49J53 Set-valued and variational analysis [See also 28B20, 47H04, 54C60, 58C06]
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Keywords
- calmness (2)
- Contingent Derivative (1)
- Elliptic MPEC (1)
- Epiconvergence (1)
- Epiderivative (1)
- M-stationarity (1)
- Mordukhovich coderivative (1)
- Pointwise Gradient Constraints (1)
- Second-order subdifferential (1)
- Strong Stationarity (1)
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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.
This paper deals with the computation of regular coderivatives of
solution maps associated with a frequently arising class of generalized equations.
The constraint sets are given by (not necessarily convex) inequalities,
and we do not assume linear independence of gradients to active constraints.
The achieved results enable us to state several versions of sharp necessary optimality
conditions in optimization problems with equilibria governed by such
generalized equations. The advantages are illustrated by means of examples.
Mathematical programs in which the constraint set is partially defined by the solutions of an elliptic variational inequality, so-called ``elliptic MPECs'', are formulated in reflexive Banach spaces. With the goal of deriving explicit first order optimality conditions amenable to the development of numerical procedures, variational analytic concepts are both applied and further developed. The paper is split into two main parts. The first part concerns the derivation of conditions in which the state constraints are assumed to be polyhedric sets. This part is then completed by two examples, the latter of which involves pointwise bilateral bounds on the gradient of the state. The second part begins with the derivation of a formula for the second order (Mosco) epiderivative of the indicator function of a general convex set. This result is then used to derive analogous conditions to those which are presented in the first part. Finally, an elliptic MPEC is considered important to the study of elasto-plasticity in which the pointwise Euclidean norm of the gradient of the state is bounded. Explicit strong stationarity conditions are provided for this problem.
In this article we compare two different calmness conditions which are
widely used in the literature on bilevel programming and on mathematical
programs with equilibrium constraints. In order to do so, we consider convex
bilevel programming as a kind of intersection between both research areas.
The so-called partial calmness concept is based on the function value
approach for describing the lower level solution set. Alternatively,
calmness in the sense of multifunctions may be considered for perturbations
of the generalized equation representing the same lower level solution set.
Both concepts allow to derive first order necessary optimality conditions
via tools of generalized differentiation introduced by Mordukhovich. They
are very different, however, concerning their range of applicability and the
form of optimality conditions obtained. The results of this paper seem to
suggest that partial calmness is considerably more restrictive than calmness
of the perturbed generalized equation. This fact is also illustrated by
means of a dicretized obstacle control problem.
The paper deals with co-derivative formulae for normal cone mappings to smooth
inequality systems. Both the regular (Linear Independence Constraint Qualification satisfied)
and nonregular (Mangasarian-Fromovitz Constraint Qualification satisfied) cases are considered.
A major part of the results relies on general transformation formulae previously obtained by
Mordukhovich and Outrata. This allows one to derive exact formulae for general smooth, regular and polyhedral, possibly
nonregular systems. In the nonregular, nonpolyhedral case a generalized transformation formula by
Mordukhovich and Outrata applies, however, a major difficulty consists in
checking a calmness condition of a certain multivalued mapping. The paper provides a translation
of this condition in terms of much easier to verify constraint qualifications. The final section is
devoted to the situation where the calmness condition is violated. A series of examples
illustrates the use and comparison of the presented formulae.