Skip to Main content Skip to Navigation
Preprints, Working Papers, ...

HOMOGENEOUS POLYNOMIALS AND SPURIOUS LOCAL MINIMA ON THE UNIT SPHERE

Abstract : We consider degree-d forms on the Euclidean unit sphere. We specialize to our setting a genericity result by Nie obtained in a more general framework. We exhibit an homogeneous polynomial Res in the coefficients of f , such that if Res(f) = 0 then all points that satisfy first-and second-order necessary optimality conditions are in fact local minima of f on the unit sphere. Then we obtain obtain a simple and compact characterization of all local minima of generic degree-d forms, 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. Finally we obtain a characterization of generic degree-d forms with no spurious local minimum on the unit sphere by using a property of gradient ideals in algebraic geometry.
Complete list of metadata

https://hal.laas.fr/hal-02966390
Contributor : Jean Bernard Lasserre <>
Submitted on : Tuesday, May 4, 2021 - 10:14:54 AM
Last modification on : Wednesday, June 9, 2021 - 10:00:20 AM

Files

spurious-new.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : hal-02966390, version 2
  • ARXIV : 2010.07066

Citation

Jean-Bernard Lasserre. HOMOGENEOUS POLYNOMIALS AND SPURIOUS LOCAL MINIMA ON THE UNIT SPHERE. 2020. ⟨hal-02966390v2⟩

Share

Metrics

Record views

50

Files downloads

21