Variational Techniques in Nonsmooth Optimization

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Finite and infinite dimensional optimization problems naturally arise in a vast array of applied and theoretical problems in the natural sciences, business and economics, management sciences, information sciences, and engineering. A common theme in modern methods, theory, and applications is nonsmoothness. Nonsmoothness arises either directly through the modeling structure or indirectly through the variational properties of the optimal value function and solution mapping. Prominent examples where nonsmoothness plays a central role is in sparsity optimization, maximum likelihood methods for robust statistics, machine learning, robust optimization, mathematical programs with equilibrium constraints (MPECs), mathematical programs with vanishing constraints (MPVCs), (generalized) Nash equilibrium problems ((G)NEPs) or eigenvalue optimization, just to name a few areas. The theoretical tools for dealing with nonsmoothness are subsumed under the label "variational analysis", which comprises convex, nonsmooth and set-valued analysis among other things. The focus of this research project lies on smoothing methods, using variational techniques for both constructing and analyzing smooth approximations of nonsmooth functions.Smoothing methods constitute a standard approach to solving nonsmooth and constrained optimization problems by solving a related sequence of unconstrained smooth approximations. The approximations are constructed so that cluster points of the solutions or stationary points of the approximating smooth problems are solutions or stationary points for the limiting nonsmooth or constrained optimization problem. In the setting of convex programming, there is now great interest in these methods for solving very large-scale problems, where first-order methods for convex nonsmooth optimization have been very successful. In a small- to medium-scale setting, however, second-order methods, in particular, semismooth Newton methods received much attention and are now being used in the infinite-dimensional setting to solve PDE-constrained optimization problems. This project emphasizes so-called epi-smoothing functions, which rely on the notionof epi-convergence of sequences of functionals, and includes two broad areas of study: The first is on second-order methods for a certain class of epi-smoothing functions, namely for those based on infimal convolution, and the second aims at adapting the concept of epi-smoothing to infinite-dimensional spaces due to the importance of epi-convergence in the function space setting. In addition to the epi-smoothing, integral convolution smoothing functions are to be analyzed in order to generalize and strengthen existing results for this important class of smoothing functions.