Dynamic probabilistic constraints under continuous random distributions
- PDF / 526,622 Bytes
- 32 Pages / 439.37 x 666.142 pts Page_size
- 92 Downloads / 288 Views
Series B
Dynamic probabilistic constraints under continuous random distributions T. González Grandón1 · R. Henrion2 · P. Pérez-Aros3 Received: 3 March 2020 / Accepted: 3 November 2020 © The Author(s) 2020
Abstract The paper investigates analytical properties of dynamic probabilistic constraints (chance constraints). The underlying random distribution is supposed to be continuous. In the first part, a general multistage model with decision rules depending on past observations of the random process is analyzed. Basic properties like (weak sequential) (semi-) continuity of the probability function or existence of solutions are studied. It turns out that the results differ significantly according to whether decision rules are embedded into Lebesgue or Sobolev spaces. In the second part, the simplest meaningful two-stage model with decision rules from L 2 is investigated. More specific properties like Lipschitz continuity and differentiability of the probability function are considered. Explicitly verifiable conditions for these properties are provided along with explicit gradient formulae in the Gaussian case. The application of such formulae in the context of necessary optimality conditions is discussed and a concrete identification of solutions presented. Keywords Dynamic probabilistic constraints · Chance constraints · Continuous distributions · Decision rules · Stochastic programming Mathematics Subject Classification 90C15 · 49K45
1 Introduction 1.1 Overview The application of probabilistic constraints (or: chance constraints) to engineering problems and their numerical solution is nowadays standard. Introduced by Charnes
This work is dedicated to the memory of Shabbir Ahmed.
B
R. Henrion [email protected]
Extended author information available on the last page of the article
123
T. González Grandón et al.
et al. [5] in a simple form (individual constrains) in 1958, their systematic theoretical and algorithmic investigation has been pioneered by Prékopa and his students starting in the Seventies (see [15] and references therein). The typical form of a probabilistic constraint is the inequality P(gi (x, ξ ) ≤ 0 (i = 1, . . . , p)) ≥ p,
(1)
where x is a decision vector, ξ is a random vector, P a probability measure and g a random constraint mapping with finitely many components. The meaning of (1) is to define a decision x as feasible if the random inequality system g(x, ·) ≤ 0 is satisfied at least with probability p ∈ (0, 1]. A modern theoretical treatment of probabilistic constraints can be found in the monograph [16, chapter 4]. The algorithmic solution of optimization problems subject to constraints (1) has been tremendously advanced within the last twenty years. Rather than providing a detailed list of references here, we want to emphasize the contribution to this development by Shabbir Ahmed (e.g., [12,13]). At the same time the traditional model (1) has been extended to broader settings such as PDE constrained optimization ([6,7,9]) or infinite random inequality systems (probust constraints,
Data Loading...
