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Active manifolds, stratifications, and convergence to local minima in nonsmooth optimization
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We show that the subgradient method converges only to local minimizers when applied to generic Lipschitz continuous and subdifferentially regular functions that are definable in an o-minimal structure. At a high level, the argument we present is appealingly transparent: we interpret the nonsmooth dynamics as an approximate Riemannian gradient method on a certain distinguished submanifold that captures the nonsmooth activity of the function. In the process, we develop new regularity conditions in nonsmooth analysis that parallel the stratification conditions of Whitney, Kuo, and Verdier and extend stochastic processes techniques of Pemantle.
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Cited by 1 Pith paper
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On exploration of an interior mirror descent flow for stochastic nonconvex constrained problem
A Riemannian subgradient differential inclusion unifies Hessian barrier and mirror descent methods and explains their spurious stationary points as stable equilibria outside the true stationary set.
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