Aggregation and discretization in multistage stochastic programming

Multistage stochastic programs have applications in many areas and support policy makers in finding rational decisions that hedge against unforeseen negative events. In order to ensure computational tractability, continuous-state stochastic programs are usually discretized; and frequently, the curse of dimensionality dictates that decision stages must be aggregated. In this article we construct two discrete, stage-aggregated stochastic programs which provide upper and lower bounds on the optimal value of the original problem. The approximate problems involve finitely many decisions and constraints, thus principally allowing for numerical solutio

Tight Probability Bounds with Pairwise Independence

Probability bounds on the sum of $n$ pairwise independent Bernoulli random variables exceeding an integer $k$ have been proposed in the literature. However, these bounds are not tight in general. In this paper, we provide three results towards finding tight probability bounds on the sum of pairwise independent Bernoulli random variables. Firstly, for $k = 1$, the tightest upper bound on the probability of the union of $n$ pairwise independent events is provided. Secondly, for $k \geq 2$, the tightest upper bound with identical marginals is provided. Lastly, for general pairwise independent Bernoulli random variables, new upper bounds are derived for $k \geq 2$, by ordering the probabilities. These bounds improve on existing bounds and are tight under certain conditions. The proofs of tightness are developed using techniques of linear optimization. Numerical examples are provided to quantify the improvement of the bounds over existing bounds.Comment: 33 pages, 4 figure

Generalized Decision Rule Approximations for Stochastic Programming via Liftings

Stochastic programming provides a versatile framework for decision-making under uncertainty, but the resulting optimization problems can be computationally demanding. It has recently been shown that, primal and dual linear decision rule approximations can yield tractable upper and lower bounds on the optimal value of a stochastic program. Unfortunately, linear decision rules often provide crude approximations that result in loose bounds. To address this problem, we propose a lifting technique that maps a given stochastic program to an equivalent problem on a higherdimensional probability space. We prove that solving the lifted problem in primal and dual linear decision rules provides tighter bounds than those obtained from applying linear decision rules to the original problem. We also show that there is a one-to-one correspondence between linear decision rules in the lifted problem and families of non-linear decision rules in the original problem. Finally, we identify structured liftings that give rise to highly flexible piecewise linear decision rules and assess their performance in the context of a stylized investment planning problem.

A complementarity approach to multistage stochastic linear programs

Das Gebiet der Stochastischen Programmierung gehört in die Problemklasse der "Entscheidungsfindung unter Unsicherheit". Anwendungen finden sich weitverbreitet in den Feldern der industriellen Produktion und der finanziellen Planung neben vielen anderen. Die Arbeit befasst sich mit der Approximation von 'Multistage Stochastic Linear Programs' (MSLP), wo einige Modelldaten als zufällig vorausgesetzt werden und sich sukzessiv in diskreter Zeit $t=1,...,T$ realisieren, wobei $T$ ein endlicher Planungshorizont sei. Entscheidungen zum Zeitpunkt $t$ sollen so gefällt werden, dass die Summe ihrer unmittelbar anfallenden Kosten und den erwarteten Recourse Kosten minimiert wird, gegeben die vorangegangenen Entscheidungen und die Information, welche bis $t$ verfügbar ist. Falls die Anzahl Szenarien endlich ist, dann lässt sich das Optimierungsproblem als Linearprogramm formulieren und auch direkt lösen, sofern diese Anzahl nicht zu gross ist. Numerische Approximationsmethoden sind häufig unumgänglich, insbesondere falls die zufälligen Daten stetig verteilt sind. Es gibt einige Methoden für den Fall $T=2$, welche auf diese Situation zugeschnitten sind. Leider stellten sich diese als unpraktisch heraus, um sie auf den Fall $T≥3$ zu erweitern, weil in diesem Fall die Auswertung eines einzelnen Recourse Funktionswertes nahezu denselben Schwierigkeitsgrad wie die Bestimmung des optimalen Zielfunktionswertes des Gesamtproblems aufweist. Da wir den Fall von stetig verteilten Daten miteinschliessen, wird MSLP als infinites Linearprogramm formuliert, welches auch eine infinite duale Form besitzt. Die Optimalitätslücke eines zulässigen primal-dual Paares kann als Erwartungswert einer nichtnegativen Zufallsvariablen ausgedrückt werden, in der Arbeit 'Komplementaritätsvariable' genannt. Eine Aggregation von Restriktionen und Entscheidungen scheint ein natürlicher Zugangzu sein, um MSLP numerisch handhabbar zu machen. Wir analysieren vor allem Modelle, bei denen jede optimale Lösung eines geeignet aggregierten Dualproblems zulässig im originalen Dualproblem ist, was auf untere Schranken führt. Danach schlagen wir einen Weg basierend auf den aggregierten Lösungen vor, wie sich rekursiv durch das Lösen einer Folge von kleinen linearen und quadratischen Subproblemen eine zulässige Entscheidungspolitik in der Originalaufgabe definieren lässt. Unter geeigneten Modellannahmen und abhängig vom Aggregationsfehler erweist sich diese Entscheidungspolitik als nahe liegend zu der aggregierten optimalen Primallösung. Ausserdem wird das Worst-Case Verhalten der Komplementaritätsvariable, welche sich aus der rekursiven Entscheidungspolitik und der aggregierten optimalen Duallösung ergibt, sowohl in Erwartung als auch in einem fast sicheren Sinn analysiert. Das letztere Resultat wird verwendet, um die Endlichkeit des vorgeschlagenen Verfeinerungsalgorithmus MSLP-APPROX nachzuweisen, welcher auf simulierten Werten der Komplementaritätsvariable basiert. Wir beweisen auch, dass - bei sukzessiver Erhöhung der Stichprobe und eines Genauigkeitsparameters von MSLP-APPROX - die (schwachen) Häufungspunkte der Lösungskandidaten das Originalproblem lösen. Um das praktische Verhalten von MSLP-APPROX zu veranschaulichen, werden im letzten Teil numerische Resultate präsentiert. The field of Stochastic Programming belongs to the problem class of "Decision-Making under Uncertainty''. Applications are widely available in the areas of industrial production and financial planning, among many others. The thesis deals with the approximation of Multistage Stochastic Linear Programs (MSLP) where some model data are assumed to be random and successively realized at time $t=1,...,T$ where $T$ is a finite planning horizon. Decisions at time $t$ should be made such that the sum of their immediate costs and the expected recourse costs is minimized, given the previous decisions and the information available up to $t$. When the number of scenarios is finite, the optimization problem can be formulated as a linear program and may also be solved directly, provided that this number is not too high. Numerical approximation methods are often inevitable, especially if the random data are continuously distributed. There are some methods available for the case $T=2$ designed for this situation. Unfortunately, they turned out to be impractical to extend to the case $T≥3$ because, in this case, the computation of a single recourse function value has almost the same degree of difficulty as determining the optimal objective value of the overall problem.Since we include the case of continuously distributed data, MSLP is expressed as an infinite linear program which also has an infinite dual form. The optimality gap of a feasible primal-dual pair is expressed as the expectation of a nonnegative random variable, in the thesis called the 'complementarity variable'. Aggregation of constraints and decisions seems to be a natural approach to make MSLP numerically manageable. We analyze particularly models where every optimal solution of a suitably aggregated dual problem is feasible in the original dual problem, leading to lower bounds. After that, based on the aggregated solutions, we propose a way to define recursively a feasible decision policy in the original primal problem by solving a sequence of small linear and quadratic subproblems. Under suitable model assumptions and depending on the aggregation error, the recursive decision policy turns out to be close to the aggregated optimal primal solution. Furthermore, the worst-case behavior of the complementarity variable resulting from the recursive decision policy and the aggregated optimal dual solution is analyzed both in expectation and in an almost sure sense. The latter result is used to prove the finiteness of the proposed refinement algorithm MSLP-APPROX which is based on simulated values of the complementarity variable. We also prove that - by successively increasing both the sample size and an accuracy parameter of MSLP-APPROX} - the (weak) accumulation points of the candidate solutions solve the original problem. In the last part, numerical results are presented in order to illustrate the practical behavior of MSLP-APPROX

Artificial Intelligence and Machine Learning Approaches to Energy Demand-Side Response: A Systematic Review

Recent years have seen an increasing interest in Demand Response (DR) as a means to provide flexibility, and hence improve the reliability of energy systems in a cost-effective way. Yet, the high complexity of the tasks associated with DR, combined with their use of large-scale data and the frequent need for near real-time de-cisions, means that Artificial Intelligence (AI) and Machine Learning (ML) — a branch of AI — have recently emerged as key technologies for enabling demand-side response. AI methods can be used to tackle various challenges, ranging from selecting the optimal set of consumers to respond, learning their attributes and pref-erences, dynamic pricing, scheduling and control of devices, learning how to incentivise participants in the DR schemes and how to reward them in a fair and economically efficient way. This work provides an overview of AI methods utilised for DR applications, based on a systematic review of over 160 papers, 40 companies and commercial initiatives, and 21 large-scale projects. The papers are classified with regards to both the AI/ML algorithm(s) used and the application area in energy DR. Next, commercial initiatives are presented (including both start-ups and established companies) and large-scale innovation projects, where AI methods have been used for energy DR. The paper concludes with a discussion of advantages and potential limitations of reviewed AI techniques for different DR tasks, and outlines directions for future research in this fast-growing area