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

A space-discretized mixed-integer linear model for air-conflict resolution with speed and heading maneuvers

International audienceAir-conflict resolution is a bottleneck of air traffic management that will soon require powerful decision-aid systems to avoid the proliferation of delays. Since reactivity is critical for this application, we develop a mixed-integer linear model based on space discretization so that complex situations can be solved in near real-time. The discretization allows us to model the problem with finite and potentially small sets of variables and constraints by focusing on important points of the planned trajectories, including the points where trajectories intersect. A major goal of this work is to use space discretization while allowing velocity and heading maneuvers. Realistic trajectories are also ensured by considering speed vectors that are continuous with respect to time, and limits on the velocity, acceleration, and yaw rate. A classical indicator of economic efficiency is then optimized by minimizing a weighted sum of fuel consumption and delay. The experimental tests confirm that the model can solve complex situations within a few seconds without incurring more than a few kilograms of extra fuel consumption per aircraft

Identifying infeasible subsets of linear inequalities that are irreducible with respect to a given subset of the inequalities

A classical problem in the study of an infeasible system of linear inequalities is to determine irreducible infeasible subsets of inequalities (IIS), i.e. infeasible subsets of inequalities whose proper subsets are feasible. In this article, we examine a particular situation where only a given subsystem is of interest for the analysis of infeasibility. For this, we define B-IISs as infeasible subsets of inequalities that are irreducible with respect to a given subsystem. It is a generalization of the definition of an IIS, since an IIS is irreducible with respect to the full system. We provide a practical characterization of infeasible subsets irreducible with respect to a subsystem, making the link with the dual polytope commonly used in the detection of IISs. We then turn to the study of the BIISs that can be obtained from the Phase I of the simplex algorithm. We answer an open question regarding the covering of the clusters of such B-IISs and deduce a practical algorithm to find these covering B-IISs. Our findings are numerically illustratedon the Netlib infeasible linear programs

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

The positive edge pricing rule for the dual simplex

International audienceIn this article, we develop the two-dimensional positive edge criterion for the dual simplex. This work extends a similar pricing rule implemented by Towhidi et al. [24] to reduce the negative effects of degeneracy in the primal simplex. In the dual simplex, degeneracy occurs when nonbasic variables have a zero reduced cost, and it may lead to pivots that do not improve the objective value. We analyze dual degeneracy to characterize a particular set of dual compatible variables such that if any of them is selected to leave the basis the pivot will be nondegenerate. The dual positive edge rule can be used to modify any pivot selection rule so as to prioritize compatible variables. The expected effect is to reduce the number of pivots during the solution of degenerate problems with the dual simplex. For the experiments, we implement the positive edge rule within the dual simplex of the COIN-OR LP solver, and combine it with both the dual Dantzig and the dual steepest edge criteria. We test our implementation on 62 instances from four well-known benchmarks for linear programming. The results show that the dual positive edge rule significantly improves on the classical pricing rules

Two decomposition algorithms for solving a minimum weight maximum clique model for the air conflict resolution problem

International audienceIn this article, we tackle the conflict resolution problem using a new variant of the minimum-weight maximum-clique model. The problem involves identifying maneuvers that maintain the required separation distance between all pairs of a set of aircraft while minimizing fuel costs. We design a graph in which the vertices correspond to a finite set of maneuvers and the edges connect conflict-free maneuvers. A maximum clique of minimal weight yields a conflict-free situation that involves all the aircraft and minimizes the costs induced. The innovation of the model is its cost structure: the costs of the vertices cannot be determined a priori, since they depend on the vertices in the clique. We formulate the problem as a mixed integer linear program. Since the modeling of the aircraft dynamics and the computation of trajectories is separated from the solution process, the model is flexible. As a consequence, our mathematical framework is valid for any hypotheses. In particular, the aircraft can perform dynamic velocity, heading, and flight-level changes. To solve instances involving a large number of aircraft spread over several flight levels, we introduce two decomposition algorithms. The first is a sequential mixed integer linear optimization procedure that iteratively refines the discretization of the maneuvers to yield a trade-off between computational time and cost. The second is a large neighborhood search heuristic that uses the first procedure as a subroutine. The best solutions for the available set of maneuvers are obtained in less than 10 seconds for instances with up to 250 aircraft randomly allocated to 20 flight levels

Improved Primal Simplex: A More General Theoretical Framework and an Extended Experimental Analysis

International audienceIn this article, we propose a general framework for an algorithm derived from the primal simplex that guarantees a strict improvement in the objective after each iteration. Our approach relies on the identification of compatible variables that ensure a nondegenerate iteration if pivoted into the basis. The problem of finding a strict improvement in the objective function is proved to be equivalent to two smaller problems respectively focusing on compatible and incompatible variables. We then show that the improved primal simplex (IPS) of Elhallaoui et al. is a particular implementation of this generic theoretical framework. The resulting new description of IPS naturally emphasizes what should be considered as necessary adaptations of the framework versus specific implementation choices. This provides original insight into IPS that allows for the identification of weaknesses and potential alternative choices that would extend the efficiency of the method to a wider set of problems. We perform experimental tests on an extended collection of data sets including instances of Mittelmann's benchmark for linear programming. The results confirm the excellent potential of IPS and highlight some of its limits while showing a path toward an improved implementation of the generic algorithm