Dynamic probabilistic constraints under continuous random distributions

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 $$L^2$$ 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.

Approximate Methods for Solving Chance-Constrained Linear Programs in Probability Measure Space

“…Recently, chance constraints in infinite dimensions have attracted a lot of attention. In [12,14,15], some essential properties, such as convexity and semi-continuity, are generalized into the chance constraints in infinite dimensions. However, the results in [12] assume that the random variable should have a log-concave density to ensure the semicontinuity.…”

“…However, the results in [12] assume that the random variable should have a log-concave density to ensure the semicontinuity. In [15], the continuity of the probability function as chance constraints is proved under the assumption of continuous random distributions. The properties of chance constraints in infinite dimensions are crucial to constructing the optimality condition and implementing convergence analysis for optimization with chance constraints in infinite dimensions.…”

Two‐stage chance‐constrained unit commitment based on optimal wind power consumption point considering battery energy storage

“…The second-stage constraints also include system power balance constraint (32), transmission line capacity constraint (33), up/down reserve deployment constraint (34), generator ramping rate constraints (35) and (36), operation constraints of BES (37)- (41) and wind power output constraint (42). Binary variables indicating the charging/discharging status of BES are also incorporated in (37) and (38). Constraint (41) imposes the re-dispatch restriction of BES between the base state and re-dispatch state.…”

“…Many efforts, such as [30,[34][35][36][37], have been devoted to handling the chance constraint. In this section, the SAA approach is applied to deal with the joint chance constraint (44), which is too complicated to be converted into a closed-form expression directly.…”

Control in Probability for SDE Models of Growth Population

“…Here, it is necessary to mention that during the last years, many authors have made an effort to incorporate probabilistic constraints in the area of control of systems of differential equations (see, e.g., [ 5 – 7 ]). It has motivated the study of new classes of probability functions in finite and infinite dimensional settings (see, e.g., [ 10 , 23 , 27 ] and the references therein).…”