

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  • Van-Bong Nguyen

    (Tay Nguyen University)

  • Thi Ngan Nguyen

    (Tay Nguyen University)

  • Ruey-Lin Sheu

    (National Cheng Kung University)

Abstract

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.

Suggested Citation

  • Van-Bong Nguyen & Thi Ngan Nguyen & Ruey-Lin Sheu, 2020. "Strong duality in minimizing a quadratic form subject to two homogeneous quadratic inequalities over the unit sphere," Journal of Global Optimization, Springer, vol. 76(1), pages 121-135, January.
    • Handle: RePEc:spr:jglopt:v:76:y:2020:i:1:d:10.1007_s10898-019-00835-5
    DOI: 10.1007/s10898-019-00835-5
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    Cited by:

    1. Mengmeng Song & Yong Xia, 2023. "Calabi-Polyak convexity theorem, Yuan’s lemma and S-lemma: extensions and applications," Journal of Global Optimization, Springer, vol. 85(3), pages 743-756, March.

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