Degenerate flag varieties of type A and C are Schubert varieties

We show that in type A or C any degenerate flag variety is in fact isomorphic to a Schubert variety in an appropriate partial flag manifold.Comment: The new version includes an appendix where we discuss desingularizations. 14 page

Degenerate flag varieties and Schubert varieties: a characteristic free approach

We consider the PBW filtrations over the integers of the irreducible highest weight modules in type A and C. We show that the associated graded modules can be realized as Demazure modules for group schemes of the same type and doubled rank. We deduce that the corresponding degenerate flag varieties are isomorphic to Schubert varieties in any characteristic.Comment: 23 pages; A few typos corrected; Authors affiliation adde

A bilevel approach for compensation and routing decisions in last-mile delivery

In last-mile delivery logistics, peer-to-peer logistic platforms play an important role in connecting senders, customers, and independent carriers to fulfill delivery requests. Since the carriers are not under the platform's control, the platform has to anticipate their reactions, while deciding how to allocate the delivery operations. Indeed, carriers' decisions largely affect the platform's revenue. In this paper, we model this problem using bilevel programming. At the upper level, the platform decides how to assign the orders to the carriers; at the lower level, each carrier solves a profitable tour problem to determine which offered requests to accept, based on her own profit maximization. Possibly, the platform can influence carriers' decisions by determining also the compensation paid for each accepted request. The two considered settings result in two different formulations: the bilevel profitable tour problem with fixed compensation margins and with margin decisions, respectively. For each of them, we propose single-level reformulations and alternative formulations where the lower-level routing variables are projected out. A branch-and-cut algorithm is proposed to solve the bilevel models, with a tailored warm-start heuristic used to speed up the solution process. Extensive computational tests are performed to compare the proposed formulations and analyze solution characteristics

Improving ADMMs for Solving Doubly Nonnegative Programs through Dual Factorization

Alternating direction methods of multipliers (ADMMs) are popular approaches to handle large scale semidefinite programs that gained attention during the past decade. In this paper, we focus on solving doubly nonnegative programs (DNN), which are semidefinite programs where the elements of the matrix variable are constrained to be nonnegative. Starting from two algorithms already proposed in the literature on conic programming, we introduce two new ADMMs by employing a factorization of the dual variable. It is well known that first order methods are not suitable to compute high precision optimal solutions, however an optimal solution of moderate precision often suffices to get high quality lower bounds on the primal optimal objective function value. We present methods to obtain such bounds by either perturbing the dual objective function value or by constructing a dual feasible solution from a dual approximate optimal solution. Both procedures can be used as a post-processing phase in our ADMMs. Numerical results for DNNs that are relaxations of the stable set problem are presented. They show the impact of using the factorization of the dual variable in order to improve the progress towards the optimal solution within an iteration of the ADMM. This decreases the number of iterations as well as the CPU time to solve the DNN to a given precision. The experiments also demonstrate that within a computationally cheap post-processing, we can compute bounds that are close to the optimal value even if the DNN was solved to moderate precision only. This makes ADMMs applicable also within a branch-and-bound algorithm.Comment: 24 pages, 8 figure

Mathematical Programming Formulations for the Collapsed k-Core Problem

In social network analysis, the size of the k-core, i.e., the maximal induced subgraph of the network with minimum degree at least k, is frequently adopted as a typical metric to evaluate the cohesiveness of a community. We address the Collapsed k-Core Problem, which seeks to find a subset of $b$ users, namely the most critical users of the network, the removal of which results in the smallest possible k-core. For the first time, both the problem of finding the k-core of a network and the Collapsed k-Core Problem are formulated using mathematical programming. On the one hand, we model the Collapsed k-Core Problem as a natural deletion-round-indexed Integer Linear formulation. On the other hand, we provide two bilevel programs for the problem, which differ in the way in which the k-core identification problem is formulated at the lower level. The first bilevel formulation is reformulated as a single-level sparse model, exploiting a Benders-like decomposition approach. To derive the second bilevel model, we provide a linear formulation for finding the k-core and use it to state the lower-level problem. We then dualize the lower level and obtain a compact Mixed-Integer Nonlinear single-level problem reformulation. We additionally derive a combinatorial lower bound on the value of the optimal solution and describe some pre-processing procedures and valid inequalities for the three formulations. The performance of the proposed formulations is compared on a set of benchmarking instances with the existing state-of-the-art solver for mixed-integer bilevel problems proposed in (Fischetti et al., A New General-Purpose Algorithm for Mixed-Integer Bilevel Linear Programs, Operations Research 65(6), 2017)

factors underlying the development of chronic temporal lobe epilepsy in autoimmune encephalitis

Abstract Purpose Limbic encephalitis (LE) is an autoimmune condition characterized by amnestic syndrome, psychiatric features and seizures. Early diagnosis and prompt treatment are crucial to avoid long-term sequelae, including psycho-cognitive deficits and persisting seizures. The aim of our study was to analyze the characteristics of 33 LE patients in order to identify possible prognostic factors associated with the development of chronic epilepsy. Methods This is a retrospective cohort study including adult patients diagnosed with LE in the period 2010–2017 and followed up for ≥12 months. Demographics, seizure semiology, EEG pattern, MRI features, CSF/serum findings were reviewed. Results All 33 LE patients (19 M/14F, mean age 61.2 years) presented seizures. Thirty subjects had memory deficits; 22 presented behavioural/mood disorders. Serum and/or CSF auto-antibodies were detected in 12 patients. In 31 subjects brain MRI at onset showed typical alterations involving temporal lobes. All patients received immunotherapy. At follow-up, 13/33 had developed chronic epilepsy; predisposing factors included delay in diagnosis (p = .009), low seizure frequency at onset (p = .02), absence of amnestic syndrome (p = .02) and absence/rarity of inter-ictal epileptic discharges on EEG (p = .06). Conclusions LE with paucisymptomatic electro-clinical presentation seemed to be associated to chronic epilepsy more than LE presenting with definite and severe "limbic syndrome"

Brivaracetam as Early Add-On Treatment in Patients with Focal Seizures: A Retrospective, Multicenter, Real-World Study

Introduction In randomized controlled trials, add-on brivaracetam (BRV) reduced seizure frequency in patients with drug-resistant focal epilepsy. Most real-world research on BRV has focused on refractory epilepsy. The aim of this analysis was to assess the 12-month effectiveness and tolerability of adjunctive BRV when used as early or late adjunctive treatment in patients included in the BRIVAracetam add-on First Italian netwoRk Study (BRIVAFIRST). Methods BRIVAFIRST was a 12-month retrospective, multicenter study including adult patients prescribed adjunctive BRV. Effectiveness outcomes included the rates of sustained seizure response, sustained seizure freedom, and treatment discontinuation. Safety and tolerability outcomes included the rate of treatment discontinuation due to adverse events (AEs) and the incidence of AEs. Data were compared for patients treated with add-on BRV after 1-2 (early add-on) and >= 3 (late add-on) prior antiseizure medications. Results A total of 1029 patients with focal epilepsy were included in the study, of whom 176 (17.1%) received BRV as early add-on treatment. The median daily dose of BRV at 12 months was 125 (100-200) mg in the early add-on group and 200 (100-200) in the late add-on group (p < 0.001). Sustained seizure response was reached by 97/161 (60.3%) of patients in the early add-on group and 286/833 (34.3%) of patients in the late add-on group (p < 0.001). Sustained seizure freedom was achieved by 51/161 (31.7%) of patients in the early add-on group and 91/833 (10.9%) of patients in the late add-on group (p < 0.001). During the 1-year study period, 29 (16.5%) patients in the early add-on group and 241 (28.3%) in the late add-on group discontinued BRV (p = 0.001). Adverse events were reported by 38.7% and 28.5% (p = 0.017) of patients who received BRV as early and late add-on treatment, respectively. Conclusion Brivaracetam was effective and well tolerated both as first add-on and late adjunctive treatment in patients with focal epilepsy

Optimisation biniveau et applications

A bilevel problem is an optimization problem where a subset of variables is constrained to be optimal for another given problem parameterized by the remaining variables. The outer problem is commonly referred to as the upper-level problem, the inner one as the lower-level problem. The first part of this dissertation concerns the key definitions, the solution approaches and the complexity of bilevel problems, as well as the study of a particular class of bilevel programs, having a quadratic lower level, the value of which is contained into an upper-level inequality constraint. Such bilevel problems can be obtained by reformulating semi-infinite programming problems with an infinite number of quadratically parametrized constraints. We propose an approach to solve this class of bilevel programs, based on the dualization of the lower-level. This approach is compared with a new cutting plane algorithm, which we prove to be convergent. The rate of convergence of this algorithm is derived under stricter assumptions and is directly related to the iteration index, which is something new w.r.t. what is usually proved in semi-infinite programming literature. We successfully test the two proposed methods on two applications: the constrained quadratic regression and a zero-sum game with cubic payoff.The second part of the thesis is devoted to practical applications. A chapter is dedicated to the aircraft conflict resolution problem. This problem essentially consists in enforcing a minimum distance between flying aircraft to avoid conflicts, using different strategies. We focus on two of them: speed regulations and heading angles changes. We present a natural semi-infinite formulation of the problem via speed regulation strategy in k dimensions. To deal with the issue of uncountably many constraints of this formulation, we reformulate it, firstly, using polynomial programming, and secondly, using bilevel programming. Then we also present a bilevel formulation of conflict resolution problem via heading angle changes in two dimensions (i.e. aircraft flying at the same altitude). In both bilevel formulations, the convexity of the lower levels allows us to derive three different single-level reformulations, using KKT conditions, Dorn’s duality, and Wolfe’s duality respectively. The single-level formulations of both problems are solved by using state-of-the-art solvers. Alternatively, we propose a cut generation algorithm to solve the bilevel problems, which fits in the general framework of the cutting plane algorithm presented in the first part. This algorithm obtains the best results in terms of computational time for most of the tested instances.Another application studied in this dissertation involves the Alternating Current (AC) Optimal Power Flow (ACOPF) problem at the lower level. The idea comes from the possibility for power generation in private households. In this scenario, we derive a bilevel problem to model the interaction between a retailer and several prosumers (consumers who can also produce, store and sell power), who interact with each other through an AC network. When, together with the ACOPF, one wants to optimally design a power transportation network with respect to line activity, an ACOPF with on/off variables on lines can be used, which yields a nonconvex mixed-integer nonlinear problem in complex numbers. We propose two convex relaxations, compared with the famous Jabr’s second-order cone relaxation.Un problème biniveau est un problème où un sous-ensemble des variables est contraint d'être optimal pour un autre problème paramétré par les variables restantes. Le problème externe est appelé problème de niveau supérieur, le problème interne le problème de niveau inférieur. La première partie de cette thèse concerne les définitions clés, les approches de solution et la complexité des problèmes biniveaux, et l'étude d'une classe particulière de problèmes biniveaux, ayant un niveau inférieur quadratique, dont la valeur est contenue dans une contrainte de niveau supérieur. Nous proposons une approche pour résoudre cette classe de problèmes, basée sur la dualisation du niveau inférieur. Cette approche est comparée à un algorithme de plans coupants, dont nous prouvons la convergence. La validité de ces deux approches est démontrée par les résultats de calcul sur deux applications: un jeu à somme nulle avec un gain cubique et une régression quadratique contrainte.La deuxième partie de la thèse est consacrée aux applications pratiques. Un chapitre est dédié au problème de résolution de conflits d'aéronefs (PRC). Ce problème consiste essentiellement à imposer une distance minimale entre les avions en vol pour éviter les conflits, en utilisant différentes stratégies. Nous nous concentrons sur deux d'entre eux: les régulations de vitesse et les changements d'angle de cap. Nous présentons une formulation de programmation semi-infinie du PRC via régulation de vitesse en k dimensions. Nous la reformulons d'une part en utilisant la programmation polynomiale et d'autre part en utilisant la programmation biniveau. Ensuite, nous présentons une formulation biniveau du PRC via changements d'angle de cap en deux dimensions. Dans les deux formulations biniveau, la convexité des niveaux inférieurs nous permet de proposer trois reformulations différentes à un seul niveau, en utilisant les conditions KKT, la dualité de Dorn et la dualité de Wolfe. Les reformulations à un seul niveau des deux problèmes sont résolues en utilisant des solveurs de l’état de l’art. Alternativement, nous proposons un algorithme de génération de coupes pour résoudre les problèmes biniveau, qui s'inscrit dans le cadre général de l'algorithme de plans coupants présenté dans la première partie. Cet algorithme obtient les meilleurs résultats en terme de temps pour la plupart des instances testées.Une autre application étudiée dans cette thèse concerne le Alternating Current (AC) Optimal Power Flow (ACOPF) au niveau inférieur. Dans un horizon temporel discrétisé fixe, un problème biniveau est derivé pour modéliser l'interaction entre un fournisseur et des prosommateurs (consommateurs qui peuvent également produire, stocker et vendre de l'électricité), qui interagissent entre eux via un réseau à courant alternatif. Lorsque, avec l'ACOPF, on veut concevoir de manière optimale un réseau de transport d'électricité par rapport à l'activité des lignes, un ACOPF avec des variables on/off sur les lignes peut être utilisé, en obtenant un problème non linéaire en variables mixtes non convexe en nombres complexes. Dans ce scénario, nous proposons deux relaxations convexes, comparées à la célèbre relaxation conique du second ordre de Jabr

Optimisation biniveau et applications

Un problème biniveau est un problème où un sous-ensemble des variables est contraint d'être optimal pour un autre problème paramétré par les variables restantes. Le problème externe est appelé problème de niveau supérieur, le problème interne le problème de niveau inférieur. La première partie de cette thèse concerne les définitions clés, les approches de solution et la complexité des problèmes biniveaux, et l'étude d'une classe particulière de problèmes biniveaux, ayant un niveau inférieur quadratique, dont la valeur est contenue dans une contrainte de niveau supérieur. Nous proposons une approche pour résoudre cette classe de problèmes, basée sur la dualisation du niveau inférieur. Cette approche est comparée à un algorithme de plans coupants, dont nous prouvons la convergence. La validité de ces deux approches est démontrée par les résultats de calcul sur deux applications: un jeu à somme nulle avec un gain cubique et une régression quadratique contrainte.La deuxième partie de la thèse est consacrée aux applications pratiques. Un chapitre est dédié au problème de résolution de conflits d'aéronefs (PRC). Ce problème consiste essentiellement à imposer une distance minimale entre les avions en vol pour éviter les conflits, en utilisant différentes stratégies. Nous nous concentrons sur deux d'entre eux: les régulations de vitesse et les changements d'angle de cap. Nous présentons une formulation de programmation semi-infinie du PRC via régulation de vitesse en k dimensions. Nous la reformulons d'une part en utilisant la programmation polynomiale et d'autre part en utilisant la programmation biniveau. Ensuite, nous présentons une formulation biniveau du PRC via changements d'angle de cap en deux dimensions. Dans les deux formulations biniveau, la convexité des niveaux inférieurs nous permet de proposer trois reformulations différentes à un seul niveau, en utilisant les conditions KKT, la dualité de Dorn et la dualité de Wolfe. Les reformulations à un seul niveau des deux problèmes sont résolues en utilisant des solveurs de l’état de l’art. Alternativement, nous proposons un algorithme de génération de coupes pour résoudre les problèmes biniveau, qui s'inscrit dans le cadre général de l'algorithme de plans coupants présenté dans la première partie. Cet algorithme obtient les meilleurs résultats en terme de temps pour la plupart des instances testées.Une autre application étudiée dans cette thèse concerne le Alternating Current (AC) Optimal Power Flow (ACOPF) au niveau inférieur. Dans un horizon temporel discrétisé fixe, un problème biniveau est derivé pour modéliser l'interaction entre un fournisseur et des prosommateurs (consommateurs qui peuvent également produire, stocker et vendre de l'électricité), qui interagissent entre eux via un réseau à courant alternatif. Lorsque, avec l'ACOPF, on veut concevoir de manière optimale un réseau de transport d'électricité par rapport à l'activité des lignes, un ACOPF avec des variables on/off sur les lignes peut être utilisé, en obtenant un problème non linéaire en variables mixtes non convexe en nombres complexes. Dans ce scénario, nous proposons deux relaxations convexes, comparées à la célèbre relaxation conique du second ordre de Jabr.A bilevel problem is an optimization problem where a subset of variables is constrained to be optimal for another given problem parameterized by the remaining variables. The outer problem is commonly referred to as the upper-level problem, the inner one as the lower-level problem. The first part of this dissertation concerns the key definitions, the solution approaches and the complexity of bilevel problems, as well as the study of a particular class of bilevel programs, having a quadratic lower level, the value of which is contained into an upper-level inequality constraint. Such bilevel problems can be obtained by reformulating semi-infinite programming problems with an infinite number of quadratically parametrized constraints. We propose an approach to solve this class of bilevel programs, based on the dualization of the lower-level. This approach is compared with a new cutting plane algorithm, which we prove to be convergent. The rate of convergence of this algorithm is derived under stricter assumptions and is directly related to the iteration index, which is something new w.r.t. what is usually proved in semi-infinite programming literature. We successfully test the two proposed methods on two applications: the constrained quadratic regression and a zero-sum game with cubic payoff.The second part of the thesis is devoted to practical applications. A chapter is dedicated to the aircraft conflict resolution problem. This problem essentially consists in enforcing a minimum distance between flying aircraft to avoid conflicts, using different strategies. We focus on two of them: speed regulations and heading angles changes. We present a natural semi-infinite formulation of the problem via speed regulation strategy in k dimensions. To deal with the issue of uncountably many constraints of this formulation, we reformulate it, firstly, using polynomial programming, and secondly, using bilevel programming. Then we also present a bilevel formulation of conflict resolution problem via heading angle changes in two dimensions (i.e. aircraft flying at the same altitude). In both bilevel formulations, the convexity of the lower levels allows us to derive three different single-level reformulations, using KKT conditions, Dorn’s duality, and Wolfe’s duality respectively. The single-level formulations of both problems are solved by using state-of-the-art solvers. Alternatively, we propose a cut generation algorithm to solve the bilevel problems, which fits in the general framework of the cutting plane algorithm presented in the first part. This algorithm obtains the best results in terms of computational time for most of the tested instances.Another application studied in this dissertation involves the Alternating Current (AC) Optimal Power Flow (ACOPF) problem at the lower level. The idea comes from the possibility for power generation in private households. In this scenario, we derive a bilevel problem to model the interaction between a retailer and several prosumers (consumers who can also produce, store and sell power), who interact with each other through an AC network. When, together with the ACOPF, one wants to optimally design a power transportation network with respect to line activity, an ACOPF with on/off variables on lines can be used, which yields a nonconvex mixed-integer nonlinear problem in complex numbers. We propose two convex relaxations, compared with the famous Jabr’s second-order cone relaxation

Optimisation biniveau et applications

A bilevel problem is an optimization problem where a subset of variables is constrained to be optimal for another given problem parameterized by the remaining variables. The outer problem is commonly referred to as the upper-level problem, the inner one as the lower-level problem. The first part of this dissertation concerns the key definitions, the solution approaches and the complexity of bilevel problems, as well as the study of a particular class of bilevel programs, having a quadratic lower level, the value of which is contained into an upper-level inequality constraint. Such bilevel problems can be obtained by reformulating semi-infinite programming problems with an infinite number of quadratically parametrized constraints. We propose an approach to solve this class of bilevel programs, based on the dualization of the lower-level. This approach is compared with a new cutting plane algorithm, which we prove to be convergent. The rate of convergence of this algorithm is derived under stricter assumptions and is directly related to the iteration index, which is something new w.r.t. what is usually proved in semi-infinite programming literature. We successfully test the two proposed methods on two applications: the constrained quadratic regression and a zero-sum game with cubic payoff.The second part of the thesis is devoted to practical applications. A chapter is dedicated to the aircraft conflict resolution problem. This problem essentially consists in enforcing a minimum distance between flying aircraft to avoid conflicts, using different strategies. We focus on two of them: speed regulations and heading angles changes. We present a natural semi-infinite formulation of the problem via speed regulation strategy in k dimensions. To deal with the issue of uncountably many constraints of this formulation, we reformulate it, firstly, using polynomial programming, and secondly, using bilevel programming. Then we also present a bilevel formulation of conflict resolution problem via heading angle changes in two dimensions (i.e. aircraft flying at the same altitude). In both bilevel formulations, the convexity of the lower levels allows us to derive three different single-level reformulations, using KKT conditions, Dorn’s duality, and Wolfe’s duality respectively. The single-level formulations of both problems are solved by using state-of-the-art solvers. Alternatively, we propose a cut generation algorithm to solve the bilevel problems, which fits in the general framework of the cutting plane algorithm presented in the first part. This algorithm obtains the best results in terms of computational time for most of the tested instances.Another application studied in this dissertation involves the Alternating Current (AC) Optimal Power Flow (ACOPF) problem at the lower level. The idea comes from the possibility for power generation in private households. In this scenario, we derive a bilevel problem to model the interaction between a retailer and several prosumers (consumers who can also produce, store and sell power), who interact with each other through an AC network. When, together with the ACOPF, one wants to optimally design a power transportation network with respect to line activity, an ACOPF with on/off variables on lines can be used, which yields a nonconvex mixed-integer nonlinear problem in complex numbers. We propose two convex relaxations, compared with the famous Jabr’s second-order cone relaxation.Un problème biniveau est un problème où un sous-ensemble des variables est contraint d'être optimal pour un autre problème paramétré par les variables restantes. Le problème externe est appelé problème de niveau supérieur, le problème interne le problème de niveau inférieur. La première partie de cette thèse concerne les définitions clés, les approches de solution et la complexité des problèmes biniveaux, et l'étude d'une classe particulière de problèmes biniveaux, ayant un niveau inférieur quadratique, dont la valeur est contenue dans une contrainte de niveau supérieur. Nous proposons une approche pour résoudre cette classe de problèmes, basée sur la dualisation du niveau inférieur. Cette approche est comparée à un algorithme de plans coupants, dont nous prouvons la convergence. La validité de ces deux approches est démontrée par les résultats de calcul sur deux applications: un jeu à somme nulle avec un gain cubique et une régression quadratique contrainte.La deuxième partie de la thèse est consacrée aux applications pratiques. Un chapitre est dédié au problème de résolution de conflits d'aéronefs (PRC). Ce problème consiste essentiellement à imposer une distance minimale entre les avions en vol pour éviter les conflits, en utilisant différentes stratégies. Nous nous concentrons sur deux d'entre eux: les régulations de vitesse et les changements d'angle de cap. Nous présentons une formulation de programmation semi-infinie du PRC via régulation de vitesse en k dimensions. Nous la reformulons d'une part en utilisant la programmation polynomiale et d'autre part en utilisant la programmation biniveau. Ensuite, nous présentons une formulation biniveau du PRC via changements d'angle de cap en deux dimensions. Dans les deux formulations biniveau, la convexité des niveaux inférieurs nous permet de proposer trois reformulations différentes à un seul niveau, en utilisant les conditions KKT, la dualité de Dorn et la dualité de Wolfe. Les reformulations à un seul niveau des deux problèmes sont résolues en utilisant des solveurs de l’état de l’art. Alternativement, nous proposons un algorithme de génération de coupes pour résoudre les problèmes biniveau, qui s'inscrit dans le cadre général de l'algorithme de plans coupants présenté dans la première partie. Cet algorithme obtient les meilleurs résultats en terme de temps pour la plupart des instances testées.Une autre application étudiée dans cette thèse concerne le Alternating Current (AC) Optimal Power Flow (ACOPF) au niveau inférieur. Dans un horizon temporel discrétisé fixe, un problème biniveau est derivé pour modéliser l'interaction entre un fournisseur et des prosommateurs (consommateurs qui peuvent également produire, stocker et vendre de l'électricité), qui interagissent entre eux via un réseau à courant alternatif. Lorsque, avec l'ACOPF, on veut concevoir de manière optimale un réseau de transport d'électricité par rapport à l'activité des lignes, un ACOPF avec des variables on/off sur les lignes peut être utilisé, en obtenant un problème non linéaire en variables mixtes non convexe en nombres complexes. Dans ce scénario, nous proposons deux relaxations convexes, comparées à la célèbre relaxation conique du second ordre de Jabr