Nonsmooth Trust Region Algorithms for Locally Lipschitz Functions on Riemannian Manifolds

This paper presents a Riemannian trust region algorithm for unconstrained optimization problems with locally Lipschitz objective functions defined on complete Riemannian manifolds. To this end we define a function Φ:TM→ℝ on the tangent bundle TM, and at the kth iteration, using the restricted function Φ|TxkM, where TxkM is the tangent space at xk, a local model function Qk that carries both first- and second-order information for the locally Lipschitz objective function f:M→ℝ on a Riemannian manifold M, is defined and minimized over a trust region. We establish the global convergence of the proposed algorithm. Moreover, using the Riemannian ε-subdifferential, a suitable model function is defined. Numerical experiments illustrate our results.ISSN:0272-4979ISSN:1464-364

Equilibria on LL-retracts in Riemannian manifolds

We introduce a class of subsets of Riemannian manifolds called the $L$-retract. Next we consider a topological degree for set-valued upper semicontinuous maps defined on open sets of compact $L$-retracts in Riemannian manifolds. Then, we present a theorem on the existence of equilibria (or zeros) of an upper semicontinuous set-valued map with nonempty closed convex values satisfying the tangency condition defined on a compact $L$-retract in a Riemannian manifold

Nonsmooth optimization and its applications

Since nonsmooth optimization problems arise in a diverse range of real-world applications, the potential impact of efficient methods for solving such problems is undeniable. Even solving difficult smooth problems sometimes requires the use of nonsmooth optimization methods, in order to either reduce the problem’s scale or simplify its structure. Accordingly, the field of nonsmooth optimization is an important area of mathematical programming that is based on by now classical concepts of variational analysis and generalized derivatives, and has developed a rich and sophisticated set of mathematical tools at the intersection of theory and practice. This special issue of ISNM is an outcome of the workshop "Nonsmooth Optimization and its Applications," which was held from May 15 to 19, 2017 at the Hausdorff Center for Mathematics, University of Bonn. The six research articles gathered here focus on recent results that highlight different aspects of nonsmooth and variational analysis, optimization methods, their convergence theory and applications