Volume 2000http://edoc.hu-berlin.de/18452/3572021-09-23T03:03:22Z2021-09-23T03:03:22ZQuantitative stability in stochastic programmingRachev, Svetlozar T.Römisch, Wernerhttp://edoc.hu-berlin.de/18452/89032020-03-07T04:14:06Z2000-12-20T00:00:00ZQuantitative stability in stochastic programming
Rachev, Svetlozar T.; Römisch, Werner
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
Quantitative stability of optimal values and solution sets to stochastic programming problems is studied when the underlying probability distribution varies in some metric space of probability measures. We give conditions that imply that a stochastic program behaves stable with respect to a minimal information (m.i.) probability metric that is naturally associated with the data of the program. Canonical metrics bounding the m.i. metric are derived for specific models, namely, for linear two-stage, mixed-integer two-stage and chance constrained models. The corresponding quantivative stability results as well as some consequences for asymptotic properties of empirical approximations extend earlier results in this direction. In particular, rates of convergence in probability are derived under metric entropy conditions. Finally, we study stability properties of stable investment portfolios having minimal risk with respect to the spectral measure and stability index of the underlying stable probability distribution.
2000-12-20T00:00:00ZThe C 3 theorem and a D 2 algorithm for large scale stochastic integer programmingSen, SuvrajeetHigle, Julia L.http://edoc.hu-berlin.de/18452/89022020-03-07T04:14:05Z2000-12-19T00:00:00ZThe C 3 theorem and a D 2 algorithm for large scale stochastic integer programming
Sen, Suvrajeet; Higle, Julia L.
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
This paper considers the two stage stochastic integer programming problems, with an emphasis on problems in which integer variables appear in the second stage. Drawing heavily on the theory of disjunctive programming, we characterize convexifications of the second stage problem and develop a decomposition-based algorithm for the solution of such problems. In particular, we verify that problems with fixed recourse are characterized by scenario-dependent second stage convexifications that have a great deal in common. We refer to this characterization as the C^3 (Common Cut Coefficients) Theorem. Based on the C^3 Theorem, we develop an algorithmic methodology that we refer to as Disjunctive Decomposition (D^2). We show that when the second stage consists of 0-1 MILP problems , we can obtain accurate second stage objective function estimates afer finitely many steps. We also set the stage for comparisions between problems in which the first stage includes only 0-1 variables and those that allow both continuous and integer variables in the first stage.
2000-12-19T00:00:00ZThe stable non-Gaussian asset allocationTokat, YesimRachev, Svetlozar T.Schwartz, Eduardohttp://edoc.hu-berlin.de/18452/89012020-03-07T04:14:05Z2000-11-28T00:00:00ZThe stable non-Gaussian asset allocation
Tokat, Yesim; Rachev, Svetlozar T.; Schwartz, Eduardo
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
We analyze a multistage stochastic asset allocation problem with decision rules. The uncertainty is modeled using economic scenarios with Gaussian and stable Paretian non-Gaussian innovations. The optimal allocations under these alternative hypothesis are compared. If the agent has very low or very high risk aversibility, then the Gaussian and stable non-Gaussian scenarios result in similar allocations. When the risk aversion of the agent is between these two extreme cases, then the two distributional assumptions may result in very different asset allocations. Our calculations suggest that the allocations may be up to 85% different depending on the utility function and the level of risk aversion of the agent.
2000-11-28T00:00:00ZAdaptive optimal stochastic trajectory planning and control (AOSTPC) for robotsMarti, Kurthttp://edoc.hu-berlin.de/18452/89002020-03-07T04:14:05Z2000-11-07T00:00:00ZAdaptive optimal stochastic trajectory planning and control (AOSTPC) for robots
Marti, Kurt
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
In optimal control of robots, the standard procedure is to determine first off-line an optimal open-loop control, using some nominal or estimated values of the model parameters, and to correct then the resulting deviation of the effective trajectory or performance of the system from the prescribed trajectory, from the prescribed performance values, resp., by on-line measurement and control actions. However, on-line measurement and control actions are expensive in general and very time-consuming, moreover, they are suitable only for rather small deviations. By adaptive optimal stochastic trajectory planning and control (AOSTPC), i.e., incorporating sequentially the available a priori and measurement information about the unknown model parameters into the optimal control design process by using stochastic optimization methods, the (conditional) mean absolute deviation between the actual and prescribed trajectory, performance, resp., can be reduced considerably, hence, more robust controls are obtained. The corresponding feedforward and feedback (PD-)controls are derived by means of sequential stochastic optimization and by using stability requirements. In addition, methods for the numerical computation of the controls in real-time are presented. Moreover, analytical estimates are given for the reduction of the tracking error, hence, for the reduction of the on-line measurement and correction expenses by applying (AOSTPC).
2000-11-07T00:00:00Z