The Analysis and Optimization of Probability Functions

A problem of probability function optimization is considered. This function represents probability that some random quantity depending on deterministic parameters does not exceed some given level. The problem is motivated by studies of safety domains and risk control problems in complex stochastic systems. For example, pollution control includes maximization of probability that some given levels of deposition at reception points are not exceeded. Optimization of probability function is performed over a given range of parameters. To solve the problem stochastic quasi-gradient method is applied under quasi-concavity assumption on functions and measures involved. Convergence and rate of convergence results are presented

Concavity and Efficient Points of Discrete Distributions in Probabilistic Programming

We consider stochastic programming problems with probabilistic constraints involving integer-valued random variables. The concept of a p-efficient point of a probability distribution is used to derive various equivalent problem formulations. Next we introduce the concept of r-concave discrete probability distributions and analyse its relevance for problems under consideration. These notions are used to derive lower and upper bounds for the optimal value of probabilistically constrained stochastic programming problems with discrete random variables. The results are illustrated with numerical examples

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

Metric regularity and quantitative stability in stochastic programs with probabilistic constraints

Necessary and sufficient conditions for metric regularity of (several joint) probabilistic constraints are derived using recent results from nonsmooth analysis. The conditions apply to fairly general nonconvex, nonsmooth probabilistic constraints and extend earlier work in this direction. Further, a verifiable sufficient condition for quadratic growth of the objective function in a more specific convex stochastic program is indicated and applied in order to obtain a new result on quantitative stability of solution sets when the underlying probability distribution is subjected to perturbations. This is used to establish a large deviation estimate for solution sets when the probability measure is replaced by empirical ones

On the Convexity of Level-sets of Probability Functions

In decision-making problems under uncertainty, probabilistic constraints are a valuable tool to express safety of decisions. They result from taking the probability measure of a given set of random inequalities depending on the decision vector. Even if the original set of inequalities is convex, this favourable property is not immediately transferred to the probabilistically constrained feasible set and may in particular depend on the chosen safety level. In this paper, we provide results guaranteeing the convexity of feasible sets to probabilistic constraints when the safety level is greater than a computable threshold. Our results extend all the existing ones and also cover the case where decision vectors belong to Banach spaces. The key idea in our approach is to reveal the level of underlying convexity in the nominal problem data (e.g., concavity of the probability function) by auxiliary transforming functions. We provide several examples illustrating our theoretical developments

Solving joint chance constrained problems using regularization and Benders' decomposition

In this paper we investigate stochastic programms with joint chance constraints. We consider discrete scenario set and reformulate the problem by adding auxiliary variables. Since the resulting problem has a difficult feasible set, we regularize it. To decrease the dependence on the scenario number, we propose a numerical method by iteratively solving a master problem while adding Benders cuts. We find the solution of the slave problem (generating the Benders cuts) in a closed form and propose a heuristic method to decrease the number of cuts. We perform a numerical study by increasing the number of scenarios and compare our solution with a solution obtained by solving the same problem with continuous distribution