We are concerned with contingent derivatives and their second-order counterparts (introduced by Ngai et al.) of set-valued mappings. Special attention is given to the development of new sum-rules for second-order contingent derivatives. To be precise, we want to find conditions under which the second-order contingent derivative of the sum of a smooth and a set-valued … Read more
When dealing with general Lipschitzian optimization problems, there are many problem classes where even weak constraint qualications fail at local minimizers. In contrast to a constraint qualification, a problem qualification does not only rely on the constraints but also on the objective function to guarantee that a local minimizer is a Karush-Kuhn-Tucker (KKT) point. For … Read more
We consider a smooth pessimistic bilevel optimization problem, where the lower-level problem is convex and satisfies the Slater constraint qualification. These assumptions ensure that the Karush-Kuhn-Tucker (KKT) reformulation of our problem is well-defined. We then introduce and study the (i) Scholtes, (ii) Lin and Fukushima, (iii) Kadrani, Dussault and Benchakroun, (iv) Steffensen and Ulbrich, and … Read more
This paper develops a comprehensive convergence analysis for generic classes of descent algorithms in nonsmooth and nonconvex optimization under several conditions of the Polyak-Lojasiewicz-Kurdyka (PLK) type. Along other results, we prove the finite termination of generic algorithms under the PLK conditions with lower exponents. Specifications are given to establish new convergence rates for inexact reduced … Read more
We consider trust-region methods for solving optimization problems where the objective is the sum of a smooth, nonconvex function and a nonsmooth, convex regularizer. We extend the global convergence theory of such methods to include worst-case complexity bounds in the case of unbounded model Hessian growth, and introduce a new, simple nonsmooth trust-region subproblem solver … Read more
This paper studies the primal-dual convergence and iteration-complexity of proximal bundle methods for solving nonsmooth problems with convex structures. More specifically, we develop a family of primal-dual proximal bundle methods for solving convex nonsmooth composite optimization problems and establish the iteration-complexity in terms of a primal-dual gap. We also propose a class of proximal bundle … Read more
This work considers the nonconvex, nonsmooth problem of minimizing a composite objective of the form $f(g(x))+h(x)$ where the inner mapping $g$ is a smooth finite summation or expectation amenable to variance reduction. In such settings, prox-linear methods can enjoy variance-reduced speed-ups despite the existence of nonsmoothness. We provide a unified convergence theory applicable to a … Read more
Many of the primal ingredients of convex optimization extend naturally from Euclidean to Hadamard spaces — nonpositively curved metric spaces like Euclidean, Hilbert, and hyperbolic spaces, metric trees, and more general CAT(0) cubical complexes. Linear structure, however, and the duality theory it supports are absent. Nonetheless, we introduce a new type of subgradient for convex … Read more
The Alternating Direction Method of Multipliers (ADMM) is widely recognized for its efficiency in solving separable optimization problems. However, its application to optimization on Riemannian manifolds remains a significant challenge. In this paper, we propose a novel inertial Riemannian gradient ADMM (iRG-ADMM) to solve Riemannian optimization problems with nonlinear constraints. Our key contributions are as … Read more
Abs-smooth functions are given by compositions of smooth functions and the evaluation of the absolute value. The linear independence kink qualification (LIKQ) is a fundamental assumption in optimization problems governed by these abs-smooth functions, generalizing the well-known LICQ from smooth optimization. In particular, provided that LIKQ holds it is possible to derive optimality conditions for … Read more