Hybridization of Nonlinear and Mixed-Integer Linear Programming for Aircraft Separation With Trajectory Recovery

International audienceThe approach presented in this article aims at finding a solution to the problem of conflict-free motion planning for multiple aircraft on the same flight level with trajectory recovery. One contribution of this work is to develop three consistent models, from a continuous-time representation to a discrete-time linear approximation. Each of these models guarantees separation at all times as well as trajectory recovery, but they are not equally difficult to solve. A new hybrid algorithm is thus developed in order to use the optimal solution of a mixed integer linear program as a starting point when solving a nonlinear formulation of the problem. The significance of this process is that it always finds a solution when the linear model is feasible while still taking into account the nonlinear nature of the problem. A test bed containing numerous data sets is then generated from three virtual scenarios. A comparative analysis with three different initialisations of the nonlinear optimisation validates the efficiency of the hybrid method

A Unified View of Piecewise Linear Neural Network Verification

The success of Deep Learning and its potential use in many safety-critical applications has motivated research on formal verification of Neural Network (NN) models. Despite the reputation of learned NN models to behave as black boxes and the theoretical hardness of proving their properties, researchers have been successful in verifying some classes of models by exploiting their piecewise linear structure and taking insights from formal methods such as Satisifiability Modulo Theory. These methods are however still far from scaling to realistic neural networks. To facilitate progress on this crucial area, we make two key contributions. First, we present a unified framework that encompasses previous methods. This analysis results in the identification of new methods that combine the strengths of multiple existing approaches, accomplishing a speedup of two orders of magnitude compared to the previous state of the art. Second, we propose a new data set of benchmarks which includes a collection of previously released testcases. We use the benchmark to provide the first experimental comparison of existing algorithms and identify the factors impacting the hardness of verification problems.Comment: Updated version of "Piecewise Linear Neural Network verification: A comparative study

Proceedings of the XIII Global Optimization Workshop: GOW'16

[Excerpt] Preface: Past Global Optimization Workshop shave been held in Sopron (1985 and 1990), Szeged (WGO, 1995), Florence (GO’99, 1999), Hanmer Springs (Let’s GO, 2001), Santorini (Frontiers in GO, 2003), San José (Go’05, 2005), Mykonos (AGO’07, 2007), Skukuza (SAGO’08, 2008), Toulouse (TOGO’10, 2010), Natal (NAGO’12, 2012) and Málaga (MAGO’14, 2014) with the aim of stimulating discussion between senior and junior researchers on the topic of Global Optimization. In 2016, the XIII Global Optimization Workshop (GOW’16) takes place in Braga and is organized by three researchers from the University of Minho. Two of them belong to the Systems Engineering and Operational Research Group from the Algoritmi Research Centre and the other to the Statistics, Applied Probability and Operational Research Group from the Centre of Mathematics. The event received more than 50 submissions from 15 countries from Europe, South America and North America. We want to express our gratitude to the invited speaker Panos Pardalos for accepting the invitation and sharing his expertise, helping us to meet the workshop objectives. GOW’16 would not have been possible without the valuable contribution from the authors and the International Scientific Committee members. We thank you all. This proceedings book intends to present an overview of the topics that will be addressed in the workshop with the goal of contributing to interesting and fruitful discussions between the authors and participants. After the event, high quality papers can be submitted to a special issue of the Journal of Global Optimization dedicated to the workshop. [...

Joint Metering and Conflict Resolution in Air Traffic Control

This paper describes a novel optimization-based approach to conflict resolution in air traffic control, based on geometric programming. The main advantage of the approach is that Geometric Programs (GPs) can also capture various metering directives issued by the traffic flow management level, in contrast to most recent methods focusing purely on aircraft separation issues. GPs can also account for some of the nonlinearities present in the formulations of conflict resolution problems, while incurring only a small penalty in computation time with respect to the fastest linear programming based approaches. Additional integer variables can be introduced to improve the quality of the obtained solutions and handle combinatorial choices, resulting in Mixed-Integer Geometric Programs (MIGPs). We present GPs and MIGPs to solve a variety of joint metering and separation scenarios, e.g. including miles-in-trail and minutes-in-trail restrictions through airspace fixes and boundaries. Simulation results demonstrate the efficiency of the approach

Comparison of Mixed-Integer Linear Models for Fuel-Optimal Air Conflict Resolution With Recovery

International audienceAny significant increase in current levels of air traffic will need the support of efficient decision-aid tools. One of the tasks of air traffic management is to modify trajectories when necessary to maintain a sufficient separation between pairs of aircraft. Several algorithms have been developed to solve this problem, but the diversity in the underlying assumptions makes it difficult to compare their performance. In this article, separation is maintained through changes of heading and velocity while minimizing a combination of fuel consumption and delay. For realistic trajectories, the speed is continuous with respect to time, the acceleration and turning rate are bounded, and the planned trajectories are recovered after the maneuvers. After describing the major modifications to existing models that are necessary to satisfy this definition of the problem, we compare three mixed integer linear programs. The first model is based on a discretization of the airspace, and the second relies on a discretization of the time horizon. The third model implements a time decomposition of the problem; it allows only one initial maneuver, and it is solved periodically with a receding horizon to build a complete trajectory. The computational tests are conducted on a benchmark of artificial instances specifically built to include complex situations. Our analysis of the results highlights the strengths and limits of each model. The time decomposition proves to be an excellent compromise

Mathematical optimization methods for aircraft conflict resolution in air traffic control

Air traffic control is a very dynamic and heavy constrained environment where many decisions need to be taken over short periods of time and in the context of uncertainty. Adopting automation under such circumstances can be a crucial initiative to reduce controller workload and improve airspace usage and capacity. Traditional methods for air traffic control have been exhaustively used in the last decades and are reaching their limits, therefore automated approaches are receiving a significant and growing attention. In this thesis, the focus is to obtain optimal aircraft trajectories to ensure flight safety in the short-term by solving optimization problems. During cruise stage, separation conditions require a minimum of 5 Nautical Miles (NM) horizontally or 1000 feet (ft) vertically between any pair of aircraft. A conflict between two or more aircraft is a loss of separation among these aircraft. Air traffic networks are organized in flight levels which are separated by at least 1000 ft, hence during cruise stage, most conflicts occur among aircraft flying at the same flight level. This thesis presents several mathematical formulations to address the aircraft conflict resolution problem and its variants. The core contribution of this research is the development of novel mixed integer programming models for the aircraft conflict resolution problem. New mathematical optimization formulations for the deterministic aircraft conflict resolution problem are analyzed and exact methods are developed. Building on this framework, richer formulations capable of accounting for aircraft trajectory prediction uncertainty and trajectory recovery are proposed. Results suggest that the formulations presented in thesis are efficient and competitive enough with the state-of-art models and they can provide an alternative solution to possibly fill some of the gaps currently present in the literature. Furthermore, the results obtained demonstrate the impact of these models in solving very denser air space scenarios and their competitiveness with state-of-the-art formulations without regarding variable discretization or non-linear components