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.
Depending on whether a mathematical program with equilibrium constraints
(MPEC) is considered in its original or its enhanced (via KKT conditions) form, the assumed qualification
conditions as well as the derived necessary optimality conditions may differ significantly. In this paper, we
study this issue when imposing one of the weakest possible qualification conditions, namely the calmness of
the perturbation mapping associated with the respective generalized equations in both forms of the MPEC.
It is well known that the calmness property allows one to derive the so-called M-stationarity conditions. The
restrictiveness of assumptions and the strength of conclusions in the two forms of the MPEC is also strongly
related to the qualification conditions on the “lower level”. For instance, even under the Linear Independence
Constraint Qualification (LICQ) for a lower level feasible set described by C 1 functions, the calmness properties
of the original and the enhanced perturbation mapping are drastically different. When passing to C 1,1 data, this
difference still remains true under the weaker Mangasarian-Fromovitz Constraint Qualification, whereas under
LICQ both the calmness assumption and the derived optimality conditions are fully equivalent for the original
and the enhanced form of the MPEC. After clarifying these relations, we provide a compilation of practically
relevant consequences of our analysis in the derivation of necessary optimality conditions. The obtained results
are finally applied to MPECs with structured equilibria.
The paper considers the computation of the probability of feasible load constellations in a stationary gas
network with uncertain demand. More precisely, a network with a single entry and several
exits with uncertain loads is studied. Feasibility of a load constellation is understood in the sense of
an existing flow meeting these loads along with given pressure bounds in the pipes.
In a first step, feasibility of deterministic exit loads is characterized algebraically and these general
conditions are specified to networks involving at most one cycle.
This prerequisite is essential for determining probabilities in a stochastic setting when exit loads
are assumed to follow some (joint) Gaussian distribution when modeling uncertain customer demand.
The key of our approach is the application of the spheric-radial decomposition of Gaussian random
vectors coupled with Quasi Monte-Carlo sampling. This approach requires an efficient algorithmic
treatment of the mentioned algebraic relations moreover depending on a scalar parameter. Numerical
results are illustrated for different network examples and demonstrate a clear superiority in terms of
precision over simple generic Monte-Carlo sampling. They lead to fairly accurate probability values
even for moderate sample size.
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.
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 with multivariate truncated Gaussian and lognormal distributions
(2016)
Many engineering problems with uncertain data, notably arising
in power management, can be formulated as optimization problems subject to
probabilistic constraints. While dealing with such constraints under continuous distributions of the underlying random parameter remains a difficult task
in general both from the numerical and theoretical point of view, quite some
progress has been made in the special case of multivariate Gaussian distributions. These are not perfectly adequate, however, in many circumstances, in
particular not, when modeling uncertain inflows to hydro reservoirs or uncertain demands in gas networks. Interesting alternatives are offered by truncations of multivariate Gaussian distributions to polyhedra or by multivariate
lognormal distributions. The paper discusses the applicability of such distributions in the context of a simple joint linear probabilistic constraint putting
the emphasis on the numerical approximation of probabilities and their gradients (w.r.t. decisions to be optimized) as well as on the convexity of the set
of feasible decisions.
In this paper, we discuss optimality conditions for optimization problems {involving} random state constraints, which are modeled in probabilistic or almost sure form. While the latter can be understood as the limiting case of the former, the derivation of optimality conditions requires substantially different approaches. We apply them to a linear elliptic partial differential equation (PDE) with random inputs.
In the probabilistic case, we rely on the spherical-radial decomposition of Gaussian random vectors in order to formulate fully explicit optimality conditions involving a spherical integral. In the almost sure case, we derive optimality conditions and compare them to a model based on robust constraints with respect to the (compact) support of the given distribution.
We consider the optimal control of a PDE with random source term subject to probabilistic or almost sure state constraints. In the main theoretical result, we provide an exact formula for the Clarke subdifferential of the probability function without a restrictive assumption made in an earlier paper. The focus of the paper is on numerical solution algorithms. As for probabilistic constraints, we apply the method of spherical radial decomposition. Almost sure constraints are dealt with a Moreau--Yosida smoothing of the constraint function accompanied by Monte Carlo sampling of the given distribution or its support or even just the boundary of its support. Moreover, one can understand the almost sure constraint as a probabilistic constraint with safety level one which offers yet another perspective. Finally, robust optimization can be applied efficiently when the support is sufficiently simple. A comparative study of these five different methodologies is carried out and illustrated.
In this paper we consider systems that are governed by linear time-discrete dynamics with an initial condition and a terminal condition for the expected values. We study optimal control problems where in the objective function a term of tracking type for the expected values and a control cost appear. In addition, the feasible states have to satisfy a conservative probabilistic constraint that requires that the probability that the trajectories remain in a given set F is greater than or equal to a given lower bound. An application are
optimal control problems related to storage management systems with uncertain in- and output. We give suffcient conditions that imply that the optimal expected trajectories remain close to a certain state that can be characterized as the solution of an optimal control problem without prescribed initial- and terminal condition. Hence we contribute to the study of the turnpike phenomenon that is well-known in mathematical economics.
We consider Mathematical Programs with Equilibrium Constraints with proba-
bilistic constraints (PMPECs). Such models have proven to be useful in modeling
electricity or gas markets subject to random parameters. Our main interest is the
derivation of Mordukhovich (M-) stationarity conditions under suitable constraint
quali...cations ensuring the calmness of the canonically perturbed generalized equation.
Applying recent results from deterministic MPECs, we identify the needed properties
of the probability function in order to derive explicit M-stationarity conditions. The
results are applied to a simple stochastic bilevel problem in an economic context.
