3 Presentations

• 
  02:00 PM - 02:30 PM

  Risk-Averse Stochastic Dual Dynamic Programming

  • Vaclav Kozmik, presenter, Charles University in Prague, Faculty of Mathematics and Physics
  • David P. Morton, The University of Texas at Austin, Graduate Program in Operations Research & Industrial Engineering

  We formulate a risk-averse multi-stage stochastic program using conditional value at risk as the risk measure. The underlying random process is assumed to be stage-wise independent, and a stochastic dual dynamic programming (SDDP) algorithm is applied. We discuss the poor performance of the standard upper bound estimator in the risk-averse setting and propose a new approach based on importance sampling, which yields improved upper bound estimators. Modest additional computational effort is required to use our new estimators. Our procedures allow for significant improvement in terms of controlling solution quality in an SDDP algorithm in the risk-averse setting. We give computational results for multi-stage asset allocation using a log-normal distribution for the asset returns.

• 
  02:30 PM - 03:00 PM

  Approximations for Multistage Stochastic Optimization Programs

  • Georg Pflug, presenter, University of Vienna

  Multistage stochastic programs are often formulated as untractable variational problems. For solving them, they have to be approximated. We review the notion of the nested distance between processes, aiming at finding approximations which are good w.r.t. this distance. The nested distance serves not only as a guidline for scenario approximation, it is also the basis for formulating ambiguity problems, i.e. problems with model uncertainty. We present algorithms for scenario generation as well as for solving ambiguous problems based on the nested distance.

• 
  03:00 PM - 03:30 PM

  On the Distance Between Stochastic Processes in Multi-Stage Stochastic Optimization Programs

  • Anna Timonina, presenter, University of Vienna

  Stochastic approximation that attracts great attention of experts in control theory is one of the most challenging, important and very often irreplaceable solution methods for multi-stage stochastic optimization programs with wide range of applications in financial and investment planning, inventory control, energy production and trading, electricity generation planning, pension fund management, supply chain management and in similar fields. In multi-stage stochastic optimization problems the amount of stage-wise available information is crucial. The traditional approach to deal with this information goes back to introduction of filtrations, i.e. increasing sequences of sigma-algebras, to which the decision must be adapted. In this research this original method is replaced with concept of nested distributions, that allows to keep the setup purely distributional but at the same time to introduce information and information constraints. We introduce the distance between stochastic process and a tree and we generalize the concept of nested distance for the case of infinite trees, i.e. for the case of two stochastic processes given by their continuous distributions. We are making a step towards to a new method for distribution quantization that is the most suitable for multi-stage stochastic optimization programs as it takes into account both the stochastic process and the stage-wise information.
  The main problems that are considered in this research are: Scenario generation: When dealing with scenario generation for the multi-stage stochastic optimization problem, the natural question that arises is how to approximate (to quantize) the continuous distribution of the stochastic process in such a way, that the nested distance between the initial distribution and its approximation is minimized. The stage-wise minimization of the Kantorovich distance gives us a well-known result for the optimal quantizers of the distributions at each stage. However, this result does not mean that the nested distance between initial and approximate problems is minimized. The aim of this research is to show that the quantizers received by the minimization of the nested distance are better (in the sense of minimal distance) than the quantizers received by the stage-wise minimization of the Kantorovich distance.
  Computational efficiency: When dealing with numerical calculation of the nested distance the question of the computational efficiency is of interest. It is necessary to understand how many values should the scenario process have at each stage in order to make the computational time small and at the same time to make the approximation good (i.e. to make the approximation error small). The compromise for this should be found.
  Applications: Stochastic programming offers a huge variety of applications. In this research we focus on the applications in the field of natural hazards risk-management.