May 15 - May 19, 2017

Venue: Lipschitz lecture hall, Mathematics Center, Endenicher Allee 60, 53115 Bonn
Organisers: Seyedehsomayeh Hosseini, Boris Mordukhovich, André Uschmajew

Description: As many problems in computer vision, robotics, signal processing and geometric mechanics are expressed as nonsmooth optimization problems, the huge impact of efficient methods to solve these problems is undeniable. Even solving difficult smooth problems sometimes leads to the use of nonsmooth optimization methods to make the problem either smaller in dimension or simpler in structure. It also may happen that a problem is analytically smooth, but numerically nonsmooth. Therefore, it is of eminent interest to develop useful computational and theoretical tools in these situations as well.

This workshop on nonsmooth optimization techniques and their applications will bring together leading experts from different research groups from all over the world to present the latest theoretical and practical achievements in this growing field, fostering collaboration and discussing future developments. Topics include, but are not limited to nonsmooth and variational analysis, optimization methods and their convergence analysis, applications in image processing, machine learning, etc.
In case of questions regarding this workshop, please contact noa2017(at)hcm.uni-bonn.de.

Michael Hintermüller: (Pre)Dualization, Dense Embeddings of Convex Sets, and Applications in Image Processing

For a class of non smooth minimization problems in Banach spaces, predualization results and their connection to dense embeddings of convex sets are discussed. Motivating applications are related to nonsmooth filters in mathematical image processing. For this problem class also some numerical aspects are highlighted including primal/dual splitting or ADMM-type methods as well as proper (numerical) dissipation reducing discretization.

Yuri Ledyaev: Multidirectional mean-value inequalities and dynamic optimization

We discuss recent results on generalized mean-value inequalities for lower semicontinuous functions for smooth Banach spaces. Previously, such inequlities have been used to establish relations between Dini derivatives and subgradients of lower semicontinuous functions, to derive metric regularity of functional relations or equivalence of viscosity and minimax solutions in theory of generalized solutions of first-order PDE. In this talk we demonstrate how new mean-value inequality can be used to provide shorter proofs of optimality conditions for  nonsmooth calculus of variation problems. We also discuss application of these results to nonsmooth dynamic optimization problems on manifolds.

Adil Baghirov: Nonsmooth DC Optimization: Methods and Applications

An unconstrained optimization problem with the objective function represented, implicitly or explicitly, as a difference of two convex functions is considered. Numerical methods for solving such problems are designed and their convergence is discussed. Applications of these methods in machine learning and regression analysis are also considered.

Helmut Gfrerer: On first-order optimality conditions for general optimization problems

The basic first-order optimality condition states that the negative gradient of the objective belongs to the regular normal cone of the feasible region. In the nonconvex case the computation of the regular normal cone is a nontrivial task, and therefore one has to confine himself with some estimates of it. It is easy to find a lower estimate of the regular normal cone  yielding so-called S-stationarity conditions. For an upper estimate one often uses  the limiting normal cone which results in so-called M-stationarity conditions. Though a comprehensive calculus is available for computing limiting normals, sometimes it is very difficult or even impossible to give a point-based representationof the limiting normal cone. In this talk we provide a tighter estimate for the regular normal cone by means of regular normals to a series on tangent cones to tangent cones of the feasible set. We present an example of a mathematical program with equilibrium constraints, where this approach yields manageable optimality conditions wheras it is impossible to write down the M-stationarity conditions.

James Burke: Variational Analysis of Convexly Generated Spectral Max Functions

The spectral abscissa is the largest real part of an eigenvalue of a matrix and the spectral radius is the largest modulus. Both are examples of spectral max functions---the maximum of a real-valued function over the spectrum of a matrix. These mappings arise in the control and stabilization of dynamical systems. In 2001, Burke and Overton characterized the regular subdifferential of the spectral abscissa and showed that the spectral abscissa is subdifferentially regular in the sense of Clarke if and only if all active eigenvalues are nonderogatory (geometric multiplicity of 1). In this talk we discuss how the same result can be obtained for the class of convexly generated spectral max functions.

Lukas Adam: Chance constrained problems: A view from variational analysis

We consider problems with joint chance constraints where the data are nonlinear, possibly nonconvex, random functions with finite distribution. Such problems are generally hard to tackle as the feasible set is a union of certain sets. We introduce integer variables and by their relaxation we arrive at an optimization problem with a MPCC-like structure. We derive strong stationarity conditions for this problem and relate them to the stationarity conditions of the original problem. Based on MPCC regularizations, we propose an iterative algorithm and show its convergence.

Coralia Cartis: Universal regularization methods -- varying the smoothness, the power and the accuracy

Adaptive cubic regularization methods have emerged as a credible alternative to linesearch and trust-region for smooth nonconvex optimization, with optimal complexity amongst second-order methods. Here we consider a general/new class of adaptive regularization methods, that use first- or higher-order local Taylor models of the objective regularized by a(ny) power of the step size and applied to convexly-constrained optimization problems.

We investigate the worst-case evaluation complexity/global rate of convergence of these algorithms, when the level of sufficient smoothness  of the objective may be unknown or may even be absent. We find that the methods accurately reflect in their complexity  the degree of smoothness of the objective and satisfy increasingly better bounds with improving accuracy of the models. The bounds vary continuously and robustly with respect to the regularisation power, the accuracy of the model and the degree of smoothness of the objective. This work is joint with Nick Gould (Rutherford Appleton Laboratory, UK) and Philippe Toint (University of Namur, Belgium).

Michael Ulbrich: A Proximal Gradient Method for Matrix Optimization Problems Arising in Ensemble Density Functional Theory

We present a proximal gradient method to solve a nonconvex optimization problem that is equivalent to the ensemble density functional theory (EDFT) model for electronic structure calculations in computational chemistry. The EDFT model is especially well suited for metallic systems. It can be cast as a matrix optimization problem with orthogonality constraints. Our algorithm uses an equivalent reformulation in which the singular values of the matrix-valued optimization variable arise in the cost function and the constraints. For the proposed proximal gradient method, convergence to stationary points is established. Numerical results show that this method can outperform the well-known self-consistent field iteration on many metallic systems.