Prox-regularity and subjets (Q2776677)

This long paper offers an exhaustive treatment of second-order analysis of prox-regular functions introduced by Rockafellar and Poliquin. The first section deals with the Moreau smoothing via the infimal convolution. Here some special classes of functions, such as paraconvex, amenable, and fully amenable functions are considered. It is also investigated how certain generalized convexity notions lead to a smooth function when the infimal convolution is applied. Section 3 deals with sets of symmetric matrices and a duality with rank-one operators. One of the main results characterizes the rank-one support as a proper, lower semi-continuous positively homogeneous degree two even function. The next section focuses on generalized second-order derivatives. A characterization of the second-order (lower) epi-derivative of parabolically regular functions is offered. Section 5 deals with quadratic supports of rank one representers. The next section is devoted to the investigation of the second-order derivatives of the Moreau envelope. In the last section, necessary and sufficient conditions are obtained for unconstrained and constrained optimization problems utilizing the notions and results of the previous sections. The main emphasis is on the properties of second-order derivatives for prox-regular functions and the associated composite function formulation.NEWLINENEWLINEFor the entire collection see [Zbl 0974.00048].