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Homogeneous polynomials and spurious local minima on the unit sphere

Abstract

We consider forms on the Euclidean unit sphere. We obtain a simple and complete characterization of all points that satisfies the standard second-order necessary condition of optimality. It is stated solely in terms of the value of (i) f, (ii) the norm of its gradient, and (iii) the first two smallest eigenvalues of its Hessian, all evaluated at the point. In fact this property also holds for twice continuous differentiable functions that are positively homogeneous. We also characterize a class of degree-d forms with no spurious local minima on \(\mathbb {S}^{n-1}\) by using a property of gradient ideals in algebraic geometry.

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Acknowledgements

The author gratefully acknowledges Professor Jiawang Nie for fruitful discussions that helped improve the paper.

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Correspondence to Jean B. Lasserre.

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Work partly funded by the AI Interdisciplinary Institute ANITI through the French “Investing for the Future PI3A”program under the Grant agreement ANR-19-PI3A-0004

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

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Lasserre, J.B. Homogeneous polynomials and spurious local minima on the unit sphere. Optim Lett (2021). https://doi.org/10.1007/s11590-021-01811-3

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Keywords

  • Homogeneous polynomials
  • Global and local minima
  • Optimization on the unit sphere