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.
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This research was supported by the Beijing Natural Science Foundation under grant Z180005, and the National Natural Science Foundation of China under grants 12171021 and 11822103.
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Song, M., Xia, Y. Calabi-Polyak convexity theorem, Yuan’s lemma and S-lemma: extensions and applications. J Glob Optim 85, 743–756 (2023). https://doi.org/10.1007/s10898-022-01225-0
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DOI: https://doi.org/10.1007/s10898-022-01225-0
Keywords
- Calabi-Polyak theorem
- Yuan’s lemma
- S-lemma
- Second-order optimality condition
- Quadratic optimization
- Hidden convexity