Qualitative Stability of Convex Programs with Probabilistic Constraints

We consider convex stochastic optimization problems with probabilistic constraints which are defined by so-called r-concave probability measures. Since the true measure is unknown in general, the problem is usually solved on the basis of estimated approximations, hence the issue of perturbation analysis arises in a natural way. For the solution set mapping and for the optimal value function, stability results are derived. In order to include the important class of empirical estimators, the perturbations are allowed to be arbitrary in the space of probability measures (in contrast to the convexity property of the original measure). All assumptions relate to the original problem. Examples show the necessity of the formulated conditions and illustrate the sharpness of results in the respective settings

Perturbation analysis of chance-constrained programs under variation of all constraint data

We consider stability of solutions to optimization problems with probabilistic constraints under perturbations of all constraint data (probability level, probability measure, deterministic constraints, random set mapping). Constraint qualifications ensuring stability are derived for each of the single parameters. Examples illustrating the necessity of the stated conditions as well as the limitations of the given results are provided

An enumerative formula for the spherical cap discrepancy

The spherical cap discrepancy is a widely used measure for how uniformly a sample of points on the sphere is distributed. Being hard to compute, this discrepancy measure is typically replaced by some lower or upper estimates when designing optimal sampling schemes for the uniform distribution on the sphere. In this paper, we provide a fully explicit, easy to implement enumerative formula for the spherical cap discrepancy. Not surprisingly, this formula is of combinatorial nature and, thus, its application is limited to spheres of small dimension and moderate sample sizes. Nonetheless, it may serve as a useful calibrating tool for testing the efficiency of sampling schemes and its explicit character might be useful also to establish necessary optimality conditions when minimizing the discrepancy with respect to a sample of given size

Optimality conditions in control problems with random state constraints in probabilistic or almost-sure form

In this paper, we discuss optimality conditions for optimization problems involving random state constraints, which are modeled in probabilistic or almost sure form. While the latter can be understood as the limiting case of the former, the derivation of optimality conditions requires substantially different approaches. We apply them to a linear elliptic partial differential equation (PDE) with random inputs. In the probabilistic case, we rely on the spherical-radial decomposition of Gaussian random vectors in order to formulate fully explicit optimality conditions involving a spherical integral. In the almost sure case, we derive optimality conditions and compare them to a model based on robust constraints with respect to the (compact) support of the given distribution

On calmness conditions in convex bilevel programming

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

Stability of solutions to chance constrained stochastic programs

Perturbations of convex chance constrained stochastic programs are considered the underlying probability distributions of which are r-concave. Verifiable sufficient conditions are established guaranteeing Hölder continuity properties of solution sets with respect to variations of the original distribution. Examples illustrate the potential, sharpness and limitations of the results

Problem-based optimal scenario generation and reduction in stochastic programming

Scenarios are indispensable ingredients for the numerical solution of stochastic programs. Earlier approaches to optimal scenario generation and reduction are based on stability arguments involving distances of probability measures. In this paper we review those ideas and suggest to make use of stability estimates based only on problem specific data. For linear two-stage stochastic programs we show that the problem-based approach to optimal scenario generation can be reformulated as best approximation problem for the expected recourse function which in turn can be rewritten as a generalized semi-infinite program. We show that the latter is convex if either right-hand sides or costs are random and can be transformed into a semi-infinite program in a number of cases. We also consider problem-based optimal scenario reduction for two-stage models and optimal scenario generation for chance constrained programs. Finally, we discuss problem-based scenario generation for the classical newsvendor problem

On properties of different notions of centers for convex cones

The points on the revolution axis of a circular cone are somewhat special: they are the "most interior'' elements of the cone. This paper addresses the issue of formalizing the concept of center for a convex cone that is not circular. Four distinct proposals are studied in detail: the incenter, the circumcenter, the inner center, and the outer center. The discussion takes place in the context of a reflexive Banach space

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 secondorder subdifferentials

Uniform boundedness of norms of convex and nonconvex processes

The lower limit of a sequence of closed convex processes is again a closed convex process. In this note we prove the following uniform boundedness principle: if the lower limit is nonempty-valued everywhere, then, starting from a certain index, the given sequence is uniformly norm-bounded. As shown with an example, the uniform boundedness principle is not true if one drops convexity. By way of illustration, we consider an application to the controllability analysis of differential inclusions