Optimization Under Probabilistic Envelope Constraints

Abstract

Chance constraints are an important modeling tool in stochastic optimization, providing probabilistic guarantees that a solution "succeeds" in satisfying a given constraint. Although they control the probability of "success," they provide no control whatsoever in the event of a "failure." That is, they do not distinguish between a slight overshoot or undershoot of the bounds and more catastrophic violation. In short, they do not capture the magnitude of violation of the bounds. This paper addresses precisely this topic, focusing on linear constraints and ellipsoidal (Gaussian-like) uncertainties. We show that the problem of requiring different probabilistic guarantees at each level of constraint violation can be reformulated as a semi-infinite optimization problem. We provide conditions that guarantee polynomial-time solvability of the resulting semi-infinite formulation. We show further that this resulting problem is what has been called a comprehensive robust optimization problem in the literature. As a byproduct, we provide tight probabilistic bounds for comprehensive robust optimization. Thus, analogously to the connection between chance constraints and robust optimization, we provide a broader connection between probabilistic envelope constraints and comprehensive robust optimization.

Ambiguous Joint Chance Constraints Under Mean and Dispersion Information

“…Calafiore and El Ghaoui [14] employ various statistical bounds to approximate robust individual chance constraints where the ambiguity set specifies structural properties such as radial symmetry, unimodality or independence. Bertsimas et al [8] use statistical hypothesis tests to approximate robust individual chance constraints when the ambiguity set has to be estimated from samples of the unknown distribution Q. Xu et al [57] employ a generalized Chebyshev inequality to derive tractable reformulations of problems with probabilistic envelope constraints, which enforce robust chance constraints at all tolerance levels ∈ [0, 1). Instead of relying on statistical results, one can also employ duality of moment problems [10] to derive tractable reformulations of robust individual chance constraints.…”

Generalized Gauss inequalities via semidefinite programming

“…Vandenberghe et al [37] later showed that (P) admits an exact reformulation as a single SDP whenever Ξ is described through linear and quadratic inequalities and P contains all distributions sharing the same first and second-order moments. The resulting generalized Chebyshev bounds are widely used across many different application domains, ranging from distributionally robust optimization [9] to chance-constrained programming [8,39,43], stochastic control [36], machine learning [16], signal processing [38], option pricing [1,13,18], portfolio selection and hedging [40,44], decision theory [33] etc.…”

A distributionally robust perspective on uncertainty quantification and chance constrained programming

“…Ambiguity sets of special interest include the Markov ambiguity set containing all distributions with known mean and support [48], the Chebyshev ambiguity set containing all distributions with known bounds on the first and second-order moments [12,14,22,31,39,46,49,51,52], the Gauss ambiguity set containing all unimodal distributions from within the Chebyshev ambiguity set [38,41], various generalized Chebyshev ambiguity sets that specify asymmetric moments [12,13,35], higher-order moments [7,30,45] or marginal moments [17,18], the median-absolute deviation ambiguity set containing all symmetric distributions with known median and mean absolute deviation [24], the Huber ambiguity set containing all distributions with known upper bound on the expected Huber loss function [15,48], the Wasserstein ambiguity set containing all distributions that are close to the empirical distribution with respect to the Wasserstein metric [19,34,40], the KullbackLeibler divergence ambiguity set and likelihood ratio ambiguity set [10,26,27,31,47] containing all distributions that are sufficiently likely to have generated a given data set, the Hoeffding ambiguity set containing all component-wise independent distributions with a box support [3,8,10], the Bernstein ambiguity set containing all distributions from within the Hoeffding ambiguity set subject to marginal moment bounds [36], several φ-divergence-based ambiguity sets [2,…”