Predicting Accurate Lagrangian Multipliers for Mixed Integer Linear Programs

Lagrangian relaxation stands among the most efficient approaches for solving a Mixed Integer Linear Programs (MILP) with difficult constraints. Given any duals for these constraints, called Lagrangian Multipliers (LMs), it returns a bound on the optimal value of the MILP, and Lagrangian methods seek the LMs giving the best such bound. But these methods generally rely on iterative algorithms resembling gradient descent to maximize the concave piecewise linear dual function: the computational burden grows quickly with the number of relaxed constraints. We introduce a deep learning approach that bypasses the descent, effectively amortizing the local, per instance, optimization. A probabilistic encoder based on a graph convolutional network computes high-dimensional representations of relaxed constraints in MILP instances. A decoder then turns these representations into LMs. We train the encoder and decoder jointly by directly optimizing the bound obtained from the predicted multipliers. Numerical experiments show that our approach closes up to 85~\% of the gap between the continuous relaxation and the best Lagrangian bound, and provides a high quality warm-start for descent based Lagrangian methods

Frailty and comorbidity burden in Atrial Fibrillation

Background: With the aging of the population, the characterization of frailty and comorbidity burden is increasingly taking on particular importance. The aims of the present study are to analyze such conditions in a population affected by Atrial Fibrillation (AF), matching it with a population without AF, and to recognize potential independent factors associated with such common cardiovascular disease. Methods: This study included subjects consecutively evaluated over 5 years at the Geriatric Outpatient Service, University Hospital of Monserrato, Cagliari, Italy. A sum of 1981 subjects met the inclusion criteria. The AF-group was made up of 330 people, and another 330 people were randomly selected to made up the non-AF-group. The sample was subjected to Comprehensive Geriatric Assessment (CGA). Results: In our sample, severe comorbidity burden (p = 0.01) and frailty status (p = 0.04) were significantly more common in patients with AF than without AF, independently on gender and age. Furthermore, the 5-years follow-up demonstrated that survival probability was significantly higher in AF-group (p = 0.03). The multivariate analysis (AUC: 0.808) showed that the presence of AF was independently positively associated with a history of coronary heart disease (OR: 2.12) and cerebrovascular disease (OR: 1.64), with the assumption of Beta Blockers (OR: 3.39), and with the number of drugs taken (OR: 1.12), and negatively associated with the assumption of antiplatelets (OR: 0.09). Conclusions: Elderly people with AF are frailer, have more severe comorbidities, and take more drugs, in particular beta blockers, than people without AF, who conversely have a higher survival probability. Furthermore, it is necessary to pay attention to antiplatelets, especially in AF-group, in order to avoid dangerous under- or over-prescriptions

A mean field analysis of two-layers neural networks with general convex loss function

Nowadays neural networks are a powerful tool, even if there are few mathematical results that explain the effectiveness of this approach.Until a few years ago, one of the powerful results guaranteed that any continuous function can be well approximated by a two-layers neural network with convex activation functions and enough hidden nodes.However this tells us nothing about the practical choice of the parameters.Typically the Stochastic Gradient Descent (SGD), or one of its variants, is used to update them.In the last years several results have been discovered in order to analyse the convergence of parameters using the SGD, in particular using the mean field approach.The key idea is to consider a risk function defined over a set of distributions of the parameters. This allows us to study the convergence through a PDE, known as distributional dynamics (DD), using common tools of mathematical analysis. Many results use a quadratic loss function, thus optimize the mean square error. In this works we extend this analysis for a general convex loss function.This generalization is fundamental, because the success of a learning problem can be enhanced by the choice of the most suitable loss function.We start by proving that the empirical distributions weakly converge to the solution of the DD for any finite time. Then we analyse the time convergence of the distributions, finding that, under suitable assumptions, the continuous distributions weakly converge to a fixed point of the DD

Semi-Amortized Models for Lagrangian Relaxation

International audienceWe present machine learning techniques to predict parameters of Lagrangian Relaxation. The solutions of these methods can be used either as approximations of the solutions returned by iterative algorithms such as subgradient descent and bundle method, or as informed starting points for such algorithms, saving many iterations. We evaluate our proposition on instances of the Multi-Commodity Fixed-Charge Network Design Problem and show the merits of our method. Lagrangian Relaxation (LR) [1] is a well known method in optimization that allows to relax some hard constraints and thus reduce the complexity of the problem while providing, in many cases, a bound to the original problem better than the one computed by the continuous relaxation. Relaxed constraints are put into the objective function as soft penalizations whose coefficients are called the Lagrangian multipliers. The goal of LR is to find optimal coefficients, i.e. coefficients that provide the optimal bound. Unfortunately, most algorithms to solve LR are based on (sub-)gradient descent and are thus slow to converge in practice. Machine Learning and Semi-Amortization In this presentation, we show that we can leverage machine learning techniques based on neural networks to predict Lagrangian Multipliers framed as Supervised and Unsupervised Learning. These predictions can be used in lieu of the Multipliers obtained by iterative algorithms. One of the key advantages is speed, where instead of dozens of steps of gradient descent a single forward pass over the network is required for each subproblem (which can well be parallelized on GPU). Unfortunately, learning accurate Multipliers is difficult and the predictions can be far from the optimal multipliers. Hence, we propose to use predictions as starting points for the exact iterative algorithms such as Volume or Bundle methods [2]. Hence our method appertains to the family of Semi-Amortized learning methods [3], that improve their prediction at inference time

Semi-Amortized Models for Lagrangian Relaxation

International audienceWe present machine learning techniques to predict parameters of Lagrangian Relaxation. The solutions of these methods can be used either as approximations of the solutions returned by iterative algorithms such as subgradient descent and bundle method, or as informed starting points for such algorithms, saving many iterations. We evaluate our proposition on instances of the Multi-Commodity Fixed-Charge Network Design Problem and show the merits of our method. Lagrangian Relaxation (LR) [1] is a well known method in optimization that allows to relax some hard constraints and thus reduce the complexity of the problem while providing, in many cases, a bound to the original problem better than the one computed by the continuous relaxation. Relaxed constraints are put into the objective function as soft penalizations whose coefficients are called the Lagrangian multipliers. The goal of LR is to find optimal coefficients, i.e. coefficients that provide the optimal bound. Unfortunately, most algorithms to solve LR are based on (sub-)gradient descent and are thus slow to converge in practice. Machine Learning and Semi-Amortization In this presentation, we show that we can leverage machine learning techniques based on neural networks to predict Lagrangian Multipliers framed as Supervised and Unsupervised Learning. These predictions can be used in lieu of the Multipliers obtained by iterative algorithms. One of the key advantages is speed, where instead of dozens of steps of gradient descent a single forward pass over the network is required for each subproblem (which can well be parallelized on GPU). Unfortunately, learning accurate Multipliers is difficult and the predictions can be far from the optimal multipliers. Hence, we propose to use predictions as starting points for the exact iterative algorithms such as Volume or Bundle methods [2]. Hence our method appertains to the family of Semi-Amortized learning methods [3], that improve their prediction at inference time

ECHO COLOR DOPPLER STUDY DEMONSTRATING AN HIGHER INCIDENCE OF LEFT VS RIGHT RENAL ARTERY STENOSIS IN OLD PEOPLE

INTRODUCTION AND AIMS: Renal vascular pathology is a topic of rising clinical interest and growing therapeutic application (atherosclerosis prevention, percutaneous transluminal renal artery angioplasty and stenting). The aim of this study was to confirm our previous scintigraphic data suggesting a higher incidence of left Renal Artery Stenosis (RAS) Vs the right one. METHODS: 341 nephropathic or hypertensive elderly patients (pts) (219 males, 122 females) mean age 71.7 ± 15.5 (years ± SD) were studied with B mode ultrasound (US) and color Doppler (CD). Renal function was evaluated by determining serum creatinine and glomerular filtration rate (eGFR), calculated using the CKD-EPI formula. The kidney size has been measured by m mode ultrasound. The presence of stenosis has been defined as estimated lumen reduction >50%. RESULTS: Mean serum creatinine levels 1.8 ± 0.9 mg/dl, eGFR 44.1 ± 25.7 ml/min, due to hypertensive or parenchymal renal disease Right Kidney (RK) Longitudinal Diameter (LD) 103.2 ± 12.4 mm (Normal Value -NV)-: 109 ± 12 mm) and Left Kidney (LK) LD (103 ± 12.9 mm - NV: 112 ± 12 mm); that’s to say there is a slight inversion in anatomic size between the kidneys, with a left kidney shrinking for ageing more pronounced in our whole group of nephropathic people. Over 65 years the LK becomes clearly smaller than RK (101.2 ± 13.2 Vs 102.3 ± 12.4 mm) kidney cortical thickness (NV: 16-18 mm) resulted 14.2 ± 3.3 mm on the RK Vs of 13.9 ± 3.3 mm of the LK, confirming a more evident parenchymal loss on this side 100 pts (29%) presented renal artery stenosis with lumen reduction >50%: (in these, 78% interested a single renal artery versus 22% involved both renal arteries) In the subgroups with a single RAS, 31% (24) interested the RKartery and 69% (54) the LK artery, p <0.001 ). Pts with single or bilateral RAS showed more evident reduction in renal function (39.2+21.61) in comparison to pts without any stenosis (46.3 ± 27.04, p = 0.029). We didn’t find any statistically significant functional difference, between pts with single or bilateral stenosis (eGFR 38.5 ± 20.40 and 41.6 ± 25.8 ml/ min respectively p: ns) Index Resistance (IR) presented symmetrically in nephroangiosclerotic pts without any lateral unbalancement in pts without critical RAS. CONCLUSIONS: Our study confirms a higher susceptibility of the left renal artery to develop atherosclerotic stenosis in the elderly. This probably derives from anatomic reasons (first of all the narrower emergency angle) inducing blood flow turbulence and atheroslerosis progression. Peripheral Nephrosclerosis, on the contrary, results a symmetric disease, probably for a predominance of metabolic over reologic causes in atherosclerosis development. The LK seems hence more susceptible to aging progressively loosing both size and function. In over 65 people the RK becomes the predominant one in size and functional performance

La vie postmédiévale des artéfacts médiévaux

Dans la perspective épistémologique des études sur la culture matérielle, le présent numéro de Perspectives médiévales est consacré aux artéfacts médiévaux, et à leur détermination par des actions, des textes ou des dispositifs qui s’en sont emparés après le Moyen Âge pour prolonger ou inventer leur existence, et leur donner une « vie ». Manuscrits et imprimés, objets sacrés et profanes, vêtements et équipements, bâtiments et fresques, machines, instruments et outils : ces cristallisations matérielles de l’époque médiévale ont été étudiées dans une vaste diversité formelle et technique, qu’elles aient été effectivement fabriquées au Moyen Âge, ou qu’elles soient présentées ou interprétées comme telles. L’approche diachronique a été privilégiée, pour mettre en évidence les valeurs d’usage de ces artefacts selon les époques et les acteurs qui s’en sont emparés. Le numéro s’attache à décrire la variété des opérations effectuées sur ces objets, et à en interroger les visées, entre anachronisme, propagande, et manipulation scientifique, et entre charge subjective et valeurs culturelles ou intellectuelles partagées. Des artéfacts en mouvance : on a envisagé la mobilité des objets dans l’espace et dans le temps ainsi que les variations de leur réception, en déclinant la notion selon quatre axes. Restaurer, refaire, contrefaire : la mouvance matérielle réfléchit à la réfection moderne des objets médiévaux, au statut variable, entre faux véritables et contrefaçons assumées. Collectionner, acheter, déplacer : la mouvance spatiale soumet à l’examen les parcours de collectionneurs, dont les politiques d’achat ont façonné une représentation du Moyen Âge appuyée sur les ensembles d’objets voyageurs dont ils ont assuré le déplacement et l’implantation, de l’Europe aux États-Unis ou du continent à l’Angleterre. Avec la mouvance herméneutique, c’est l’usage par le présent des objets du passé médiéval pour écrire l’histoire actuelle ou penser son identité qui est analysé. Enfin, la mouvance discursive met au jour l’usage des artefacts médiévaux pour définir cette fois le Moyen Âge comme période historique, méprisée ou adulée, en contraste ou en continuité avec l’Antiquité ou l’art paléochrétien. L’ambition de ce numéro est ainsi de mieux saisir en quoi et comment la réception postmédiévale des artéfacts médiévaux a façonné et conditionné notre vision actuelle du Moyen Âge, à partir de l’usage moderne ou contemporain des objets médiévaux ou dits tels