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We study perturbations of a stochastic program with a probabilistic constraint and r-concave original probability distribution. First we improve our earlier results substantially and provide conditions implying Hölder continuity properties of the solution sets w.r.t. the Kolmogorov distance of probability distributions. Secondly, we derive an upper Lipschitz continuity property for solution sets under more restrictive conditions on the original program and on the perturbed probability measures. The latter analysis applies to linear-quadratic models and is based on work by Bonnans and Shapiro. The stability results are illustrated by numerical tests showing the different asymptotic behaviour of parametric and nonparametric estimates in a program with a normal probabilistic constraint.
The paper provides a structural analysis of the feasible set defined by linear probabilistic constraints. Emphasis is laid on single (individual) probabilistic constraints. A classical convexity result by Van de Panne/Popp and Kataoka is extended to a broader class of distributions and to more general functions of the decision vector. The range of probability levels for which convexity can be expected is exactly identified. Apart from convexity, also nontriviality and compactness of the
feasible set are precisely characterized at the same time. The relation between feasible sets with negative and with nonnegative right-hand side is revealed. Finally, an existence result is formulated for the more difficult case of joint probabilistic constraints.
Discrete approximations to chance constrained and mixed-integer two-stage stochastic programs require moderately sized scenario
sets. The relevant distances of (multivariate) probability
distributions for deriving quantitative stability results for such stochastic programs are $\mathcal{B}$-discrepancies, where the class $\mathcal{B}$ of Borel sets depends on their structural properties.
Hence, the optimal scenario reduction problem for such models is stated with respect to $\mathcal{B}$-discrepancies. In this paper,
upper and lower bounds, and some explicit solutions for optimal scenario reduction problems are derived. In addition, we develop
heuristic algorithms for determining nearly optimally reduced probability measures, discuss the case of the cell discrepancy (or
Kolmogorov metric) in some detail and provide some numerical experience.
This paper concerns second-order analysis for a remarkable class of variational systems in finite-dimensional and infinite-dimensional spaces, which is particularly important for the study of optimization and equilibrium problems with equilibrium constraints. Systems of this type are described via variational inequalities over polyhedral convex sets and allow us to provide a comprehensive local analysis by using appropriate generalized differentiation of the normal cone mappings for such sets. In this paper we efficiently compute the required coderivatives of the normal cone mappings exclusively via the initial data of polyhedral sets in reflexive Banach spaces. This provides the main tools of second-order variational analysis allowing us, in particular, to derive necessary and sufficient conditions for robust Lipschitzian stability of solution maps to parameterized variational inequalities with evaluating the exact bound of the corresponding Lipschitzian moduli. The efficient coderivative calculations and characterizations of robust stability obtained in this paper are the first results in the literature for the problems under consideration in infinite-dimensional spaces. Most of them are also new in finite dimensions.
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.
In this paper, a model for (joint) dynamic chance constraints is proposed and applied to an optimization problem in water reservoir management. The model relies on discretization of the decision variables but keeps the probability distribution
continuous. Our approach relies on calculating probabilities of rectangles which is particularly useful in the presence of independent random variables but works for a moderate number of stages equally well in case of correlated variables. Numerical results are provided for two and three stages.
In this paper, we consider the characterization of strong stationary solutions to
equilibrium problems with equilibrium constraints (EPECs). Assuming that the underlying
generalized equation satisfies strong regularity in the sense of Robinson, an explicit
multiplier-based stationarity condition can be derived. This is applied then
to an equilibrium model arising from ISO-regulated electricity spot markets.
On probabilistic constraints induced by rectangular sets and multivariate normal distributions
(2009)
In this paper, we consider optimization problems under probabilistic constraints which are defined by two-sided
inequalities for the underlying normally distributed random vector. As a main step
for an algorithmic solution of such problems, we derive a derivative formula for (normal) probabilities
of rectangles as functions of their lower or upper bounds. This formula allows to reduce the calculus
of such derivatives to the calculus of (normal) probabilities of rectangles themselves thus generalizing a
similar well-known statement for multivariate normal distribution functions. As an application, we consider
a problem from water reservoir management. One of the outcomes of the problem solution is that the
(still frequently encountered) use of simple individual probabilistic can completely fail. In contrast, the
(more difficult) use of joint probabilistic constraints which heavily depends on the derivative formula mentioned
before yields very reasonable and robust solutions over the whole time horizon considered.
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.
An optimal control problem to find the fastest collision-free trajectory of a robot surrounded by obstacles is presented.
The collision avoidance is based on linear programming arguments and expressed as state constraints. The optimal control problem is
solved with a sequential programming method. In order to decrease the number of unknowns and constraints a backface culling active set
strategy is added to the resolution technique.