Standard Bundle Methods: Untrusted Models and Duality

We review the basic ideas underlying the vast family of algorithms for nonsmooth convex optimization known as "bundle methods|. In a nutshell, these approaches are based on constructing models of the function, but lack of continuity of first-order information implies that these models cannot be trusted, not even close to an optimum. Therefore, many different forms of stabilization have been proposed to try to avoid being led to areas where the model is so inaccurate as to result in almost useless steps. In the development of these methods, duality arguments are useful, if not outright necessary, to better analyze the behaviour of the algorithms. Also, in many relevant applications the function at hand is itself a dual one, so that duality allows to map back algorithmic concepts and results into a "primal space" where they can be exploited; in turn, structure in that space can be exploited to improve the algorithms' behaviour, e.g. by developing better models. We present an updated picture of the many developments around the basic idea along at least three different axes: form of the stabilization, form of the model, and approximate evaluation of the function

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

Designing the Liver Allocation Hierarchy: Incorporating Equity and Uncertainty

Liver transplantation is the only available therapy for any acute or chronic condition resulting in irreversible liver dysfunction. The liver allocation system in the U.S. is administered by the United Network for Organ Sharing (UNOS), a scientific and educational nonprofit organization. The main components of the organ procurement and transplant network are Organ Procurement Organizations (OPOs), which are collections of transplant centers responsible for maintaining local waiting lists, harvesting donated organs and carrying out transplants. Currently in the U.S., OPOs are grouped into 11 regions to facilitate organ allocation, and a three-tier mechanism is utilized that aims to reduce organ preservation time and transport distance to maintain organ quality, while giving sicker patients higher priority. Livers are scarce and perishable resources that rapidly lose viability, which makes their transport distance a crucial factor in transplant outcomes. When a liver becomes available, it is matched with patients on the waiting list according to a complex mechanism that gives priority to patients within the harvesting OPO and region. Transplants at the regional level accounted for more than 50% of all transplants since 2000.This dissertation focuses on the design of regions for liver allocation hierarchy, and includes optimization models that incorporate geographic equity as well as uncertainty throughout the analysis. We employ multi-objective optimization algorithms that involve solving parametric integer programs to balance two possibly conflicting objectives in the system: maximizing efficiency, as measured by the number of viability adjusted transplants, and maximizing geographic equity, as measured by the minimum rate of organ flow into individual OPOs from outside of their own local area. Our results show that efficiency improvements of up to 6% or equity gains of about 70% can be achieved when compared to the current performance of the system by redesigning the regional configuration for the national liver allocation hierarchy.We also introduce a stochastic programming framework to capture the uncertainty of the system by considering scenarios that correspond to different snapshots of the national waiting list and maximize the expected benefit from liver transplants under this stochastic view of the system. We explore many algorithmic and computational strategies including sampling methods, column generation strategies, branching and integer-solution generation procedures, to aid the solution process of the resulting large-scale integer programs. We also explore an OPO-based extension to our two-stage stochastic programming framework that lends itself to more extensive computational testing. The regional configurations obtained using these models are estimated to increase expected life-time gained per transplant operation by up to 7% when compared to the current system.This dissertation also focuses on the general question of designing efficient algorithms that combine column and cut generation to solve large-scale two-stage stochastic linear programs. We introduce a flexible method to combine column generation and the L-shaped method for two-stage stochastic linear programming. We explore the performance of various algorithm designs that employ stabilization subroutines for strengthening both column and cut generation to effectively avoid degeneracy. We study two-stage stochastic versions of the cutting stock and multi-commodity network flow problems to analyze the performances of algorithms in this context

Techniques for solving Nonlinear Programming Problems with Emphasis on Interior Point Methods and Optimal Control Problems

The primary focus of this work is a thorough research into the current available techniques for solving nonlinear programming problems. Emphasis is placed on interior-point methods and the connection between optimal control problems and nonlinear programming is explored. The document contains a detailed discussion of nonlinear programming, introducing different methods used to find solutions to NLP problems and then describing a large variety of algorithms from the literature. These descriptions make use of a unified notation, highlighting key algorithmic differences between solvers. Specifically, the variations in problem formulation, chosen merit functions, ways of determining stepsize and dealing with nonconvexity are shown. Comparisons between reported results on standard test sets are made. The work also contains an understanding of optimal control problems, beginning with an introduction to Hamiltonians, based on their background in calculus of variations and Newtonian mechanics. Several small real-life problems are taken from the literature and it is shown that they can be modelled as optimal control problems so that Hamiltonian theory and Pontryagin's maximum principle can be used to solve them. This is followed by an explanation of how Runge-Kutta discretization schemes can be used to transform optimal control problems into nonlinear programs, making the wide range of NLP solvers available for their solution. A large focus of this work is on the interior point LP and QP solver hopdm. The aim has been to extend the solver so that the logic behind it can be used for solving nonlinear programming problems. The decisions which were made when converting hopdm into an nlp solver have been listed and explained. This includes a discussion of implementational details required for any interior point method, such as maintenance of centrality and choice of barrier parameter. hopdm has successfully been used as the basis for an SQP solver which is able to solve approximately 85% of the CUTE set and work has been carried out into extending it into an interior point NLP solver

Fuelling the zero-emissions road freight of the future: routing of mobile fuellers

The future of zero-emissions road freight is closely tied to the sufficient availability of new and clean fuel options such as electricity and Hydrogen. In goods distribution using Electric Commercial Vehicles (ECVs) and Hydrogen Fuel Cell Vehicles (HFCVs) a major challenge in the transition period would pertain to their limited autonomy and scarce and unevenly distributed refuelling stations. One viable solution to facilitate and speed up the adoption of ECVs/HFCVs by logistics, however, is to get the fuel to the point where it is needed (instead of diverting the route of delivery vehicles to refuelling stations) using "Mobile Fuellers (MFs)". These are mobile battery swapping/recharging vans or mobile Hydrogen fuellers that can travel to a running ECV/HFCV to provide the fuel they require to complete their delivery routes at a rendezvous time and space. In this presentation, new vehicle routing models will be presented for a third party company that provides MF services. In the proposed problem variant, the MF provider company receives routing plans of multiple customer companies and has to design routes for a fleet of capacitated MFs that have to synchronise their routes with the running vehicles to deliver the required amount of fuel on-the-fly. This presentation will discuss and compare several mathematical models based on different business models and collaborative logistics scenarios

Theoretical Concepts of Quantum Mechanics

Quantum theory as a scientific revolution profoundly influenced human thought about the universe and governed forces of nature. Perhaps the historical development of quantum mechanics mimics the history of human scientific struggles from their beginning. This book, which brought together an international community of invited authors, represents a rich account of foundation, scientific history of quantum mechanics, relativistic quantum mechanics and field theory, and different methods to solve the Schrodinger equation. We wish for this collected volume to become an important reference for students and researchers

A Polyhedral Study of Mixed 0-1 Set

We consider a variant of the well-known single node fixed charge network flow set with constant capacities. This set arises from the relaxation of more general mixed integer sets such as lot-sizing problems with multiple suppliers. We provide a complete polyhedral characterization of the convex hull of the given set