On Approximations of Data-Driven Chance Constrained Programs over Wasserstein Balls

Distributionally robust chance constrained programs minimize a deterministic cost function subject to the satisfaction of one or more safety conditions with high probability, given that the probability distribution of the uncertain problem parameters affecting the safety condition(s) is only known to belong to some ambiguity set. We study three popular approximation schemes for distributionally robust chance constrained programs over Wasserstein balls, where the ambiguity set contains all probability distributions within a certain Wasserstein distance to a reference distribution. The first approximation replaces the chance constraint with a bound on the conditional value-at-risk, the second approximation decouples different safety conditions via Bonferroni's inequality, and the third approximation restricts the expected violation of the safety condition(s) so that the chance constraint is satisfied. We show that the conditional value-at-risk approximation can be characterized as a tight convex approximation, which complements earlier findings on classical (non-robust) chance constraints, and we offer a novel interpretation in terms of transportation savings. We also show that the three approximations can perform arbitrarily poorly in data-driven settings, and that they are generally incomparable with each other.Comment: arXiv admin note: substantial text overlap with arXiv:1809.0021

Chance Constrained Optimization for Targeted Internet Advertising

We introduce a chance constrained optimization model for the fulfillment of guaranteed display Internet advertising campaigns. The proposed formulation for the allocation of display inventory takes into account the uncertainty of the supply of Internet viewers. We discuss and present theoretical and computational features of the model via Monte Carlo sampling and convex approximations. Theoretical upper and lower bounds are presented along with a numerical substantiation

A scenario approach for non-convex control design

Randomized optimization is an established tool for control design with modulated robustness. While for uncertain convex programs there exist randomized approaches with efficient sampling, this is not the case for non-convex problems. Approaches based on statistical learning theory are applicable to non-convex problems, but they usually are conservative in terms of performance and require high sample complexity to achieve the desired probabilistic guarantees. In this paper, we derive a novel scenario approach for a wide class of random non-convex programs, with a sample complexity similar to that of uncertain convex programs and with probabilistic guarantees that hold not only for the optimal solution of the scenario program, but for all feasible solutions inside a set of a-priori chosen complexity. We also address measure-theoretic issues for uncertain convex and non-convex programs. Among the family of non-convex control- design problems that can be addressed via randomization, we apply our scenario approach to randomized Model Predictive Control for chance-constrained nonlinear control-affine systems.Comment: Submitted to IEEE Transactions on Automatic Contro

Data-Driven Chance Constrained Optimization under Wasserstein Ambiguity Sets

We present a data-driven approach for distributionally robust chance constrained optimization problems (DRCCPs). We consider the case where the decision maker has access to a finite number of samples or realizations of the uncertainty. The chance constraint is then required to hold for all distributions that are close to the empirical distribution constructed from the samples (where the distance between two distributions is defined via the Wasserstein metric). We first reformulate DRCCPs under data-driven Wasserstein ambiguity sets and a general class of constraint functions. When the feasibility set of the chance constraint program is replaced by its convex inner approximation, we present a convex reformulation of the program and show its tractability when the constraint function is affine in both the decision variable and the uncertainty. For constraint functions concave in the uncertainty, we show that a cutting-surface algorithm converges to an approximate solution of the convex inner approximation of DRCCPs. Finally, for constraint functions convex in the uncertainty, we compare the feasibility set with other sample-based approaches for chance constrained programs.Comment: A shorter version is submitted to the American Control Conference, 201

A distributionally robust perspective on uncertainty quantification and chance constrained programming

The objective of uncertainty quantification is to certify that a given physical, engineering or economic system satisfies multiple safety conditions with high probability. A more ambitious goal is to actively influence the system so as to guarantee and maintain its safety, a scenario which can be modeled through a chance constrained program. In this paper we assume that the parameters of the system are governed by an ambiguous distribution that is only known to belong to an ambiguity set characterized through generalized moment bounds and structural properties such as symmetry, unimodality or independence patterns. We delineate the watershed between tractability and intractability in ambiguity-averse uncertainty quantification and chance constrained programming. Using tools from distributionally robust optimization, we derive explicit conic reformulations for tractable problem classes and suggest efficiently computable conservative approximations for intractable ones