In the nonconvex case, it is proved that quadratic growth in conjunction with a regularity property for the active gradients suffices, and it is proved that quadratic growth in conjunction with a regularity property for the active gradients suffices.
Abstract
We explore a new growth condition for Lipschitz functions around local minimizers, recently proposed in the context of Goldstein-subgradient-based algorithms and their behavior in practice. On generic examples, these algorithms are often observed to converge approximately linearly. We focus on objectives that are piecewise twice continuously differentiable. In that case, the new growth condition holds when the objective is, in addition, strongly convex. In the nonconvex case, we prove that quadratic growth in conjunction with a regularity property for the active gradients suffices. Computational experiments illustrate the importance of the assumptions.
This work analyzes a stochastic Newton optimization scheme for locating the unique global minimizer of a general nonconvex objective function and proves global almost sure convergence.
A model-based minimization method is developed under a relative Lipschitz condition and a framework with convergence guarantees is extended to the setting where the distance generating function and its gradient are accessible only through a stochastic oracle.
We investigate the optimization problem of minimizing a nonsmooth function that satisfies a nonsmooth version of the descent lemma over a nonempty and closed but not necessarily convex set. The objective function belongs to the class of upper-$\mathcal{C}^2$ functions, whereas the constraints may promote a sparse or lo...
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We prove by means of the Moreau-Yosida regularization an existence result for perturbed degenerate state-dependent sweeping processes with convex and nonconvex moving sets. The moving sets vary in a Lipschitz continuous way with respect to the truncated Hausdorff distance. The theoretical results are applied to obtain...
D. Narváez, Emilio Vilches· Journal of Convex Analysis· 2 citations· ⚡1
Using advanced concepts and techniques from convex and variational analysis (such as the notion of twice epi-differentiability), we establish, under a set of appropriate conditions (including a polyhedricity assumption), that the solution to a parameterized nonlinear variational inequality associated with a positively...
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