324 research outputs found
Numerics and Fractals
Local iterated function systems are an important generalisation of the
standard (global) iterated function systems (IFSs). For a particular class of
mappings, their fixed points are the graphs of local fractal functions and
these functions themselves are known to be the fixed points of an associated
Read-Bajactarevi\'c operator. This paper establishes existence and properties
of local fractal functions and discusses how they are computed. In particular,
it is shown that piecewise polynomials are a special case of local fractal
functions. Finally, we develop a method to compute the components of a local
IFS from data or (partial differential) equations.Comment: version 2: minor updates and section 6.1 rewritten, arXiv admin note:
substantial text overlap with arXiv:1309.0243. text overlap with
arXiv:1309.024
Nonlocal error bounds for piecewise affine functions
The paper is devoted to a detailed analysis of nonlocal error bounds for
nonconvex piecewise affine functions. We both improve some existing results on
error bounds for such functions and present completely new necessary and/or
sufficient conditions for a piecewise affine function to have an error bound on
various types of bounded and unbounded sets. In particular, we show that any
piecewise affine function has an error bound on an arbitrary bounded set and
provide several types of easily verifiable sufficient conditions for such
functions to have an error bound on unbounded sets. We also present general
necessary and sufficient conditions for a piecewise affine function to have an
error bound on a finite union of polyhedral sets (in particular, to have a
global error bound), whose derivation reveals a structure of sublevel sets and
recession functions of piecewise affine functions
Second-order subdifferential calculus with applications to tilt stability in optimization
The paper concerns the second-order generalized differentiation theory of
variational analysis and new applications of this theory to some problems of
constrained optimization in finitedimensional spaces. The main attention is
paid to the so-called (full and partial) second-order subdifferentials of
extended-real-valued functions, which are dual-type constructions generated by
coderivatives of frst-order subdifferential mappings. We develop an extended
second-order subdifferential calculus and analyze the basic second-order
qualification condition ensuring the fulfillment of the principal secondorder
chain rule for strongly and fully amenable compositions. The calculus results
obtained in this way and computing the second-order subdifferentials for
piecewise linear-quadratic functions and their major specifications are applied
then to the study of tilt stability of local minimizers for important classes
of problems in constrained optimization that include, in particular, problems
of nonlinear programming and certain classes of extended nonlinear programs
described in composite terms
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