Variational Analysis

By Bastian Goldlücke. 2014

4,554 citations recorded by the source corpus.

Subfields: Computer Networks and Communications, Geometry and Topology, Statistics and Probability, Mathematical Optimization

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Citations in the corpus

Frank-Wolfe Splitting via Augmented Lagrangian Method (2018, background): “The smoothness of the augmented dual function is due to the duality between strong convexity and strong smoothness (Rockafellar and Wets, 1998).”
Certifying Some Distributional Robustness with Principled Adversarial Training (2017, methodology): “Recalling Rockafellar and Wets [57, Def.”
Optimality conditions for a controlled sweeping process with applications to the crowd motion model (2015, background): “9) enjoys comprehensive calculus rules based of variational/extremal principles of variational analysis; see [19, 22] for more details and references.”
Statistics of Robust Optimization: A Generalized Empirical Likelihood Approach (2016, background): “We show this in two ways: ﬁrst, by considering uniform convergence of the robust objective (4a) to the population risk E P 0 [ ℓ ( x ; ξ )] over compacta (Section 5.1), and second by leveraging epigraphical convergence [69] to allow unbounded feasible region X when ℓ ( ; ξ ) is convex (Section 5.2).”
Accelerated Methods for Non-Convex Optimization (2016, background): “If the problem is locally non-convex, the Hessian must have a negative eigenvalue.”
On the Linear Convergence of a Proximal Gradient Method for a Class of Nonsmooth Convex Minimization Problems (2013, background): “For any convex function ϕ(x) (possibly nonsmooth), the Moreau–Yoshida proximity operator [15] prox ϕ : R n (cid:4)→ R n is deﬁned as Since (cid:3)·− x 2 is strongly convex and ϕ( · ) is convex, the minimizer of (6) exists and is unique, so the prox-operator prox ϕ is well-deﬁned.”
Nonsmooth optimization via quasi-Newton methods (2012, methodology): “We use ∂ f (x) to denote the Clarke subdifferential [9,45] of f at x , which for locally Lipschitz f is simply the convex hull of the limits of gradients of f evaluated at sequences converging to x [6, Theorem 6.”
Second-order growth, tilt stability, and metric regularity of the subdifferential (2013, background): “We refer the reader to [17, 26] for the latter subdifferential construction and to [26] for properties and applications of fully amenable functions.”