doi 10.1287/moor.1040.0094

Abstract. We consider a chance constrained problem, where one seeks to minimize a convex objective over solutions satisfying, with a given close to one probability, a system of randomly perturbed convex constraints. This problem may happen to be computationally intractable; our goal is to build its computationally tractable approximation, i.e., an efficiently solvable deterministic optimization program with the feasible set contained in the chance constrained problem. We construct a general class of such convex conservative approximations of the corresponding chance constrained problem. Moreover, under the assumptions that the constraints are affine in the perturbations and the entries in the perturbation vector are independent-of-each-other random variables, we build a large deviation-type approximation, referred to as “Bernstein approximation, ” of the chance constrained problem. This approximation is convex and efficiently solvable. We propose a simulation-based scheme for bounding the optimal value in the chance constrained problem and report numerical experiments aimed at comparing the Bernstein and well-known scenario approximation approaches. Finally, we extend our construction to the case of ambiguous chance constrained problems, where the random perturbations are independent with the collection of distributions known to belong to a given convex compact set rather than to be known exactly, while the chance constraint should be satisfied for every distribution given by this set.

In this paper, we introduce an approach for constructing uncertainty sets for robust optimization using new deviation measures for bounded random variables known as the forward and backward de-viations. These deviation measures capture distributional asymmetry and lead to better approxima-tions of chance constraints. We also propose a tractable robust optimization approach for obtaining robust solutions to a class of stochastic linear optimization problems where the risk of infeasibility can be tolerated as a tradeoff to improve upon the objective value. An attractive feature of the framework is the computational scalability to multiperiod models. We show an application of the framework for solving a project management problem with uncertain activity completion time.

In this paper we study ambiguous chance constrained problems where the distributions of the random parameters in the problem are themselves uncertain. We primarily focus on the special case where the uncertainty set Q of the distributions is of the form Q = {Q : # p (Q, Q 0 ) # #}, where # p denotes the Prohorov metric. The ambiguous chance constrained problem is approximated by a robust sampled problem where each constraint is a robust constraint centered at a sample drawn according to the central measure Q 0 . The main contribution of this paper is to show that the robust sampled problem is a good approximation for the ambiguous chance constrained problem with high probability. This result is established using the Strassen-Dudley Representation Theorem that states that when the distributions of two random variables are close in the Prohorov metric one can construct a coupling of the random variables such that the samples are close with high probability. We also show that the robust sampled problem can be solved e#ciently both in theory and in practice. 1

We consider a chance constraint Prob{ξ: A(x, ξ) ∈ K} ≥ 1 − ɛ (x is the decision vector, ξ is a random perturbation, K is a closed convex cone, and A(·, ·) is bilinear). While important for many applications in Optimization and Control, chance constraints typically are “computationally intractable”, which makes it necessary to look for their tractable approximations. We present these approximations for the cases when the underlying conic constraint A(x, ξ) ∈ K is (a) scalar inequality, or (b) conic quadratic inequality, or (c) linear matrix inequality, and discuss the level of conservativeness of the approximations. 1 The problem Consider a randomly perturbed convex constraint in the conic form: k� Aξ,σ(x) = A0(x) + σ ξiAi(x) ∈ K, (1) where i=1 • Ai(·) are affine mappings from R n to finite-dimensional real vector space E, and x ∈ R n is the decision vector; • ξi are scalar random perturbations satisfying the relations (a) : ξi are mutually independent; (b) : E {ξi} = 0; (c) : E � exp{ξ 2 i /4} � ≤ √ 2 (2) Cases of primary interest: — ξi ∼ N (0, 1) (“Gaussian noise”; the absolute constants in (2.c) come exactly from the desire to make the relation valid for the standard Gaussian perturbations); — E{ξi} = 0, |ξi | ≤ 1 (“bounded random noise”). • σ ≥ 0 is the level of perturbations, • K is a closed pointed convex cone in E. Cases of primary interest: — E = R, K = R+; here (1) is a scalar linear inequality;

Randomized constraint sampling has recently been proposed as an approach for approximating solutions to optimization problems when the number of constraints is intractable – say, a googol or even infinity. The idea is to define a probability distribution ψ over the set of constraints and to sample a subset

In this paper we survey the primary research, both theoretical and applied, in the field of Robust Optimization (RO). Our focus will be on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying the most prominent theoretical results of RO over the past decade, we will also present some recent results linking RO to adaptable models for multi-stage decision-making problems. Finally, we will highlight successful applications of RO across a wide spectrum of domains, including, but not limited to, finance, statistics, learning, and engineering.

Linear programs with joint probabilistic constraints (PCLP) are difficult to solve because the feasible region is not convex. We consider a special case of PCLP in which only the right-hand side is random and this random vector has a finite distribution. We give a mixed-integer programming formulation for this special case and study the relaxation corresponding to a single row of the probabilistic constraint. We obtain two strengthened formulations. As a byproduct of this analysis, we obtain new results for the previously studied mixing set, subject to an additional knapsack inequality. We present computational results which indicate that by using our strengthened formulations, instances that are considerably larger than have been considered before can be solved to optimality.

doi 10.1287/opre.1080.0646

Large templated factor graphs with complex structure that changes during inference have been shown to provide state-of-the-art experimental results on tasks such as identity uncertainty and information integration. However, learning parameters in these models is difficult because computing the gradients require expensive inference routines. In this paper we propose an online algorithm that instead learns preferences over hypotheses from the gradients between the atomic steps of inference. Although there are a combinatorial number of ranking constraints over the entire hypothesis space, a connection to the frameworks of sampled convex programs reveals a polynomial bound on the number of rankings that need to be satisfied in practice. We further apply ideas of passive aggressive algorithms to our update rules, enabling us to extend recent work in confidenceweighted classification to structured prediction problems. We compare our algorithm to structured perceptron, contrastive divergence, and persistent contrastive divergence, demonstrating substantial error reductions on two real-world problems (20 % over contrastive divergence).