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On the algorithmic solution of optimization problems subject to probabilistic/robust (probust) constraints
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.
On the algorithmic solution of optimization problems subject to probabilistic/robust (probust) constraints
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.
Joint Model of Probabilistic-Robust (Probust) Constraints Applied to Gas Network Optimization
Optimization problems under uncertain conditions abound in many real-life applications. While 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 applied in case solutions are sought that are feasible for all realizations of uncertainties within some predefined 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. 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 finding 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.
Probabilistic constraints via SQP solver: Application to a renewable energy management problem
The aim of this paper is to illustrate the efficient solution of nonlinear optimization problems with joint probabilistic constraints by means of an SQP method. Here, the random vector is assumed to obey some multivariate Gaussian distribution. The numerical solution approach is applied to a renewable energy management problem. We consider a coupled system of hydro and wind power production used in order to satisfy some local demand of energy and to sell/buy excessive or missing energy on a day-ahead and intraday market, respectively. A short term planning horizon of 2 days is considered and only wind power is assumed to be random. In the first part of the paper, we develop an appropriate optimization problem involving a probabilistic constraint reflecting demand satisfaction. Major attention will be payed to formulate this probabilistic constraint not directly in terms of random wind energy produced but rather in terms of random wind speed, in order to benefit from a large data base for identifying an appropriate distribution of the random parameter. The second part presents some details on integrating Genz' code for Gaussian probabilities of rectangles into the environment of the SQP solver SNOPT. The procedure is validated by means of a simplified optimization problem which by its convex structure allows to estimate the gap between the numerical and theoretical optimal values, respectively. In the last part, numerical results are presented and discussed for the original (nonconvex) optimization problem.
A turnpike property for optimal control problems with dynamic probabilistic constraints
In this paper we consider systems that are governed by linear time-discrete dynamics with an initial condition, additive random perturbations in each step and a terminal condition for the expected values. We study optimal control problems where the objective function consists of a term of tracking type for the expected values and a control cost. In addition, the feasible states have to satisfy a conservative probabilistic constraint that requires that the probability that the trajectories remain in a given set F is greater than or equal to a given lower bound. An application are optimal control problems related to storage management systems with uncertain in- and output. We give sufficient conditions that imply that the optimal expected trajectories remain close to a certain state that can be characterized as the solution of an optimal control problem without prescribed initial- and terminal condition. In this way we contribute to the study of the turnpike phenomenon that is well-known in mathematical economics and make a step towards the extension of the turnpike theory to problems with probabilistic constraints.