A Sample Approximation Approach for Optimization with Probabilistic Constraints

Abstract

We study approximations of optimization problems with probabilistic constraints in which the original distribution of the underlying random vector is replaced with an empirical distribution obtained from a random sample. We show that such a sample approximation problem with a risk level larger than the required risk level will yield a lower bound to the true optimal value with probability approaching one exponentially fast. This leads to an a priori estimate of the sample size required to have high confidence that the sample approximation will yield a lower bound. We then provide conditions under which solving a sample approximation problem with a risk level smaller than the required risk level will yield feasible solutions to the original problem with high probability. Once again, we obtain a priori estimates on the sample size required to obtain high confidence that the sample approximation problem will yield a feasible solution to the original problem. Finally, we present numerical illustrations of how these results can be used to obtain feasible solutions and optimality bounds for optimization problems with probabilistic constraints.

Branch-and-cut approaches for chance-constrained formulations of reliable network design problems

“…In this case, we assume that we have used Monte Carlo sampling to obtain a sample average approximation (SAA) having a moderate number of scenarios, and that the problem (1) that we wish to solve is this SAA problem. Because the set of feasible solutions of this problem is finite, the results of [27] imply that the probability that such an SAA problem yields an optimal solution to the original problem approaches one exponentially fast with sample size. The SAA problem can also be used to derive statistical confidence intervals on the optimal value of the true problem [32].…”

“…For fomulation (27), which uses only probabilistic graph cut inequalities, we found it helpful to also use a simple greedy heuristic to attempt to find more violated inequalities. The heuristic greedily includes edges in a set C sequentially, in increasing order ofx e values as long as e∈Cx e ≤ 6 andx e ≤ 0.9.…”

“…We now present computational results of our branch-and-cut methods for solving formulations (21) and (27) for chance-constrained fully-connected network design problems. We test the proposed algorithms on random graphs constructed from deterministic graphs in the Survivable Network Design Data Library (SNDLIB) [34].…”

Models and algorithms for network design problems

“…However, since it takes into account the binary restriction on x (by using (x q ij ) 2 = x q ij ), it may be used together with (7.11) to provide a stronger continuous relaxation. Note nally that linearizing (7.16) requires at least |Q| additional variables and 2|Q| additional constraints for each (i, j) ∈ A, see Hansen and Meyer (2009). Discretization and deterministic equivalent Sampling a scenario set Ω that approximates the continuous distribution d in an acceptable way, see Luedtke and Ahmed (2008) and Pagnoncelli et al (2009), among others, we can write a deterministic equivalent for (7.14): (7.18) z binary, where components of vector M are numbers large enough. However, (7.17) and (7.18) yield a very di cult problem because (7.17) contains a large number of constraints and features big-M coe cients.…”

From CVaR to Uncertainty Set: Implications in Joint Chance-Constrained Optimization

“…Indeed, sampling approximation of the chance constrained problem has been studied theoretically in Calafiore and Campi [11,12], Ergodan and Iyengar [17] and Luedtke and Ahmed [22]. These methods require roughly about O( n ) constraint duplications to yield a highly reliable solution with respect to its feasibility (see Calafiore and Campi [12]) as well as optimality (see Luedtke and Ahmed [22]). However, it may be computationally prohibitive to solve large problems or to solve problems under high feasibility requirement.…”