7 research outputs found
A simple formula for the second-order subdifferential of maximum functions
We derive a simple formula for the second-order subdifferential of the maximum of coordinates which allows us to construct this set immediately from its argument and the direction to which it is applied. This formula can be combined with a chain rule recently proved by Mordukhovich and Rockafellar [9] in order to derive a similarly simple formula for the extended partial second-order subdifferential of finite maxima of smooth functions. Analogous formulae can be derived immediately for the full and conventional partial second-order subdifferentials
Second-order subdifferential of 1- and maximum norm
We derive formulae for the second-order subdifferential of polyhedral norms. These formulae are fully explicit in terms of initial data. In a first step we rely on the explicit formula for the coderivative of normal cone mapping to polyhedra. Though being explicit, this formula is quite involved and difficult to apply. Therefore, we derive simple formulae for the 1-norm and -- making use of a recently obtained formula for the second-order subdifferential of the maximum function -- for the maximum norm
Second-order subdifferential of 1- and maximum norm
We derive formulae for the second-order subdifferential of polyhedral
norms. These formulae are fully explicit in terms of initial data. In a first
step we rely on the explicit formula for the coderivative of normal cone
mapping to polyhedra. Though being explicit, this formula is quite involved
and difficult to apply. Therefore, we derive simple formulae for the 1-norm
an
On the Aubin property of solution maps to parameterized variational systems with implicit constraints
In the paper, a new sufficient condition for the Aubin property to a class of parameterized variational systems is derived. In these systems, the constraints depend both on the parameter as well as on the decision variable itself and they include, e.g. parameter-dependent quasi-variational inequalities and implicit complementarity problems. The result is based on a general condition ensuring the Aubin property of implicitly defined multifunctions which employs the recently introduced notion of the directional limiting coderivative. Our final condition can be verified, however, without an explicit computation of these coderivatives. The procedure is illustrated by an example. © 2019, © 2019 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.The research of the first author was supported by the Austrian Science Fund (FWF) under grant P29190-N32. The research of the second author was supported by the Grant Agency of the Czech Republic, Project 17-04301S and the Australian Research Council, Project 10.13039/501100000923DP160100854
On computation of limiting coderivatives of the normal-cone mapping to inequality systems and their applications
The paper concerns the computation of the limiting coderivative of the normalcone mapping related to C² inequality constraints under weak qualification conditions. The obtained results are applied to verify the Aubin property of solution maps to a class of parameterized generalized equations
Bilevel programming: reformulations, regularity, and stationarity
We have considered the bilevel programming problem in the case where the lower-level problem admits more than one optimal solution. It is well-known in the literature that in such a situation, the problem is ill-posed from the view point of scalar objective optimization. Thus the optimistic and pessimistic approaches have been suggested earlier in the literature to deal with it in this case. In the thesis, we have developed a unified approach to derive necessary optimality conditions for both the optimistic and pessimistic bilevel programs, which is based on advanced tools from variational analysis. We have obtained various constraint qualifications and stationarity conditions depending on some constructive representations of the solution set-valued mapping of the follower’s problem. In the auxiliary developments, we have provided rules for the generalized differentiation and robust Lipschitzian properties for the lower-level solution setvalued map, which are of a fundamental interest for other areas of nonlinear and nonsmooth optimization.
Some of the results of the aforementioned theory have then been applied to derive stationarity conditions for some well-known transportation problems having the bilevel structure
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