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The paper provides a condition for differentiability as well as an equivalent criterion
for Lipschitz continuity of singular normal distributions. Such distributions are of interest,
for instance, in stochastic optimization problems with probabilistic constraints, where
a comparatively small (nondegenerate-) normally distributed random vector induces a large
number of linear inequality constraints (e.g. networks with stochastic demands). The criterion
for Lipschitz continuity is established for the class of quasi-concave distributions which
the singular normal distribution belongs to.
Polyhedral discrepancies are relevant for the quantitative stability
of mixed-integer two-stage and chance constrained stochastic programs. We
study the problem of optimal scenario reduction for a discrete probability
distribution with respect to certain polyhedral discrepancies and develop
algorithms for determining the optimally reduced distribution approximately.
Encouraging numerical experience for optimal scenario reduction is provided.
Modeling several competitive leaders and followers acting in an electricity market
leads to coupled systems of mathematical programs with equilibrium constraints,
called equilibrium problems with equilibrium constraints (EPECs). We consider
a simplified model for competition in electricity markets under uncertainty of demand
in an electricity network
as a (stochastic) multi-leader-follower game. First order necessary conditions are
developed for the corresponding stochastic EPEC based on a result of Outrata.
For applying the general result an explicit representation of the co-derivative of
the normal cone mapping to a polyhedron is derived. Later the
co-derivative formula is used for verifying constraint qualifications and for identifying
$M$-stationary solutions of the stochastic EPEC if the demand is represented by a
finite number of scenarios.
Stability and Sensitivity of Optimization Problems with First Order Stochastic Dominance Constraints
(2007)
We analyze the stability and sensitivity of stochastic optimization problems with stochastic dominance constraints of first order. We consider general perturbations of the underlying probability measures in the space of regular measures equipped with a suitable discrepancy distance. We show that the graph of the feasible set mapping is closed under rather general assumptions. We obtain conditions for the continuity of the optimal value and upper-semicontinuity of the optimal solutions, as well as quantitative stability estimates of Lipschitz type.
Furthermore, we analyze the sensitivity of the optimal value and obtain upper and lower bounds for the directional
derivatives of the optimal value. The estimates are formulated in terms of the dual utility functions associated with the
dominance constraints.
We investigate the convexity of chance constraints with independent random variables. It will be shown, how concavity properties of the mapping related to the decision vector have to be combined with a suitable property of decrease for the marginal densities in order to arrive at convexity of the feasible set for large enough probability levels. It turns out that the required decrease can be verified for most prominent density functions. The results are applied then, to derive convexity of linear chance constraints with normally distributed stochastic coefficients when assuming independence of the rows of the coefficient matrix.
In this paper, we deal with a hydraulic reservoir optimization problem with uncertainty on
inflows in a joint chance constrained programming setting. In particular, we will consider inflows with
a persistency effect, following a causal time series model, and examine the impact of the ”Gaussian”
assumption for such inflows. We present an iterative algorithm for solving similarly structured joint
chance constrained programming problems that requires a Slater point and the computation of gradients.
Several alternatives to the joint chance constraint problem are presented. In particular, we present an
individual chance constraint problem and a robust model. We illustrate the interest of joint chance
constrained programming by comparing results obtained on a realistic hydro-valley with those obtained
from the alternative models. Despite the fact that the alternative models often require less hypothesis
on the law of the inflows, we show that they yield conservative and costly solutions. The simpler models,
such as the individual chance constraint one, are shown to yield insufficient robustness and are therefore
not useful. We therefore conclude that Joint Chance Constrained programming appears as a technique
offering a good trade-off between cost and robustness and can be tractable for complex realistic models.
We provide lower estimates for the norm of gradients of Gaussian
distribution functions and apply the results obtained to a special class of
probabilistically constrained optimization problems. In particular, it is shown
how the precision of computing gradients in such problems can be controlled
by the precision of function values for Gaussian distribution functions. Moreover,
a sensitivity result for optimal values with respect to perturbations of the
underlying random vector is derived. It is shown that the so-called maximal
increasing slope of the optimal value with respect to the Kolmogorov distance
between original and perturbed distribution can be estimated explicitly from
the input data of the problem.
This paper deals with the computation of regular coderivatives of
solution maps associated with a frequently arising class of generalized equations.
The constraint sets are given by (not necessarily convex) inequalities,
and we do not assume linear independence of gradients to active constraints.
The achieved results enable us to state several versions of sharp necessary optimality
conditions in optimization problems with equilibria governed by such
generalized equations. The advantages are illustrated by means of examples.
A mixed-integer stochastic nonlinear optimization problem with joint probabilistic constraints
(2013)
We illustrate the solution of a mixed-integer stochastic nonlinear optimization problem in an application of power management. In this application, a coupled system consisting of a hydro power station and a wind farm is considered. The objective is to satisfy the local energy demand and sell any surplus energy on a spot market for a short time horizon. Generation of wind energy is assumed to be random, so that demand satisfaction is modeled by a joint probabilistic constraint taking into account the multivariate distribution. The turbine is forced to either operate between given positive limits or to be shut down. This introduces additional binary decisions. The numerical solution procedure is presented and results are illustrated.
In this paper a condition number for linear-quadratic two-stage stochastic optimization problems is introduced as the Lipschitz modulus of the multifunction assigning to a (discrete) probability distribution the solution set of the problem. Being the outer norm of the Mordukhovich coderivative of this multifunction, the condition number can be estimated from above explicitly in terms of the problem data by applying appropriate calculus rules. Here, a chain rule for the extended partial second-order subdifferential recently proved by Mordukhovich and Rockafellar plays a crucial role. The obtained results are illustrated for the example of two-stage stochastic optimization problems with simple recourse.