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Calabi-Polyak convexity theorem, Yuan’s lemma and S-lemma: extensions and applications

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  • Mengmeng Song

    (Beihang University)

  • Yong Xia

    (Beihang University)

Abstract

We extend the Calabi-Polyak theorem on the convexity of joint numerical range from three to any number of matrices on condition that each of them is a linear combination of three matrices having a positive definite linear combination. Our new result covers the fundamental Dines’s theorem. As applications, the further extended Yuan’s lemma and S-lemma are presented. The former is used to establish a more generalized assumption under which the standard second-order necessary optimality condition holds at the local minimizer in nonlinear programming, and the latter reveals hidden convexity of the homogeneous quadratic optimization problem with two bilateral quadratic constraints and its fractional extension.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:85:y:2023:i:3:d:10.1007_s10898-022-01225-0
    DOI: 10.1007/s10898-022-01225-0
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    References listed on IDEAS

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    4. 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.
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