Duality for mixed-integer convex minimization

We extend in two ways the standard Karush–Kuhn–Tucker optimality conditions to problems with a convex objective, convex functional constraints, and the extra requirement that some of the variables must be integral. While the standard Karush–Kuhn–Tucker conditions involve separating hyperplanes, our extension is based on mixed-integer-free polyhedra. Our optimality conditions allow us to define an exact dual of our original mixed-integer convex problem

On subadditive duality for conic mixed-integer programs

In this paper, we show that the subadditive dual of a feasible conic mixed-integer program (MIP) is a strong dual whenever it is feasible. Moreover, we show that this dual feasibility condition is equivalent to feasibility of the conic dual of the continuous relaxation of the conic MIP. In addition, we prove that all known conditions and other 'natural' conditions for strong duality, such as strict mixed-integer feasibility, boundedness of the feasible set or essentially strict feasibility imply that the subadditive dual is feasible. As an intermediate result, we extend the so-called 'finiteness property' from full-dimensional convex sets to intersections of full-dimensional convex sets and Dirichlet convex sets

On Subadditive Duality for Conic Mixed-Integer Programs

In this paper, we show that the subadditive dual of a feasible conic mixed-integer program (MIP) is a strong dual whenever it is feasible. Moreover, we show that this dual feasibility condition is equivalent to feasibility of the conic dual of the continuous relaxation of the conic MIP. In addition, we prove that all known conditions and other 'natural' conditions for strong duality, such as strict mixed-integer feasibility, boundedness of the feasible set or essentially strict feasibility imply that the subadditive dual is feasible. As an intermediate result, we extend the so-called 'finiteness property' from full-dimensional convex sets to intersections of full-dimensional convex sets and Dirichlet convex sets

On subadditive duality for conic mixed-integer programs

In this paper, we show that the subadditive dual of a feasible conic mixed-integer program (MIP) is a strong dual whenever it is feasible. Moreover, we show that this dual feasibility condition is equivalent to feasibility of the conic dual of the continuous relaxation of the conic MIP. In addition, we prove that all known conditions and other 'natural' conditions for strong duality, such as strict mixed-integer feasibility, boundedness of the feasible set or essentially strict feasibility imply that the subadditive dual is feasible. As an intermediate result, we extend the so-called 'finiteness property' from full-dimensional convex sets to intersections of full-dimensional convex sets and Dirichlet convex sets

Structure Exploitation in Mixed-Integer Optimization with Applications to Energy Systems

Das Ziel dieser Arbeit ist neue numerische Methoden für gemischt-ganzzahlige Optimierungsprobleme zu entwickeln um eine verbesserte Geschwindigkeit und Skalierbarkeit zu erreichen. Dies erfolgt durch Ausnutzung gängiger Problemstrukturen wie separierbarkeit oder Turnpike-eigenschaften. Methoden, die diese Strukturen ausnutzen können, wurden bereits im Bereich der verteilten Optimierung und optimalen Steuerung entwickelt, sie sind jedoch nicht direkt auf gemischt-ganztägige Probleme anwendbar. Um verteilte Rechenressourcen zur Lösung von gemischt-ganzzahligen Problemen nutzen zu können, sind neue Methoden erforderlich. Zu diesem Zweck werden verschiedene Erweiterungen bestehender Methoden sowie neuartige Techniken zur gemischt-ganzzahligen Optimierung vorgestellt. Benchmark-Probleme aus Strom- und Energiesystemen werden verwendet, um zu demonstrieren, dass die vorgestellten Methoden zu schnelleren Laufzeiten führen und die Lösung großer Probleme ermöglichen, die sonst nicht zentral gelöst werden können. Die vorliegende Arbeit enthält die folgenden Beiträge: - Eine Erweiterung des Augmented Lagrangian Alternating Direction Inexact Newton-Algorithmus zur verteilten Optimierung für gemischt-ganzzahlige Probleme. - Ein neuer, teilweise-verteilter Optimierungsalgorithmus für die gemischt-ganzzahlige Optimierung basierend auf äußeren Approximationsverfahren. - Ein neuer Optimierungsalgorithmus für die verteilte gemischt-ganzzahlige Optimierung, der auf branch-and-bound Verfahren basiert. - Eine erste Untersuchung von Turnpike-Eigenschaften bei Optimalsteuerungsproblemen mit gemischten-Ganzzahligen Entscheidungsgrößen und ein spezieller Algorithmus zur Lösung dieser Probleme. - Eine neue Branch-and-Bound Heuristik, die a priori Probleminformationen effizienter nutzt als aktuelle Warmstarttechniken. Schließlich wird gezeigt, dass die Ergebnisse der vorgestellten Optimierungsalgorithmen für verteilte gemischt-ganzzahlige Optimierung stark Partitionierungsabhängig sind. Zu diesem Zweck wird auch eine Untersuchung von Partitionierungsmethoden für die verteilte Optimierung vorgestellt

Energy Management Systems and Potential Applications of Quantum Computing in the Energy Sector

The combined use of technologies plays a key role in the energy transition towards a green and sustainable economy, driven by the European Green Deal initiatives and the Paris Agreement to achieve climate neutrality in the European Union (EU) by 2050. Indeed, all viable solutions with no barriers to innovation should be considered if a fair, cost-effective, competitive, and green transition is to be ensured.Energy hubs enable the synergy of different forms of energy by exploiting their specific vir-tues. However, their management in an integrated context must be entrusted to automated manage-ment systems capable of making real-time decisions.This PhD thesis aims to assess the main potential applications of quantum computing to the energy sector in the current development scenario of quantum technologies, as well as provide the elements for modelling an energy hub and managing uncertainties.The thesis is organized as follows. Chapter 1 provides an introduction to energy manage-ment systems. The concept of an energy hub and its mathematical modelling are introduced in chap-ter 2. Chapter 3 introduces the fundamentals of energy supply. Chapter 4 examines potential use cases for quantum computing in the energy sector. Chapter 5 addresses the modelling of uncertain parameters. Chapter 6 concludes the thesis with a case study of two urban districts modelled as mul-ticarrier energy hubs connected by a multicarrier energy infrastructure providing electricity, gas and hydrogen. The conclusions are drawn in chapter 7. The appendices with additional insights enrich the thesis, which is full of comments and bibliographical references