Volume 2011http://edoc.hu-berlin.de/18452/3682023-12-11T23:01:11Z2023-12-11T23:01:11ZOn the Geometry of Acceptability FunctionalsPichler, Aloishttp://edoc.hu-berlin.de/18452/90722020-03-07T04:14:42Z2011-11-28T00:00:00ZOn the Geometry of Acceptability Functionals
Pichler, Alois
http://dx.doi.org/10.18452/8420
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
In this paper we discuss the geometry of acceptability functionals or risk measures. The dependenceof the random variable is investigated ﬁrst. The main contribution and focus of this paper is to studyhow acceptability functionals vary whenever the underlying probability measure is perturbed. It turns out that the Wasserstein distance provides a valuable notion of distance, and may acceptability functionals allow a precise quantiﬁcation in terms of this distance.
2011-11-28T00:00:00ZMultistage OptimizationPflug, Georg Ch.Pichler, Aloishttp://edoc.hu-berlin.de/18452/90712020-03-07T04:14:41Z2011-11-28T00:00:00ZMultistage Optimization
Pflug, Georg Ch.; Pichler, Alois
http://dx.doi.org/10.18452/8419
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
We provide a new identity for the multistage Average Value-at-Risk. The identity is based on the conditional Average Value-at-Risk at random level, which is introduced. It is of interest in situations, where the information available increases over time, so it is – among other applications – customized to multistage optimization. The identity relates to dynamic programming and is adapted to problemswhich involve the Average Value-at-Risk in its objective. We elaborate further dynamic programming equations for speciﬁc multistage optimization problems and derive a characterizing martingale property for the value function. The concept solves a particular aspect of time consistency and is adapted for situations, wheredecisions are planned and executed consecutively in subsequent instants of time. We discuss theapproach for other risk measures, which are in frequent use for decision making under uncertainty,particularly for ﬁnancial decisions.
2011-11-28T00:00:00ZA gradient formula for linear chance constraints under Gaussian distributionHenrion, RenéMöller, Andrishttp://edoc.hu-berlin.de/18452/90702020-03-07T04:14:41Z2011-09-13T00:00:00ZA gradient formula for linear chance constraints under Gaussian distribution
Henrion, René; Möller, Andris
http://dx.doi.org/10.18452/8418
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
We provide an explicit gradient formula for linear chance constraints under a (possibly singular) multivariate Gaussian distribution. This formula allows one to reduce the calculus of gradients to the calculus of values of the same type of chance constraints (in smaller dimension and with different distribution parameters). This is an important aspect for the numerical solution of stochastic optimization problems because existing efficient codes for e.g., calculating singular Gaussian distributions or regular Gaussian probabilities of polyhedra can be employed to calculate gradients at thesame time. Moreover, the precision of gradients can be controlled by that of function values, which is a great advantage over using ﬁnite difference approximations. Finally, higher order derivatives are easily derived explicitly. The use of the obtained formula is illustrated for an example of a stochastic transportation network.
2011-09-13T00:00:00ZStochastic programs without duality gapsPennanen, TeemuPerkkiö, Ari-Pekkahttp://edoc.hu-berlin.de/18452/90692020-03-07T04:14:41Z2011-08-02T00:00:00ZStochastic programs without duality gaps
Pennanen, Teemu; Perkkiö, Ari-Pekka
http://dx.doi.org/10.18452/8417
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
This paper studies dynamic stochastic optimization problems parametrizedby a random variable. Such problems arise in many applications in operations research and mathematical ﬁnance. We give sufficient conditionsfor the existence of solutions and the absence of a duality gap. Our proofuses extended dynamic programming equations, whose validity is established under new relaxed conditions that generalize certain no-arbitrageconditions from mathematical ﬁnance.
2011-08-02T00:00:00Z