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

A joint model of probabilistic/robust constraints for gas transport management in stationary networks

We present a novel mathematical algorithm to assist gas network operators in managing uncertainty, while increasing reliability of transmission and supply. As a result, we solve an optimization problem with a joint probabilistic constraint over an infinite system of random inequalities. Such models arise in the presence of uncertain parameters having partially stochastic and partially nonstochastic character. The application that drives this new approach is a stationary network with uncertain demand (which are stochastic due to the possibility of fitting statistical distributions based on historical measurements) and with uncertain roughness coefficients in the pipes (which are uncertain but non-stochastic due to a lack of attainable measurements). We study the sensitivity of local uncertainties in the roughness coefficients and their impact on a highly reliable network operation. In particular, we are going to answer the question, what is the maximum uncertainty that is allowed (shaping a maximal uncertainty set) around nominal roughness coefficients, such that random demands in a stationary gas network can be satisfied at given high probability level for no matter which realization of true roughness coefficients within the uncertainty set. One ends up with a constraint, which is probabilistic with respect to the load of gas and robust with respect to the roughness coefficients. We demonstrate how such constraints can be dealt with in the framework of the so-called spheric-radial decomposition of multivariate Gaussian distributions. The numerical solution of a corresponding optimization problem is illustrated. The results might assist the network operator with the implementation of cost-intensive roughness measurements

A joint model of probabilistic/robust constraints for gas transport management in stationary networks

We present a novel mathematical algorithm to assist gas network operators in managing uncertainty, while increasing reliability of transmission and supply. As a result, we solve an optimization problem with a joint probabilistic constraint over an infinite system of random inequalities. Such models arise in the presence of uncertain parameters having partially stochastic and partially non-stochastic character. The application that drives this new approach is a stationary network with uncertain demand (which are stochastic due to the possibility of fitting statistical distributions based on historical measurements) and with uncertain roughness coefficients in the pipes (which are uncertain but non-stochastic due to a lack of attainable measurements). We study the sensitivity of local uncertainties in the roughness coefficients and their impact on a highly reliable network operation. In particular, we are going to answer the question, what is the maximum uncertainty that is allowed (shaping a 'maximal' uncertainty set) around nominal roughness coefficients, such that random demands in a stationary gas network can be satisfied at given high probability level for no matter which realization of true roughness coefficients within the uncertainty set. One ends up with a constraint, which is probabilistic with respect to the load of gas and robust with respect to the roughness coefficients. We demonstrate how such constraints can be dealt with in the framework of the so-called spheric-radial decomposition of multivariate Gaussian distributions. The numerical solution of a corresponding optimization problem is illustrated. The results might assist the network operator with the implementation of cost-intensive roughness measurements

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 capacity maximization in a stationary gas network

The question for the capacity of a given gas network, i.e., determining the maximal amount of gas that can be transported by a given network, appears as an essential question that network operators and political administrations are regularly faced with. In that context we present a novel mathematical approach to assist gas network operators in managing uncertainty with respect to the demand and in exposing free network capacities while increasing reliability of transmission and supply. The approach is based on the rigorous examination of optimization problems with nonlinear probabilistic constraints. As consequence we deal with solving an optimization problem with joint probabilistic constraints over an infinite system of random inequalities. We will show that the inequality system can be reduced to a finite one in the situation of considering a tree network topology. A detailed study of the problem of maximizing free booked capacities in a stationary gas network is presented that comes up with an algebraic model involving Kirchhoff's first and second laws. The focus will be on both the theoretical and numerical side. We are going to validate a kind of rank two constraint qualification implying the differentiability of the considered capacity problem. At the numerical side we are going to solve the problem using a projected gradient decent method, where the function and gradient evaluations of the probabilistic constraints are performed by the approach of spheric-radial decomposition applied for multivariate Gaussian random variables and more general distributions

(Sub-) Gradient formulae for probability functions of random inequality systems under Gaussian distribution

We consider probability functions of parameter-dependent random inequality systems under Gaussian distribution. As a main result, we provide an upper estimate for the Clarke subdifferential of such probability functions without imposing compactness conditions. A constraint qualification ensuring continuous differentiability is formulated. Explicit formulae are derived from the general result in case of linear random inequality systems. In the case of a constant coefficient matrix an upper estimate for even the smaller Mordukhovich subdifferential is proven

On probabilistic capacity maximization in a stationary gas network

The question for the capacity of a given gas network, i.e., determining the maximal amount of gas that can be transported by a given network, appears as an essential question that network operators and political administrations are regularly faced with. In that context we present a novel the demand and in exposing free network capacities while increasing reliability of transmission and supply. The approach is based on the rigorous examination of optimization problems with nonlinear probabilistic constraints. As consequence we deal with solving an optimization problem with joint probabilistic constraints over an infinite system of random inequalities. We will show that the inequality system can be reduced to a finite one in the situation of considering a tree network topology. A detailed study of the problem of maximizing free booked capacities in a stationary gas network is presented that comes up with an algebraic model involving Kirchhoffs first and second laws. The focus will be on both the theoretical and numerical side. We are going to validate a kind of rank two constraint qualification implying the differentiability of the considered capacity problem. At the numerical side we are going to solve the problem using a projected gradient decent method, where the function and gradient evaluations of the probabilistic constraints are performed by the approach of spheric-radial decomposition applied for multivariate Gaussian random variables and more general distributions

Generalized gradients for probabilistic/robust (probust) constraints

Probability functions are a powerful modelling tool when seeking to account for uncertainty in optimization problems. In practice, such uncertainty may result from different sources for which unequal information is available. A convenient combination with ideas from robust optimization then leads to probust functions, i.e., probability functions acting on generalized semi-infinite inequality systems. In this paper we employ the powerful variational tools developed by Boris Mordukhovich to study generalized differentiation of such probust functions. We also provide explicit outer estimates of the generalized subdifferentials in terms of nominal data

Subdifferential characterization of probability functions under Gaussian distribution

Probability functions figure prominently in optimization problems of engineering. They may be nonsmooth even if all input data are smooth. This fact motivates the consideration of subdifferentials for such typically just continuous functions. The aim of this paper is to provide subdifferential formulae of such functions in the case of Gaussian distributions for possibly infinite-dimensional decision variables and nonsmooth (locally Lipschitzian) input data. These formulae are based on the spheric-radial decomposition of Gaussian random vectors on the one hand and on a cone of directions of moderate growth on the other. By successively adding additional hypotheses, conditions are satisfied under which the probability function is locally Lipschitzian or even differentiable

Inverse Problems and Data Assimilation

These notes are designed with the aim of providing a clear and concise introduction to the subjects of Inverse Problems and Data Assimilation, and their inter-relations, together with citations to some relevant literature in this area. The first half of the notes is dedicated to studying the Bayesian framework for inverse problems. Techniques such as importance sampling and Markov Chain Monte Carlo (MCMC) methods are introduced; these methods have the desirable property that in the limit of an infinite number of samples they reproduce the full posterior distribution. Since it is often computationally intensive to implement these methods, especially in high dimensional problems, approximate techniques such as approximating the posterior by a Dirac or a Gaussian distribution are discussed. The second half of the notes cover data assimilation. This refers to a particular class of inverse problems in which the unknown parameter is the initial condition of a dynamical system, and in the stochastic dynamics case the subsequent states of the system, and the data comprises partial and noisy observations of that (possibly stochastic) dynamical system. We will also demonstrate that methods developed in data assimilation may be employed to study generic inverse problems, by introducing an artificial time to generate a sequence of probability measures interpolating from the prior to the posterior