| Summary: | In this dissertation we discuss certain aspects of a parametric regularization technique which is based on recent work by R. Goebel. For proper, lower semicontinuous, and convex functions, this regularization is self-dual with respect to convex conjugation, and a simple extension of this smoothing exhibits the same feature when applied to proper, closed, and saddle functions.
In Chapter 1 we give a introduction to convex and saddle function theory, which includes new results on the convergence of saddle function values that were not previously available in the form presented. In Chapter 2, we define the regularization and extend some of the properties previously shown in the convex case to the saddle one. Furthermore, we investigate the properties of this regularization without convexity assumptions. In particular, we show that for a prox-bounded function the family of infimal values of the regularization converges to the infimal values of the given function, even when the given function might not have a minimizer. Also we show that for a general type of prox-regular functions the regularization is locally convex, even though their Moreau envelope might fail to have this property. Moreover, we apply the regularization technique to Lagrangians of convex optimization problems in two different settings, and describe the convergence of the associated saddle values and the value functions.
We also employ the regularization in fully convex problems in calculus of variations, in Chapter 3, in the setting studied by R. Rockafellar and P. Wolenski. In this case, we extend a result by Rockafellar on the Lipschitz continuity of the proximal mapping of the value function jointly in the time and state variables, which in turn implies the same regularity for the gradient of the self-dual regularization.
Finally, we attach a software code to use with SCAT (Symbolic Convex Analysis Toolbox) in order to symbolically compute the regularization for functions of one variable.
|
|---|
