We propose to approximate two-stage distributionally robust programs with binary recourse decisions by their associated K-adaptability problems, which pre-select K candidate secondstage policies here-and-now and implement the best of these policies once the uncertain parameters have been observed. We analyze the approximation quality and the computational complexity of the K-adaptability problem, and we derive explicit mixed-integer linear programming reformulations. We also provide efficient procedures for bounding the probabilities with which each of the K second-stage policies is selected.
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.
Adaptive robust optimization problems are usually solved approximately by restricting the adaptive decisions to simple parametric decision rules. However, the corresponding approximation error can be substantial. In this paper we show that two-stage robust and distributionally robust linear programs can often be reformulated exactly as conic programs that scale polynomially with the problem dimensions.Specifically, when the ambiguity set constitutes a 2-Wasserstein ball centered at a discrete distribution, then the distributionally robust linear program is equivalent to a copositive program (if the problem has complete recourse) or can be approximated arbitrarily closely by a sequence of copositive programs (if the problem has sufficiently expensive recourse). These results directly extend to the classical robust setting and motivate strong tractable approximations of two-stage problems based on semidefinite approximations of the copositive cone. We also demonstrate that the two-stage distributionally robust optimization problem is equivalent to a tractable linear program when the ambiguity set constitutes a 1-Wasserstein ball centered at a discrete distribution and there are no support constraints. arXiv:1609.07505v2 [math.OC]
We study joint chance constraints where the distribution of the uncertain parameters is only known to belong to an ambiguity set characterized by the mean and support of the uncertainties and by an upper bound on their dispersion. This setting gives rise to pessimistic (optimistic) ambiguous chance constraints, which require the corresponding classical chance constraints to be satisfied for every (for at least one) distribution in the ambiguity set. We demonstrate that the pessimistic joint chance constraints are conic representable and thus computationally tractable if (i) the constraint coefficients of the decisions are deterministic, (ii) the support set of the uncertain parameters is a cone, and (iii) their dispersion function is positively homogeneous. We also show that tractability is lost as soon as either of the conditions (i), (ii) or (iii) is relaxed in the mildest possible way. We further prove that the optimistic joint chance constraints are tractable if and only if (i) holds. To showcase the power of our tractability results, we solve large-scale project management and image reconstruction models to global optimality.
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