Joint dynamic probabilistic constraints with projected linear decision rules

We consider multistage stochastic linear optimization problems combining joint dynamic probabilistic constraints with hard constraints. We develop a method for projecting decision rules onto hard constraints of wait-and-see type. We establish the relation between the original (infinite dimensional) problem and approximating problems working with projections from different subclasses of decision policies. Considering the subclass of linear decision rules and a generalized linear model for the underlying stochastic process with noises that are Gaussian or truncated Gaussian, we show that the value and gradient of the objective and constraint functions of the approximating problems can be computed analytically

Gradient formulae for nonlinear probabilistic constraints with Gaussian and Gaussian-like distributions

Probabilistic constraints represent a major model of stochastic optimization. A possible approach for solving probabilistically constrained optimization problems consists in applying nonlinear programming methods. In order to do so, one has to provide sufficiently precise approximations for values and gradients of probability functions. For linear probabilistic constraints under Gaussian distribution this can be successfully done by analytically reducing these values and gradients to values of Gaussian distribution functions and computing the latter, for instance, by Genz' code. For nonlinear models one may fall back on the spherical-radial decomposition of Gaussian random vectors and apply, for instance, De'ak's sampling scheme for the uniform distribution on the sphere in order to compute values of corresponding probability functions. The present paper demonstrates how the same sampling scheme can be used in order to simultaneously compute gradients of these probability functions. More precisely, we prove a formula representing these gradients in the Gaussian case as a certain integral over the sphere again. Later, the result is extended to alternative distributions with an emphasis on the multivariate Student (or T-) distribution

Author's personal copy Discrete Optimization A model for dynamic chance constraints in hydro power reservoir management

a b s t r a c t In this paper, a model for (joint) dynamic chance constraints is proposed and applied to an optimization problem in water reservoir management. The model relies on discretization of the decision variables but keeps the probability distribution continuous. Our approach relies on calculating probabilities of rectangles which is particularly useful in the presence of independent random variables but works equally well in the case of correlated variables. Numerical results are provided for two and three stages

Joint dynamic probabilistic constraints with projected linear decision rules

We consider multistage stochastic linear optimization problems combining joint dynamic probabilistic constraints with hard constraints. We develop a method for projecting decision rules onto hard constraints of wait-and-see type. We establish the relation between the original (infinite dimensional) problem and approximating problems working with projections from different subclasses of decision policies. Considering the subclass of linear decision rules and a generalized linear model for the underlying stochastic process with noises that are Gaussian or truncated Gaussian, we show that the value and gradient of the objective and constraint functions of the approximating problems can be computed analytically

On probabilistic constraints induced by rectangular sets and multivariate normal distributions

In this paper, we consider optimization problems under probabilistic constraints which aredefined by two-sided inequalities for the underlying normally distributed random vector. Asa main step for an algorithmic solution of such problems, we derive a derivative formula for(normal) probabilities of rectangles as functions of their lower or upper bounds. This formulaallows to reduce the calculus of such derivatives to the calculus of (normal) probabilitiesof rectangles themselves thus generalizing a similar well-known statement for multivariatenormal distribution functions. As an application, we consider a problem from water reservoirmanagement. One of the outcomes of the problem solution is that the (still frequentlyencountered) use of simple individual probabilistic can completely fail. In contrast, the (more difficult) use of joint probabilistic constraints which heavily depends on the derivative formula mentioned before yields very reasonable and robust solutions over the whole time horizon considered