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Nonconvex Weak Sharp Minima on Riemannian Manifolds

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Abstract

We establish some necessary conditions (of the primal and dual types) for the set of weak sharp minima of a nonconvex optimization problem on a Riemannian manifold. Here, we provide a generalization of some characterizations of weak sharp minima for convex problems on Riemannian manifold introduced by Li et al. (SIAM J Optim 21(4):1523–1560, 2011) for nonconvex problems. We use the theory of the Fréchet and limiting subdifferentials on Riemannian manifold to give some necessary conditions of the dual type. We also consider a theory of contingent directional derivative and a notion of contingent cone on Riemannian manifold to give some necessary conditions of the primal type. Several definitions have been provided for the contingent cone on Riemannian manifold. We show that these definitions, under some modifications, are equivalent. We establish a lemma about the local behavior of a distance function. We use this lemma to establish some necessary conditions by expressing the Fréchet subdifferential (contingent directional derivative) of a distance function on a Riemannian manifold in terms of normal cones (contingent cones). As an application, we show how one can use weak sharp minima property to model a Cheeger-type constant of a graph as an optimization problem on a Stiefel manifold.

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Notes

  1. See: https://mathoverflow.net/q/301064.

  2. The key point is to show that for each vector \(u := (u_1, \ldots , u_n)\in \mathbb {R}^n\) with \(\Vert u\Vert _{2}=1\), there is a nonempty subset \(A \subset \{ v_i :u_i \ne 0 \}\) such that \(\frac{|\partial A|}{\sqrt{|A|}} \le \Vert \nabla u \Vert _1\).

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Acknowledgements

The authors thank Sharif University of Technology for supporting this work. The first author is grateful to Amir Daneshgar, Mostafa Einollahzadeh, Mehdi Shaeiri, and Mostafa Kiyaee for constructive discussions.

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  1. Sharif University of Technology, Tehran, Iran

    Mohammad Mahdi Karkhaneei & Nezam Mahdavi-Amiri

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  1. Mohammad Mahdi Karkhaneei
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  2. Nezam Mahdavi-Amiri
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Correspondence to Nezam Mahdavi-Amiri.

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Communicated by Alexandru Kristály.

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Karkhaneei, M.M., Mahdavi-Amiri, N. Nonconvex Weak Sharp Minima on Riemannian Manifolds. J Optim Theory Appl 183, 85–104 (2019). https://doi.org/10.1007/s10957-019-01539-2

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