TY - JOUR A1 - Berthold, Holger A1 - Heitsch, Holger A1 - Henrion, René A1 - Schwientek, Jan T1 - On the algorithmic solution of optimization problems subject to probabilistic/robust (probust) constraints N2 - We present an adaptive grid refinement algorithm to solve probabilistic optimization problems with infinitely many random constraints. Using a bilevel approach, we iteratively aggregate inequalities that provide most information not in a geometric but in a probabilistic sense. This conceptual idea, for which a convergence proof is provided, is then adapted to an implementable algorithm. The efficiency of our approach when compared to naive methods based on uniform grid refinement is illustrated for a numerical test example as well as for a water reservoir problem with joint probabilistic filling level constraints. KW - probabilistic constraints KW - probust constraints KW - chance constraints KW - bilevel optimization KW - semi-infinite optimization Y1 - 2021 U6 - https://doi.org/10.1007/s00186-021-00764-8 ER - TY - INPR A1 - Branda, Martin A1 - Henrion, René A1 - Pištěk, Miroslav T1 - Value at risk approach to producer's best response in electricity market with uncertain demand N2 - We deal with several sources of uncertainty in electricity markets. The independent system operator (ISO) maximizes the social welfare using chance constraints to hedge against discrepancies between the estimated and real electricity demand. We find an explicit solution of the ISO problem, and use it to tackle the problem of a producer. In our model, production as well as income of a producer are determined based on the estimated electricity demand predicted by the ISO, that is unknown to producers. Thus, each producer is hedging against the uncertainty of prediction of the demand using the value-at-risk approach. To illustrate our results, a numerical study of a producer's best response given a historical distribution of both estimated and real electricity demand is provided. KW - electricity market KW - multi-leader-common-follower game KW - stochastic demand KW - day-ahead bidding KW - chance constraints Y1 - 2021 ER - TY - JOUR A1 - Branda, Martin A1 - Henrion, René A1 - Pistek, Miroslav T1 - Producer’s Best Response in Pay-as-clear Day-ahead Electricity Market with Uncertain Demand N2 - We deal with several sources of uncertainty in electricity markets. The independent system operator (ISO) maximizes the social welfare using chance constraints to hedge against discrepancies between the estimated and real electricity demand. We find an explicit solution of the ISO problem, and use it to tackle the problem of a producer. In our model, production as well as income of a producer are determined based on the estimated electricity demand predicted by the ISO, that is unknown to producers. Thus, each producer is hedging against the uncertainty of prediction of the demand using the value-at-risk approach. To illustrate our results, a numerical study of a producer’s best response given a historical distribution of both estimated and real electricity demand is provided. KW - electricity market KW - multi-leader-common-follower game KW - stochastic demand KW - day-ahead bidding KW - chance constraints Y1 - 2020 ER - TY - JOUR A1 - Adelhütte, Dennis A1 - Aßmann, Denis A1 - Gonzàlez Grandòn, Tatiana A1 - Gugat, Martin A1 - Heitsch, Holger A1 - Liers, Frauke A1 - Henrion, René A1 - Nitsche, Sabrina A1 - Schultz, Rüdiger A1 - Stingl, Michael A1 - Wintergerst, David T1 - Joint model of probabilistic/robust (probust) constraints applied to gas network optimization N2 - Optimization tasks under uncertain conditions abound in many real-life applications. Whereas solution approaches for probabilistic constraints are often developed in case the uncertainties can be assumed to follow a certain probability distribution, robust approaches are usually used in case solutions are sought that are feasible for all realizations of uncertainties within some pre-defined uncertainty set. As many applications contain different types of uncertainties that require robust as well as probabilistic treatments, we deal with a class of joint probabilistic/robust constraints as its appears in optimization problems under uncertainty. Focusing on complex uncertain gas network optimization problems, we show the relevance of this class of problems for the task of maximizing free booked capacities in an algebraic model for a stationary gas network. We furthermore present approaches for their solution. Finally, we study the problem of controlling a transient system that is governed by the wave equation. The task consists in determining controls such that a certain robustness measure remains below some given upper bound, with high probability. KW - robust optimization KW - chance constraints KW - optimal control KW - spheric-radial decomposition Y1 - 2017 U6 - https://doi.org/10.1007/s10013-020-00434-y ER - TY - JOUR A1 - Bernhard, Daniela A1 - Heitsch, Holger A1 - Henrion, René A1 - Liers, Frauke A1 - Stingl, Michael A1 - Uihlein, Andrian A1 - Zipf, Viktor T1 - Continuous stochastic gradient and spherical radial decomposition N2 - In this paper, a new method is presented for solving chance-constrained optimization problems. The method combines the well-established Spherical-Radial Decomposition approach with the Continuous Stochastic Gradient method. While the Continuous Stochastic Gradient method has been successfully applied to chance-constrained problems in the past, only the combination with the Spherical-Radial Decomposition allows to avoid smoothing of the integrand. In this chapter, we prove this fact for a relevant class of chance-constrained problems and apply the resulting method to the capacity maximization problem for gas networks. KW - chance constraints KW - continuous stochastic gradient KW - spheric-radial decomposition Y1 - ER -