On the representation of the search region in multi-objective optimization

Given a finite set $N$ of feasible points of a multi-objective optimization (MOO) problem, the search region corresponds to the part of the objective space containing all the points that are not dominated by any point of $N$, i.e. the part of the objective space which may contain further nondominated points. In this paper, we consider a representation of the search region by a set of tight local upper bounds (in the minimization case) that can be derived from the points of $N$. Local upper bounds play an important role in methods for generating or approximating the nondominated set of an MOO problem, yet few works in the field of MOO address their efficient incremental determination. We relate this issue to the state of the art in computational geometry and provide several equivalent definitions of local upper bounds that are meaningful in MOO. We discuss the complexity of this representation in arbitrary dimension, which yields an improved upper bound on the number of solver calls in epsilon-constraint-like methods to generate the nondominated set of a discrete MOO problem. We analyze and enhance a first incremental approach which operates by eliminating redundancies among local upper bounds. We also study some properties of local upper bounds, especially concerning the issue of redundant local upper bounds, that give rise to a new incremental approach which avoids such redundancies. Finally, the complexities of the incremental approaches are compared from the theoretical and empirical points of view.Comment: 27 pages, to appear in European Journal of Operational Researc

Approches de résolution exacte et approchée en optimisation combinatoire multi-objectif, application au problème de l'arbre couvrant de poids minimal

This thesis deals with several aspects related to solving multi-objective problems, without restriction to the bi-objective case. We consider exact solving, which generates the nondominated set, and approximate solving, which computes an approximation of the nondominated set with a priori guarantee on the quality.We first consider the determination of an explicit representation of the search region. The search region, defined with respect to a set of known feasible points, excludes from the objective space the part which is dominated by these points. Future efforts to find all nondominated points should therefore be concentrated on the search region.Then we review branch and bound and ranking algorithms and we propose a new hybrid approach for the determination of the nondominated set. We show how the proposed method can be adapted to generate an approximation of the nondominated set. This approach is instantiated on the minimum spanning tree problem. We review several properties of this problem which enable us to specialize some procedures of the proposed approach and integrate specific preprocessing rules. This approach is finally supported through experimental results.On s'attache dans cette thèse à plusieurs aspects liés à la résolution de problèmes multi-objectifs, sans se limiter au cas biobjectif. Nous considérons la résolution exacte, dans le sens de la détermination de l'ensemble des points non dominés, ainsi que la résolution approchée dans laquelle on cherche une approximation de cet ensemble dont la qualité est garantie a priori.Nous nous intéressons d'abord au problème de la détermination d'une représentation explicite de la région de recherche. La région de recherche, étant donné un ensemble de points réalisables connus, exclut la partie de l'espace des objectifs que dominent ces points et constitue donc la partie de l'espace des objectifs où les efforts futurs doivent être concentrés dans la perspective de déterminer tous les points non dominés.Puis nous considérons le recours aux algorithmes de séparation et évaluation ainsi qu'aux algorithmes de ranking afin de proposer une nouvelle méthode hybride de détermination de l'ensemble des points non dominés. Nous montrons que celle-ci peut également servir à obtenir une approximation de l'ensemble des points non dominés. Cette méthode est implantée pour le problème de l'arbre couvrant de poids minimal. Les quelques propriétés de ce problème que nous passons en revue nous permettent de spécialiser certaines procédures et d'intégrer des prétraitements spécifiques. L'intérêt de cette approche est alors soutenu à l'aide de résultats expérimentaux

Static and dynamic posture control in postlingual cochlear implanted patients: effects of dual-tasking, visual and auditory inputs suppression

Posture control is based on central integration of multisensory inputs, and on internal representation of body orientation in space. This multisensory feedback regulates posture control and continuously updates the internal model of body's position which in turn forwards motor commands adapted to the environmental context and constraints. The peripheral localization of the vestibular system, close to the cochlea, makes vestibular damage possible following cochlear implant (CI) surgery. Impaired vestibular function in CI patients, if any, may have a strong impact on posture stability. The simple postural task of quiet standing is generally paired with cognitive activity in most day life conditions, leading therefore to competition for attentional resources in dual-tasking, and increased risk of fall particularly in patients with impaired vestibular function. This study was aimed at evaluating the effects of postlingual cochlear implantation on posture control in adult deaf patients. Possible impairment of vestibular function was assessed by comparing the postural performance of patients to that of age-matched healthy subjects during a simple postural task performed in static (stable platform) and dynamic (platform in translation) conditions, and during dual-tasking with a visual or auditory memory task. Postural tests were done in eyes open (EO) and eyes closed (EC) conditions, with the CI activated (ON) or not (OFF). Results showed that the postural performance of the CI patients strongly differed from the controls, mainly in the EC condition. The CI patients showed significantly reduced limits of stability and increased postural instability in static conditions. In dynamic conditions, they spent considerably more energy to maintain equilibrium, and their head was stabilized neither in space nor on trunk: they behaved dynamically without vision like an inverted pendulum while the controls showed a whole body rigidification strategy. Hearing (prosthesis on) as well as dual-tasking did not really improve the dynamic postural performance of the CI patients. We conclude that CI patients become strongly visual dependent mainly in challenging postural conditions, a result they have to be awarded of particularly when getting older