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OPTIMIZATION ON THE EUCLIDEAN UNIT SPHERE

Abstract : We consider the problem of minimizing a continuously differentiable function f of m linear forms in n variables on the Euclidean unit sphere. We show that this problem is equivalent to minimizing the same function of related m linear forms (but now in m variables) on the Euclidean unit ball. When the linear forms are known, this results in a drastic reduction in problem size whenever m ≪ n and allows to solve potentially large scale non-convex such problems. We also provide a test to detect when a polynomial is a polynomial in a fixed number of forms. Finally, we identify two classes of functions with no spurious local minima on the sphere: (i) quasi-convex polynomials of odd degree and (ii) nonnegative and homogeneous functions. Finally, odd degreed forms have only nonpositive local minima and at most (d − 1) m are strictly negative.
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Preprints, Working Papers, ...
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https://hal.laas.fr/hal-03291242
Contributor : Jean Bernard Lasserre <>
Submitted on : Monday, July 19, 2021 - 4:42:13 PM
Last modification on : Monday, July 26, 2021 - 2:03:44 PM

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  • HAL Id : hal-03291242, version 1

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Jean-Bernard Lasserre. OPTIMIZATION ON THE EUCLIDEAN UNIT SPHERE. 2021. ⟨hal-03291242⟩

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