Polyhedral Newton-min algorithms for complementarity problems

Abstract : The semismooth Newton method is a very efficient approach for computing a zero of a large class of nonsmooth equations. When the initial iterate is sufficiently close to a regular zero and the function is strongly semismooth, the generated sequence converges quadratically to that zero, while the iteration only requires to solve a linear system. If the first iterate is far away from a zero, however, it is difficult to force its convergence using linesearch or trust regions because a semismooth Newton direction may not be a descent direction of the associated least-square merit function, unlike when the function is differentiable. We explore this question in the particular case of a nonsmooth equation reformulation of the nonlinear complementarity problem, using the minimum function. We propose a globally convergent algorithm using a modification of a semismooth Newton direction that makes it a descent direction of the least-square function. Instead of requiring that the direction satisfies a linear system, it must be a feasible point of a convex polyhedron; hence, it can be computed in polynomial time. This polyhedron is defined by the often very few inequalities, obtained by linearizing pairs of functions that have close negative values at the current iterate; hence, somehow, the algorithm feels the proximity of a “negative kink” of the minimum function and acts accordingly. In order to avoid as often as possible the extra cost of having to find a feasible point of a polyhedron, a hybrid algorithm is also proposed, in which the Newton-min direction is accepted if a sufficient-descent-like criterion is satisfied, which is often the case in practice. Global convergence to regular points is proved

Algorithmes de Newton-min polyédriques pour les problèmes de complémentarité

The semismooth Newton method is a very efficient approach for computing a zero of a large class of nonsmooth equations. When the initial iterate is sufficiently close to a regular zero and the function is strongly semismooth, the generated sequence converges quadratically to that zero, while the iteration only requires to solve a linear system.If the first iterate is far away from a zero, however, it is difficult to force its convergence using linesearch or trust regions because a semismooth Newton direction may not be a descent direction of the associated least-square merit function, unlike when the function is differentiable. We explore this question in the particular case of a nonsmooth equation reformulation of the nonlinear complementarity problem, using the minimum function. We propose a globally convergent algorithm using a modification of a semismooth Newton direction that makes it a descent direction of the least-square function. Instead of requiring that the direction satisfies a linear system, it must be a feasible point of a convex polyhedron; hence, it can be computed in polynomial time. This polyhedron is defined by the often very few inequalities, obtained by linearizing pairs of functions that have close negative values at the current iterate; hence, somehow, the algorithm feels the proximity of a "bad kink" of the minimum function and acts accordingly.In order to avoid as often as possible the extra cost of having to find a feasible point of a polyhedron, a hybrid algorithm is also proposed, in which the Newton-min direction is accepted if a sufficient-descent-like criterion is satisfied, which is often the case in practice. Global convergence to regular points is proved; the notion of regularity is associated with the algorithm and is analysed with care.L'algorithme de Newton semi-lisse est très efficace pour calculer un zéro d'une large classe d'équations non lisses. Lorsque le premier itéré est suffisamment proche d'un zéro régulier et si la fonction est fortement semi-lisse, la suite générée converge quadratiquement vers ce zéro, alors que l'itération ne requière que la résolution d'un système linéaire.Cependant, si le premier itéré est éloigné d'un zéro, il est difficile de forcer sa convergence par recherche linéaire ou régions de confiance, parce que la direction de Newton semi-lisse n'est pas nécessairement une direction de descente de la fonction de moindres-carrés associée, contrairement au cas où la fonction à annuler est différentiable. Nous explorons cette question dans le cas particulier d'une reformulation par équation non lisse du problème de complémentarité non linéaire, en utilisant la fonction minimum. Nous proposons un algorithme globalement convergent, utilisant une direction de Newton semi-lisse modifiée, qui est de descente pour la fonction de moindres-carrés. Au lieu de requérir la satisfaction d'un système linéaire, cette direction doit être intérieur à un polyèdre convexe, ce qui peut se calculer en temps polynomial. Ce polyèdre est défini par souvent très peu d'inégalités, obtenus en linéarisant des couples de fonctions qui ont des valeurs négatives proches à l'itéré courant; donc, d'une certaine manière, l'algorithme est capable d'estimer la proximité des "mauvais plis" de la fonction minimum et d'agir en conséquence.De manière à éviter au si souvent que possible le coût supplémentaire lié au calcul d'un point admissible de polyèdre, un algorithme hybride est également proposé, dans lequel la direction de Newton-min est acceptée si un critère de décroissance suffisante est vérifié, ce qui est souvent le cas en pratique. La convergence globale vers des points régulier est démontrée; la notion de régularité est associée à l'algorithme et est analysée avec soin

Historical Changes and Current Distribution of Caribou, Rangifer tarandus, in Quebec

We examined published historical information, reports on aerial surveys conducted since 1953, and harvest data collected since 1971 to describe changes in the distribution and abundance of Caribou (Rangifer tarandus) in Québec. The southern limit of the Caribou distribution diminished considerably in the late 19th century, and the decline in numbers probably continued until the 1960s and 1970s east of the 62nd meridian. South of the 49th parallel, only four small populations still persist. Despite the fact that all Caribou of the province were assigned to the same sub-species (R. t. caribou), three ecotypes with specific habitats and behaviour are found. The Barren-Ground ecotype, the only migratory form, is found north of the 52nd parallel. This ecotype currently occupies ≈ 255 000 km2 in fall and winter, mainly in the ecological subzones of the forest tundra and the taiga. The Barren-Ground Caribou was characterized by a very low abundance from the end of the 19th century until the mid-1950s, but increased markedly thereafter reaching over a million individuals at the beginning of the 1990s. Populations of the Mountain ecotype have been identified in the southeastern and, possibly, in the northeastern parts of the province. The latter Mountain population is virtually unknown. The southeastern population is sedentary and uses mainly the boreal forest. This population has decreased over the last century and currently numbers only ≈ 140 individuals. Finally, the Forest-Dwelling ecotype is found discontinuously, mainly between the 49th and 55th parallels. Its current distribution covers ≈ 235 000 km2, mainly east of the 72nd meridian. This sedentary ecotype is found almost exclusively in the boreal forest, principally in areas with long forest fire cycles. Its abundance has also decreased over the years. Large Forest-Dwelling populations still persisted during the 1950s and 1960s, but they apparently disappeared. The current abundance is not known precisely, but based on density estimates and considering the current distribution, it probably does not exceed 3000 individuals. Current data are insufficient to identify precisely the causes of the population decline, although hunting seems to be an important proximal cause.Nous avons utilisé les données historiques publiées, les rapports d’inventaires aériens réalisés depuis 1953 et les statistiques de récolte sportive colligées depuis 1971 pour décrire les changements dans la répartition et l’abondance du Caribou (Rangifer tarandus) au Québec. La limite méridionale de l’aire de répartition a beaucoup diminué à la fin du 19e siècle et la régression s’est probablement poursuivie durant les années 1960 et 1970 à l’est du 62e méridien. Au sud du 49e parallèle, on ne retrouve plus que quatre petites populations. Bien que tous les caribous du Québec soient considérés appartenir à la même sous-espèce (R. t. caribou), on distingue trois écotypes fréquentant des milieux différents et arborant des comportements spécifiques. Au nord du 52e parallèle, on retrouve l’écotype Toundrique, lequel est migrateur. Ces Caribous se répartissent sur ≈ 255 000 km2 durant l’automne et l’hiver, principalement dans les sous-zones écologiques de la toundra forestière et de la taïga. Cet écotype était peu abondant entre la fin du 19e siècle et le milieu des années 1950, mais il s’est accru considérablement pour atteindre plus d’un million d’individus au début des années 1990. Une population de l’écotype Montagnard est présente au sud-est de la province et une autre existe possiblement au nord-est. Cette dernière n’est pas bien connue. Celle du sud-est utilise principalement la forêt boréale. Cette population sédentaire a diminué considérablement depuis une centaine d’années et elle ne compte plus qu’environ 140 individus. Finalement, l’écotype Forestier est présent de façon discontinue, principalement entre les 49e et 55e parallèles. Ces Caribous sont également sédentaires. On les retrouve presque exclusivement en forêt boréale, principalement là où le cycle des feux de forêt est long. Leur répartition actuelle couvre ≈ 234 000 km2, principalement à l’est du 72e méridien. D’importantes populations forestières existaient encore durant les années 1950 et 1960, mais elles semblent avoir disparu. L’abondance actuelle n’est pas connue mais elle pourrait difficilement dépasser 3000 individus si l’on se base sur les estimations de la densité et de l’aire de répartition. Les données disponibles sont insuffisantes pour identifier les causes exactes des diminutions d’effectifs bien que la chasse semble une cause proximale importante

Winter severity modulates the benefits of using a habitat temporally uncoupled from browsing

Resources whose abundance is not affected by the density of the consumer population, namely donor-controlled resources, are ubiquitous. Donor-controlled resources can act as food subsidies when they sustain consumer populations at higher densities than what would be predicted without donorcontrolled dynamics. Herbivore populations that have access to food subsidies may reach and maintain high densities, with potential major ecological and economic consequences. A better understanding of the roles of food subsidies on temperate herbivores will likely be achieved by simultaneously taking into account other drivers of demographic variations such as winter severity. Here, we tested the hypothesis that the use of a donor-controlled food resource that may act as a food subsidy, namely balsam fir (Abies balsamea), and winter severity act together to shape the patterns of overwinter mass loss in a large herbivore population (white-tailed deer, Odocoileus virginianus). We monitored weather conditions, diet, habitat use, and mass loss of female deer during two highly contrasted winters. During an exceptionally milder winter, characterized by shallower snow depth and warmer windchill temperatures, female deer shifted their diet toward resources usually covered by snow during typical winters. Surprisingly, the rate of body mass loss remained similar during the milder and the harsher winter. The rate of body mass loss rather decreased with the use of balsam fir stands during the harsher winter, but increased with that same variable during the milder winter. Our study revealed that deer can alleviate overwinter mass loss by using a donor-controlled habitat type temporally uncoupled from browsing, but that this benefit is climate dependent. This study represents an additional step to address the largely unexplored concept of how temporal uncoupling between resources and consumer dynamics may contribute to sustain consumer populations at higher densities than predicted without considering donor-controlled dynamics

An Aerial Survey Technique for the Forest-Dwelling Ecotype of Woodland Caribou, Rangifer tarandus caribou

Accurate and precise population estimates for the forest-dwelling ecotype of Woodland Caribou (Rangifer tarandus caribou) are very difficult to obtain because these Caribou are found at very low densities and in small herds dispersed over large areas. In order to suggest a standardized method, data from aerial surveys conducted in 1991 and 1993 (12 000 km2 blocks) were used to simulate various survey scenarios. Simulations showed that all the major groups of Caribou would have to be found and counted to obtain a confidence interval of ± 20% (α = 0.10). We tested this technique in a survey carried out in winter 1999 in a 42 539 km2 study site, opting for a total coverage carried out in two phases. In phase one, we used an airplane, flying north-south transects spaced 2.1 km apart so as to detect most Caribou track networks. In phase two, a helicopter was used to count and determine the sex and age classes (calves/adults) of Caribou found in phase one. Using 20 radio-collared Caribou, the visibility rate of Caribou groups (phase one) and that of Caribou within the groups (phase two) were estimated at 0.90 and 0.94 respectively for an overall rate of 0.85 (SE = 0.08; α = 0.10). The corrected density was estimated at 1.6 Caribou per 100 km2 with a 15% confidence interval (α = 0.10). The survey cost approximately $4/km2, which is lower than that of two previous surveys ($7/km2). Two main factors contributed to diminish costs: (1) the use of long-range airplanes (5-7 hours flying range) in phase one to minimize travel between the airports and the study site, and (2) the use of helicopters only in phase two for counting and determining the age and sex of the Caribou.Il est très difficile d’obtenir des estimations de population exactes et précises pour l’écotype forestier du Caribou des bois (Rangifer tarandus caribou) parce qu’on le retrouve en très faibles densités et qu’il est distribué en petites hardes réparties sur de vastes superficies. Les résultats de deux inventaires aériens réalisés en 1991 et 1993 (12 000 km2) ont été utilisés pour simuler divers scénarios d’inventaire afin de suggérer une méthode standardisée. Les simulations ont montré qu’il fallait trouver et recenser tous les groupes principaux pour obtenir un intervalle de confiance de ± 20 % (α = 0,10). Nous avons testé cette approche dans un site d’étude de 42 539 km2 où nous avons opté pour un plan en deux phases. En phase un, l’avion a été utilisé pour couvrir totalement le site d’étude selon des virées équidistantes de 2,1 km afin de détecter la plupart des réseaux de pistes. L’hélicoptère fut utilisé en phase deux pour dénombrer et sexer les Caribous dans les réseaux de pistes détectés en phase un. D’après 20 Caribous munis de colliers émetteurs, le taux de visibilité global était de 0,85 (SE = 0,08; α = 0,10), soit 0,90 en phase 1 et 0,94 en phase 2. La densité corrigée était de 1,6 Caribou par 100 km2 avec une erreur relative de 15 % (α = 0,10). L’inventaire a coûté 4 $/km2, ce qui est inférieur aux montants investis lors des inventaires antérieurs (7$/km2). La diminution des coûts est attribuable à deux facteurs principaux : (1) l’utilisation d’avions à grand rayon d’action (5-7 heures d’autonomie) pour minimiser les déplacements en phase un; (2) l’emploi d’hélicoptères exclusivement pour le dénombrement et le sexage des caribous

A rewriting of the relation between the acolinearity of annihilation photons and their energy in the context of positron emission tomography

Acolinearity of the annihilation photons observed in Positron Emission Tomography (PET) is described as following a Gaussian distribution. However, it is never explicitly said if it refers to the amplitude of the acolinearity angle or its 2D distribution relative to the case without acolinearity (herein defined as the acolinearity deviation). Since the former is obtained by integrating the latter, a wrong interpretation would lead to very different results. The paper of Shibuya et al. (2007), differs from the previous studies since it is based on the precise measurement of the energy of the annihilation photons. They also show that acolinearity follows a Gaussian distribution in the context of PET. However, their notation, which relies on being on the plane where the two annihilation photons travel, could mean that their observation refers to the amplitude of the acolinearity angle. If that understanding is correct, it would mean that acolinearity deviation follows a 2D Gaussian distribution divided by the norm of its argument. Thus, we revisited the proof presented in Shibuya et al. (2007) by using an explicit description of the acolinearity in the 3D unit sphere

On the stopping criterion for numerical methods for linear systems with additive Gaussian noise

We consider the inversion of a linear operator with centered Gaussian white noise by MAP estimation with a Gaussian prior distribution on the solution. The actual estimator is computed approximately by a numerical method. We propose a relation between the stationarity measure of this approximate solution to the mean square error of the exact solution. This relation enables the formulation of a stopping test for the numerical method, met only by iterates that satisfy chosen statistical properties. We extend this development to Gibbs priors using a quadratic extrapolation of the log-likelihood maximized by the MAP estimator