Jérémy Omer

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

Air-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.

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

In 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) 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.

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

In this paper, 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 model uses a different cost structure compared to classical clique search problems: 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, our mathematical framework is valid for any hypotheses on the aircraft dynamics and any choice of the available maneuvers. 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 programming 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 ten seconds for instances with up to 250 aircraft randomly allocated to bisten flight levels.

A linear programming decomposition focusing on the span of the nondegenerate columns

The improved primal simplex (IPS) was recently developed by Elhalaloui et al. to take advantage of degeneracy when solving linear programs with the primal simplex. It implements a dynamic constraint reduction based on the compatible columns, i.e., those that belong to the span of a given subset of basic columns including the nondegenerate ones. The identification of the compatible variables may however be computationally costly and a large number of linear programs are solved to enlarge the subset of basic variables. In this article, we first show how the positive edge criterion of Raymond et al. can be included in IPS for a fast identification of the compatible variables. Our algorithm then proceeds through a series of reduction and augmentation phases until optimality is reached. In a reduction phase, we identify compatible variables and focus on them to make quick progress toward optimality. During an augmentation phase, we compute one greatest normalized improving direction and select a subset of variables that should be considered in the reduced problem. Compared with IPS, the linear program that is solved to find this direction involves the data of the original constraint matrix. This new algorithm is tested over Mittelmann’s benchmark for linear programming and on instances arising from industrial applications. The results show that the new algorithm outperforms the primal simplex of CPLEX on most highly degenerate instances in which a sufficient number of nonbasic variables are compatible. In contrast, IPS has difficulties on the eleven largest Mittelmann instances.

Solving the Air Conflict Resolution Problem Under Uncertainty Using an Iterative Biobjective Mixed Integer Programming Approach

In this paper, we tackle the aircraft conflict resolution problem under uncertainties. We consider errors due to the wind effect, the imprecision of aircraft speed prediction, and the delay in the execution of maneuvers. Using a geometrical approach, we derive an analytical expression for the minimum distance between aircraft, along with the corresponding probability of conflict. These expressions are incorporated into an existing deterministic model for conflict resolution. This model solves the problem as a maximum clique of minimum weight in a graph whose vertices represent possible maneuvers and where edges link conflict-free maneuvers of different aircraft. We then present a solution procedure focusing on two criteria, namely, fuel efficiency and the probability of reissuing maneuvers in the future: we iteratively generate Pareto front solutions to provide the controller with a set of possible solutions where she can choose the one corresponding the most to her preferences. Intensive Monte Carlo simu...

A New Variant of the Minimum-Weight Maximum-Cardinality Clique Problem to Solve Conflicts between Aircraft

In this article, we formulate a new variant of the problem of finding a maximum clique of minimum weight in a graph applied to the detection and resolution of conflicts between aircraft. The innovation of the model relies on the cost structure: the cost of the vertices cannot be determined a priori, since they depend on the vertices in the clique. We apply this formulation to the resolution of conflicts between aircraft by building a graph whose vertices correpond to a set of maneuvers and whose edges link conflict-free maneuvers. A maximum clique of minimal weight yields a conflict-free situation involving all aircraft and minimizing the costs induced. We solve the problem as a mixed integer linear program. Simulations on a benchmark of complex instances highlight computational times smaller than 20 seconds for situations involving up to 20 aircraft.

The positive edge pricing rule for the dual simplex

In this paper, we develop the two-dimensional positive edge criterion for the dual simplex. This work extends a similar pricing rule implemented by Towhidi et al. (2014) 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.