Deterministic approximations of probability inequalities

Abstract

A simple general framework for deriving explicit deterministic approximations of probability inequalities of the form P(~/> a) ~< ~ is presented. These approximations are based on limited parametric information about the involved random variables (such as their mean, variance, range or upper bound values). First the case of a single random variable ~ is analysed, followed by the cases n of independent and dependent summands f = ~ ~i. As examples of possible applications, a stochas-1 tic extension of the "knapsack problem" and the stochastic linear programming problem with separate chance-constraints are investigated: we provide approximate deterministie surrogates for these problems.Zusammenfassung: Es wird ein Rahmen zur Ableitung expliziter deterministischer Approximation fiir Wahrscheinlichkeitsungleichungen der Form P(~ ~ a) ~< a angegeben. Diese Approximationen basieren auf begrenzter parametrischer Information tiber die beteiligten Zufallsvariablen (wie ihr Erwartungswert, Varianz, Wertebereich oder obere Schranken). Zuerst wird der Fail einer Zufallsn variablen f analysiert, sodannwerdenSummenvonunabhgngigenSummanden ~ = Y, ~ibetrachtet. i=1 Als Beispiele ftir m6gliche Anwendungen wird eine stochastische Erweiterung des Rucksackproblems untersucht sowie stochastische lineare Programme mit separablen Wahrscheinlichkeitsrestriktionen. Fiir diese Probleme werden n/iherungsweise deterministische Ersatzprobleme angegeben.

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

“…Due to the positively homogeneous property of Theorem 3.1(b), we scale α by a positive constant so that it is feasible in Problem (26). Hence, the result follows.…”

“…Other forms of deterministic approximation of an individual chance constrained problem includes using Chebyshev's inequality, Bernstein's inequality, or Hoeffding's inequality to bound the probability of violating individual constraints. See, for example, Pintér [26].…”

Convex relaxations of chance constrained optimization problems

“…Following [5] and [7], where the authors used a similar exponential function to construct efficiently computable convex restrictions to chance constraints, we refer to the convex relaxations corresponding to φ β as "Bernstein" relaxations in appreciation of the connections to the work of S. N. Bernstein in classical large deviation theory.…”

“…An additional requirement is that the approximation should be relatively easily solvable. One class of developments along this direction focuses on constructing a deterministic convex programming restriction to the chance constraint that can be optimized efficiently using standard methods [5,7]. Another class of approaches solve deterministic optimization problems, built from samples of the random vector, that are guaranteed to produce feasible solutions with high probability [2,4].…”

“…In this paper we propose a deterministic lower bounding technique by constructing a convex relaxation of the chance constrained problem. The approach is similar to the construction used in [5,7] to develop a convex restriction of the problem. Following [5,7] we show that a version of the proposed scheme leads to a tractable convex relaxation when the random vector ξ has independent components and the constraint function g is affine in ξ.…”