49 research outputs found
About [q]-regularity properties of collections of sets
We examine three primal space local Hoelder type regularity properties of
finite collections of sets, namely, [q]-semiregularity, [q]-subregularity, and
uniform [q]-regularity as well as their quantitative characterizations.
Equivalent metric characterizations of the three mentioned regularity
properties as well as a sufficient condition of [q]-subregularity in terms of
Frechet normals are established. The relationships between [q]-regularity
properties of collections of sets and the corresponding regularity properties
of set-valued mappings are discussed.Comment: arXiv admin note: substantial text overlap with arXiv:1309.700
Error Bounds and Holder Metric Subregularity
The Holder setting of the metric subregularity property of set-valued
mappings between general metric or Banach/Asplund spaces is investigated in the
framework of the theory of error bounds for extended real-valued functions of
two variables. A classification scheme for the general Holder metric
subregularity criteria is presented. The criteria are formulated in terms of
several kinds of primal and subdifferential slopes.Comment: 32 pages. arXiv admin note: substantial text overlap with
arXiv:1405.113
On implicit variables in optimization theory
Implicit variables of a mathematical program are variables which do not need
to be optimized but are used to model feasibility conditions. They frequently
appear in several different problem classes of optimization theory comprising
bilevel programming, evaluated multiobjective optimization, or nonlinear
optimization problems with slack variables. In order to deal with implicit
variables, they are often interpreted as explicit ones. Here, we first point
out that this is a light-headed approach which induces artificial locally
optimal solutions. Afterwards, we derive various Mordukhovich-stationarity-type
necessary optimality conditions which correspond to treating the implicit
variables as explicit ones on the one hand, or using them only implicitly to
model the constraints on the other. A detailed comparison of the obtained
stationarity conditions as well as the associated underlying constraint
qualifications will be provided. Overall, we proceed in a fairly general
setting relying on modern tools of variational analysis. Finally, we apply our
findings to different well-known problem classes of mathematical optimization
in order to visualize the obtained theory.Comment: 33 page