Marcel Mongeau

Optimization of aircraft container loading

We address the problem of loading as much freight as possible in an aircraft while balancing the load in order to minimize fuel consumption and to satisfy stability/safety requirements. Our formulation methodology permits to solve the problem on a PC, within ten min, by off-the-shelf integer linear programming software. This method decides which containers to load (and in which compartment) and which to leave on the ground.

Comparison of public-domain software for black box global optimization ∗

We instance our experience with six public-domain global optimization software products and report comparative computational results obtained on a set of eleven test problems. The techniques used by the software under study include integral global optimization, genetic algorithms, simulated annealing, clustering, random search, continuation, Bayesian, tunneling, and multi-level methods. The test set contains practical problems: least median of squares regression, protein folding, and multidimensional scaling. These include non-differentiable, and also discontinuous objective functions, some with an exponential number of local minima. The dimension of the search space ranges from 1 to 20. We evaluate the software in view of engineers addressing black box global optimization problems, i.e. problems with an objective function whose explicit form is unknown and whose evaluation is costly. Such an objective function is common in industry. It is for instance given under the form of computer programmes involving a simulation

A Generic Global Optimization Algorithmfor the Chemical and Phase EquilibriumProblem

This paper addresses the problem of finding the number, K, of phases present at equilibrium and their composition, in a chemical mixture of ns substances. This corresponds to the global minimum of the Gibbs free energy of the system, subject to constraints representing mb independent conserved quantities, where mb=ns when no reaction is possible and mb ≤ ne +1 when reaction is possible and ne is the number of elements present. After surveying previous work in the field and pointing out the main issues, we extend the necessary and sufficient condition for global optimality based on the ‘reaction tangent-plane criterion’, to the case involving different thermodynamical models (multiple phase classes). We then present an algorithmic approach that reduces this global optimization problem (involving a search space of mb(ns-1) dimensions) to a finite sequence of local optimization steps inK(ns-1) -space, K ≤ mb, and global optimization steps in (ns-1)-space. The global step uses the tangent-plane criterion to determine whether the current solution is optimal, and, if it is not, it finds an improved feasible solution either with the same number of phases or with one added phase. The global step also determines what class of phase (e.g. liquid or vapour) is to be added, if any phase is to be added. Given a local minimization procedure returning a Kuhn–Tucker point and a global optimization procedure (for a lower-dimensional search space) returning a global minimum, the algorithm is proved to converge to a global minimum in a finite number of the above local and global steps. The theory is supported by encouraging computational results.

Automatic decrease of the penalty parameter in exact penalty function methods

This paper presents an analysis of the involvement of the penalty parameter in exact penalty function methods that yields modifications to the standard outer loop which decreases the penalty parameter (typically dividing it by a constant). The procedure presented is based on the simple idea of making explicit the dependence of the penalty function upon the penalty parameter and is illustrated on a linear programming problem with the l1 exact penalty function and an active-set approach. The procedure decreases the penalty parameter, when needed, to the maximal value allowing the inner minimization algorithm to leave the current iterate. It moreover avoids unnecessary calculations in the iteration following the step in which the penalty parameter is decreased. We report on preliminary computational results which show that this method can require fewer iterations than the standard way to update the penalty parameter. This approach permits a better understanding of the performance of exact penalty methods.

Flight Control System Architecture Optimization for Fly-By-Wire Airliners

The design problem of a flight control system on a large fly-by-wire airliner is to find combinations of actuator(s), power circuit(s), and computer(s) for each control surface, to fulfill the constraints imposed by the safety regulations, while keeping the resulting system weight as low as possible. The trend toward more electrical aircraft makes it harder and harder to determine, in a reasonable computer time, optimal architectures solely by traditional trial-and-error methods. This paper introduces a flight control architecture optimization process,intended as a decision aid for system engineers at early stages of the flight control architecture definition. We present an optimization model for the design process, based on a safety constraint and a weight criterion, that allows the exploitation of traditional design rules in a systematic manner. We start by reducing the initial search domain through introducing the notion of surface possible architecture, which takes into account technological constraints and empirical practices. Then, we use an adaptation of branch-and-bound methods to solve the remaining discrete optimization problem. Finally, an application to the Airbus A340 roll control system is addressed. An exact optimum is found among 10 14 possible architectures in less than 25 min on a standard desktop computer. Our methodology is currently under the process of industrial implementation at Airbus, where it will be used in the early design stage as a decision-analysis tool.

A Hybrid Metaheuristic Optimization Algorithm for Strategic Planning of 4D Aircraft Trajectories at the Continental Scale

Global air-traffic demand is continuously increasing. To handle such a tremendous traffic volume while maintaining at least the same level of safety, a more efficient strategic trajectory planning is necessary. In this work, we present a strategic trajectory planning methodology which aims to minimize interaction between aircraft at the European-continent scale. In addition, we propose a preliminary study that takes into account uncertainties of aircraft positions in the horizontal plane. The proposed methodology separates aircraft by modifying their trajectories and departure times. This route/departure-time assignment problem is modeled as a mixed-integer optimization problem. Due to the very high combinatorics involved in the continent-scale context (involving more than 30,000 flights), we develop and implement a hybrid-metaheuristic optimization algorithm. In addition, we present a computationally-efficient interaction detection method for large trajectory sets. The proposed methodology is successfully implemented and tested on a full-day simulated air traffic over the European airspace, yielding to an interaction-free trajectory plan.

Discontinuous piecewise linear optimization

A theoretical framework and a practical algorithm are presented to solve discontinuous piecewise linear optimization problems dealing with functions for which theridges are known. A penalty approach allows one to consider such problems subject to a wide range of constraints involving piecewise linear functions. Although the theory is expounded in detail in the special case of discontinuous piecewiselinear functions, it is straightforwardly extendable, using standard nonlinear programming techniques, tononlinear (discontinuous piecewise differentiable) functions.The descent algorithm which is elaborated uses active-set and projected gradient approaches. It is a generalization of the ideas used by Conn to deal with nonsmoothness in thel1 exact penalty function, and it is based on the notion ofdecomposition of a function into a smooth and a nonsmooth part. The constrained case is reduced to the unconstrained minimization of a (piecewise linear)l1 exact penalty function. We also discuss how the algorithm is modified when it encounters degenerate points. Preliminary numerical results are presented: the algorithm is applied to discontinuous optimization problems from models in industrial engineering.

Global Optimization for the Chemical and Phase Equilibrium Problem using Interval Analysis

This paper addresses the problem of minimizing the Gibbs free energy in the n c -component, multi-phase chemical and phase equilibrium problem involving different thermodynamic models. The algorithmic approach used is based on the tangent-plane criterion of Gibbs: the global optimization problem considered, which involves a search space of n(n + 1) dimensions, is reduced to a finite sequence of global optimization steps in n-space, and local optimization steps in nK-space, K ≤ n + 1.

Computing Eigenelements of Real Symmetric Matrices via Optimization

In certain circumstances, it is advantageous to use an optimization approach in order to solve the generalized eigenproblem, Ax = λBx, where A and B are real symmetric matrices and B is positive definite. In particular, this is the case when the matrices A and B are very large the computational cost, prohibitive, of solving, with high accuracy, systems of equations involving these matrices. Usually, the optimization approach involves optimizing the Rayleigh quotient.We first propose alternative objective functions to solve the (generalized) eigenproblem via (unconstrained) optimization, and we describe the variational properties of these functions.We then introduce some optimization algorithms (based on one of these formulations) designed to compute the largest eigenpair. According to preliminary numerical experiments, this work could lead the way to practical methods for computing the largest eigenpair of a (very) large symmetric matrix (pair).

Solving air traffic conflict problems via local continuous optimization

This paper first introduces an original trajectory model using B-splines and a new semi-infinite programming formulation of the separation constraint involved in air traffic conflict problems. A new continuous optimization formulation of the tactical conflict-resolution problem is then proposed. It involves very few optimization variables in that one needs only one optimization variable to determine each aircraft trajectory. Encouraging numerical experiments show that this approach is viable on realistic test problems. Not only does one not need to rely on the traditional, discretized, combinatorial optimization approaches to this problem, but, moreover, local continuous optimization methods, which require relatively fewer iterations and thereby fewer costly function evaluations, are shown to improve the performance of the overall global optimization of this non-convex problem.