In this paper, we introduce and consider the concept of the prox-regularity of a multifunction. We mainly study the metric subregularity of a generalized equation defined by a proximal closed multifunction between two Hilbert spaces. Using proximal analysis techniques, we provide sufficient and/or necessary conditions for such a generalized equation to have the metric subregularity in Hilbert s...

We obtain a characterization of the proximal normal cone to prox-regular subset Riemannian manifold and some properties Bouligand tangent cones these sets are presented. Moreover, we show that on an open neighborhood set, metric projection is locally Lipschitz it directionally differentiable at boundary points set. Finally, necessary condition for curve be minimizing in set derived.

B. S. MORDUKHOVICH and T. T. A. NGHIA Abstract. The paper introduces and characterizes new notions of Lipschitzian and Hölderian full stability of solutions to general parametric variational systems described via partial subdifferential of prox-regular functions acting in finite-dimensional and Hilbert spaces. These notions, postulated certain quantitative properties of single-valued localizati...

We derive new prox-functions on the simplex from additive random utility models of discrete choice. They are convex conjugates corresponding surplus functions. In particular, we explicitly convexity parameter choice associated with generalized extreme value models, and specifically nested logit models. Incorporated into subgradient schemes, lead to a probabilistic interpretations iteration step...

We provide tight upper and lower bounds on the complexity of minimizing the average of m convex functions using gradient and prox oracles of the component functions. We show a significant gap between the complexity of deterministic vs randomized optimization. For smooth functions, we show that accelerated gradient descent (AGD) and an accelerated variant of SVRG are optimal in the deterministic...

Here we study non-convex composite optimization: first, a finite-sum of smooth but non-convex functions, and second, a general function that admits a simple proximal mapping. Most research on stochastic methods for composite optimization assumes convexity or strong convexity of each function. In this paper, we extend this problem into the non-convex setting using variance reduction techniques, ...

A local convergence result for abstract descent methods is proved. The sequence of iterates is attracted by a local (or global) minimum, stays in its neighborhood and converges. This result allows algorithms to exploit local properties of the objective function: The gradient of the Moreau envelope of a prox-regular functions is locally Lipschitz continuous and expressible in terms of the proxim...

This paper is devoted on the one hand to the study of specific properties of an evolution variational inequality, holding in the Hilbert setting. We give on the other hand a general criterion for Lyapunov pairs of this dynamical system and some results on the asymptotic behaviour of the solution. Key-words: Differential inclusions Evolution variational inequalities Proximal normal cone Uniforml...