Robust optimization with incremental recourse

In this paper, we consider an adaptive approach to address optimization problems with uncertain cost parameters. Here, the decision maker selects an initial decision, observes the realization of the uncertain cost parameters, and then is permitted to modify the initial decision. We treat the uncertainty using the framework of robust optimization in which uncertain parameters lie within a given set. The decision maker optimizes so as to develop the best cost guarantee in terms of the worst-case analysis. The recourse decision is ``incremental"; that is, the decision maker is permitted to change the initial solution by a small fixed amount. We refer to the resulting problem as the robust incremental problem. We study robust incremental variants of several optimization problems. We show that the robust incremental counterpart of a linear program is itself a linear program if the uncertainty set is polyhedral. Hence, it is solvable in polynomial time. We establish the NP-hardness for robust incremental linear programming for the case of a discrete uncertainty set. We show that the robust incremental shortest path problem is NP-complete when costs are chosen from a polyhedral uncertainty set, even in the case that only one new arc may be added to the initial path. We also address the complexity of several special cases of the robust incremental shortest path problem and the robust incremental minimum spanning tree problem

Robust Fluid Processing Networks

Fluid models provide a tractable and useful approach in approximating multiclass processing networks. However, they ignore the inherent stochasticity in arrival and service processes. To address this shortcoming, we develop a robust fluid approach to the control of processing networks. We provide insights into the mathematical structure, modeling power, tractability, and performance of the resulting model. Specifically, we show that the robust fluid model preserves the computational tractability of the classical fluid problem and retains its original structure. From the robust fluid model, we derive a (scheduling) policy that regulates how fluid from various classes is processed at the servers of the network. We present simulation results to compare the performance of our policies to several commonly used traditional methods. The results demonstrate that our robust fluid policies are near-optimal (when the optimal can be computed) and outperform policies obtained directly from the fluid model and heuristic alternatives (when it is computationally intractable to compute the optimal).National Science Foundation (U.S.) (Grant CNS-1239021)National Science Foundation (U.S.) (Grant IIS-1237022)United States. Army Research Office (Grant W911NF-11-1-0227)United States. Army Research Office (Grant W911NF-12-1-0390)United States. Office of Naval Research (Grant N00014-10-1-0952

Continuous-time Dynamic Shortest Paths with Negative Transit Times

We consider the dynamic shortest path problem in the continuous-time model because of its importance. This problem has been extensively studied in the literature. But so far, all contributions to this problem are based on the assumption that all transit times are strictly positive. However, in order to study dynamic network flows it is essential to support negative transit times since they occur quite naturally in residual networks. In this paper we extend the work of Philpott [SIAM Control Opt., 1994, pp. 538-552] to the case of arbitrary (also negative) transit times. In particular, we study a corresponding linear program in space of measures and characterize its extreme points. We show a one-to-one correspondence between extreme points and dynamic paths. Further, under certain assumptions, we prove the existence of an optimal extreme point to the linear program and establish a strong duality result. We also present counterexamples to show that strong duality only holds under these assumptions

Continuous and Discrete Flows Over Time: A General Model Based on Measure Theory

Network flows over time form a fascinating area of research. They model the temporal dynamics of network flow problems occurring in a wide variety of applications. Research in this area has been pursued in two different and mainly independent directions with respect to time modeling: discrete and continuous time models. In this paper we deploy measure theory in order to introduce a general model of network flows over time combining both discrete and continuous aspects into a single model. Here, the flow on each arc is modeled as a Borel measure on the real line (time axis) which assigns to each suitable subset a real value, interpreted as the amount of flow entering the arc over the subset. We focus on the maximum flow problem formulated in a network where capacities on arcs are also given as Borel measures and storage might be allowed at the nodes of the network. We generalize the concept of cuts to the case of these Borel Flows and extend the famous MaxFlow-MinCut Theorem

On Solving Continuous-time Dynamic Network Flows

Temporal dynamics is a crucial feature of network flow problems occurring in many practical applications. Important characteristics of real-world networks such as arc capacities, transit times, transit and storage costs, demands and supplies etc. are subject to fluctuations over time. Consequently, also flow on arcs can change over time which leads to so-called dynamic network flows. While time is a continuous entity by nature, discrete time models are often used for modeling dynamic network flows as the resulting problems are in general much easier to handle computationally

Dynamic flows with time-varying network parameters: Optimality conditions and strong duality

Dynamic network flow problems model the temporal evolution of flows over time and also consider changes of network parameters such as capacities, costs, supplies, and demands over time. These problems have been extensively studied in the past because of their important role in real world applications such as transport, traffic, and logistics. This has led to many results, but the more challenging continuous time model still lacks some of the key features such as network related optimality conditions and algorithms that are available in the static case

Dynamische Flüsse in zeitabhängigen Netzwerken

Dynamische Netzwerkflüsse spielen eine wichtige Rolle in vielen Bereichen wie beispielsweise Transport, Verkehr und Logistik. Ihre zeitliche Dimension ermöglicht die realistische Modellierung von vielen realen Anwendungen. Dynamische Flüsse wurden bislang nur in Szenarien betrachtet, in denen die Eingebeparameter vorgegeben und nicht von der Zeit abhängig sind. Für viele Anwendungen ist dies zu statisch und es wäre wünschenswert, wenn das Netzwerkmodell nicht nur zeitabhängige Flüsse, sondern auch zeitabhängige Netzwerkparameter berücksichten würde. Sich ändernde Netzwerkcharakteristiken wie Kapazitäten, Kosten und Knotenbalancen wurden jedoch in der Literatur bislang nicht ausreichend betrachtet. Der Hauptgrund dafür ist, dass die entstehenden dynamischen Flussprobleme sehr schwierig zu analysieren und zu lösen sind; insbesondere, wenn die Zeit stetig und nicht diskret modelliert wird. In dieser Dissertation behandeln wir eine allgemeine Klasse von dynamischen Flussproblemen mit zeitveränderlichen Kapzitäten, Kosten und Bedarfen und entwickeln Lösungsalgorithmen für diese Probleme. Der Fokus liegt dabei auf kontinuierlichen Modelle, da diese - im Gegensatz zu diskreten Modellen - bisher in der Literatur wenig behandelt wurden. Daher wurde nur geringer Fortschritt darin erzielt, dynamische Flüsse mit zeitveränderlichen Parametern mit einem kontinuierlichen Zeitansatz zu lösen, obwohl ein kontinuierlicher Ansatz in realen Anwendungen unerlässlich ist. Das Ziel dieser Arbeit besteht darin, ein kontinuierliches Analogon zu den grundlegenden Konzepten und Techniken der statischen Netzwerkflüsse zu entwickeln. Wir präsentieren zwei Algorithmen basierend auf einem Diskretisierungansatz, welche gegen eine optimale Loesung konvergieren. Beide Algorithmen erzeugen approximierte nicht extremale Loesungen, obwohl ein Extrempunkt in der Praxis aufgrund seiner klaren und einfachen Struktur zu bevorzugen wäre. Zur Lösung dieses Problems wurde ein sogenannter Purifikationsalgorithmus entwickelt, welcher ausgehend von einer beliebigen zulässigen Loesung einen extremale Lösung erzeugt, ohne den Zielfunktionswert zu verschlechtern. Der Theorie dynamischer Flüsse mit zeitabhängigen Netzwerkparametern fehlen weiterhin grundlegende Aspekte, wie Optimalitätskriterien und Lösungsalgorithmen, welche im statischen Fall vorhanden sind. Dies motiviert die Betrachtung des kontinuierlichen Kürzeste-Wege-Problems. Dieses Problem tritt als Teilproblem bei der Erkennung negativer Kreise im Residualgraphen auf und führt somit zu einem Optimalitätskriterium für dynamische Netzwerkflüsse. Wir formulieren das kontinuierliche Kürzeste-Wege-Problem als lineares Programm im Raum der Maße. Ausgehend von diesem linearen Programm charakterisieren wir die Extrempunkte der zulässigen Region und zeigen, dass sie dynamischen Wegen entsprechen. Wir betrachten ebenfalls ein duales Programm und leiten Zusammenhänge mit dem primalen Programm her. Unter Anderem beweisen wir ein starkes Dualitätstheorem. Ausgehend von der Analyse des kontinuierlichen Kürzeste-Wege-Problems leiten wir drei Optimalitätskriterien für eine sehr allgemeine Klasse dynamischer Flussprobleme her. Diese basieren auf reduzierte Kosten, negative Kreise und starke Dualität. Weiterhin behandeln wir einen generischen Negativen-Kreis-Eliminations-Algorithmus der auf dem entsprechenden Optimalitätskriterium basiert.Network flow problems form a large area of optimization and are central problems in operations research, computer science, applied mathematics, and many fields of engineering. They arise not only naturally in the analysis and design of large systems, such as communication, transportation, and manufacturing systems, but also in situations that apparently are quite unrelated to networks. Network flow problems have been investigated and expanded by many researchers from various point of views. The time taken to traverse an arc is typically assumed to be zero in network flows. However, time plays a vital role and temporal dynamics is a crucial feature of network flow problems occurring in many practical applications Moreover, important characteristics of real-world networks such as arc costs and capacities, demands and supplies etc. are subject to fluctuations over time. Consequently, also flow on arcs can change over time which leads to so-called dynamic network flows (also sometimes called network flows over time). In general, dynamic network flows have three aspects which distinguish them from the traditional models. Firstly, flow values on arcs change over time, which are called dynamic flows (or flows over time). Secondly, traversal of flow along an arc takes a finite time determined by the so-called transit time. Thirdly, storage is allowed at the nodes of the network for later transshipment. In this thesis we study a general class of dynamic flow problems with time-varying capacities, costs, supplies, and demands and develop algorithms for solving such problems. We mainly concentrate on a continuous time models since - in contrast to the discrete time model - little progress has been made in solving dynamic flow problems with time-varying parameters in a continuous time model despite its importance as a realistic model in real-world applications. Our aim is to mark the transition from static to dynamic network flows by developing the continuous-time analogue of those concepts and techniques which are the cornerstones of static network flows. Specifically, we develop two algorithms based on a discretization approach that compute respectively at least converge to an optimum solution. Both algorithms lead to approximate interior solutions, while one would like to have extreme point solutions. They usually have a considerably simpler structure than arbitrary feasible solutions and are more meaningful in practice. We therefore also present a purification algorithm for our dynamic flow problems, that is, an algorithm which produces an extreme point solution as output without degrading the objective function value when it is given an arbitrary feasible solution as input. The continuous time theory of dynamic network flows still lacks some of the key features such as network related optimality conditions and algorithms that are available in the static case. This is our motivation to study the continuous-time shortest path problem since this problem appears as a subproblem when we want to test, via an algorithm for shortest paths over time, the presence of negative cycles in the residual network and thus to develop network-based optimality conditions for our dynamic network flows. The continuous-time shortest path problem is formulated as a linear program in measure spaces. We characterize extreme point of the feasible region of the linear program and show that they correspond to dynamic paths. We also consider a dual problem and derive some results between the problem and its dual, specifically, a strong duality theorem. We use the results of the continuous-time shortest path problem to establish a reduced cost optimality condition, a negative cycle optimality condition, and a strong duality result for a very general class of dynamic network flows. We also discuss a generic negative cycle-canceling algorithm resulting from the corresponding optimality criterion