Solving variational inequalities defined on a domain with infinitely many linear constraints

We study a variational inequality problem whose domain is defined by infinitely many linear inequalities. A discretization method and an analytic center based inexact cutting plane method are proposed. Under proper assumptions, the convergence results for both methods are given. We also provide numerical examples to illustrate the proposed method

Design and Analysis of Vehicle Sharing Programs: A Systems Approach

Transit, touted as a solution to urban mobility problems, cannot match the addictive flexibility of the automobile. 86% of all trips in the U.S. are in personal vehicles. A more recent approach to reduce automobile dependence is through the use of Vehicle Sharing Programs (VSPs). A VSP involves a fleet of vehicles located strategically at stations across the transportation network. In its most flexible form, users are free to check out vehicles at any station and return the vehicle at a station close to their destination. Vehicle fleets are comprised of bicycles, cars or electric vehicles. Such systems offer innovative solutions to the larger mobility problem and can have positive impacts on the transportation system as a whole by reducing urban congestion. This dissertation employs a network modeling framework to quantitatively design and operate VSPs. At the strategic level, the problem of determining the optimal VSP configuration is studied. A bilevel optimization model and associated solution methods are developed and implemented for a large-scale case study in Washington D.C. The model explicitly considers the intermodalism, and views the VSP as a `last-mile' connection of an existing transit network. At the operational level, by transferring control of vehicles to the user for improved system flexibility, exceptional logistical challenges are placed on operators who must ensure adequate vehicle stock (and parking slots) at each station to service all demand. Since demand in the short-term can be asymmetric (flow from one station to another is seldom equal to flow in the opposing direction), service providers need to redistribute vehicles to correct this imbalance. A chance-constrained program is developed that generates least-cost redistribution plans such that most demand in the near future is met. Since the program has a non-convex feasible region, two methods for its solution are developed. The model is applied to a real-world car-sharing system in Singapore where the value of accounting for inherent stochasticities is demonstrated. The framework is used to characterize the efficiency of Velib, a large-scale bicycle sharing system in Paris, France

Homotopy Methods to Compute Equilibria in Game Theory

This paper presents a complete survey of the use of homotopy methods in game theory.Homotopies allow for a robust computation of game-theoretic equilibria and their refinements. Homotopies are also suitable to compute equilibria that are selected by variousselection theories. We present all relevant techniques underlying homotopy algorithms.We give detailed expositions of the Lemke-Howson algorithm and the Van den Elzen-Talman algorithm to compute Nash equilibria in 2-person games, and the Herings-Vanden Elzen, Herings-Peeters, and McKelvey-Palfrey algorithms to compute Nash equilibriain general n-person games.operations research and management science;

Interior Point Methods 25 Years Later

Interior point methods for optimization have been around for more than 25 years now. Their presence has shaken up the field of optimization. Interior point methods for linear and (convex) quadratic programming display several features which make them particularly attractive for very large scale optimization. Among the most impressive of them are their low-degree polynomial worst-case complexity and an unrivalled ability to deliver optimal solutions in an almost constant number of iterations which depends very little, if at all, on the problem dimension. Interior point methods are competitive when dealing with small problems of dimensions below one million constraints and variables and are beyond competition when applied to large problems of dimensions going into millions of constraints and variables. In this survey we will discuss several issues related to interior point methods including the proof of the worst-case complexity result, the reasons for their amazingly fast practi-cal convergence and the features responsible for their ability to solve very large problems. The ever-growing sizes of optimization problems impose new requirements on optimizatio

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