Stetige äußere Subdifferentiale und deren Anwendung zur Optimierung lokal Lipschitz-stetiger Funktionen

Description

The theory of subdifferential provides adequate methods and tools to put descent methods into practice.

In the last forty years R.T. Rockafellar, F.H. Clarke, V.F. Demyanov, A.M. Rubinov and B.S. Mordukhovich amongst others developed different concepts to construct a subdifferential of convex or local Lipschitz continuous functions. In application it is often difficult to decide on a suitable subdifferential to construct a descent method. Furthermore there is often no exact information about the whole subdifferential for local Lipschitz continuous functions. A further problem is the lack of information about surroundings of a subdifferential. That means you can’t draw a conclusion regarding the characteristics of the cost function in the surroundings of a certain point even if the whole information of the subdifferential in this point is known.

In this thesis a new strategy for optimization problems with local Lipschitz continuous cost functions is introduced. Especially the semismoothness of the cost function won’t be taken for granted. The continuous outer subdifferentials we developed which remedy the lack of information of a subdifferential build the fundament of this strategy. A descent method based on this continuous subdifferentials will be developed and its convergence will be proved. In this context it will be analyzed which features of a subdifferential are essential for optimizing non-smooth functions. Thereby we answer the question for a suitable selection of a subdifferential. Another part of this thesis deals with the construction of continuous outer subdifferentials. Here the class of marginal functions is emphasized. These difficulties that appear during construction will be discussed by academic examples.

In the last part oh this thesis we introduce a new method to minimize local Lipschitz continuous optimal value functions (BTO). The basis are Bundle-Trust-Region-ideas for convex cost functions and Armijo’s rule line search for smooth functions. The idea to construct a model function is adapted from Bundle-Trust-Region-Methods.

This idea bases upon approximate continuous outer subdifferential, previously introduced in this thesis. Moreover we generalize the strategy of adapting Trust-Region-Ratio and present a new method to improve successively the model function. Concluding the global convergence of this BTOmethod will be proved.