Sherali-Adams Relaxations and Indistinguishability in Counting Logics

Two graphs with adjacency matrices $\mathbf{A}$ and $\mathbf{B}$ are isomorphic if there exists a permutation matrix $\mathbf{P}$ for which the identity $\mathbf{P}^{\mathrm{T}} \mathbf{A} \mathbf{P} = \mathbf{B}$ holds. Multiplying through by $\mathbf{P}$ and relaxing the permutation matrix to a doubly stochastic matrix leads to the linear programming relaxation known as fractional isomorphism. We show that the levels of the Sherali--Adams (SA) hierarchy of linear programming relaxations applied to fractional isomorphism interleave in power with the levels of a well-known color-refinement heuristic for graph isomorphism called the Weisfeiler--Lehman algorithm, or, equivalently, with the levels of indistinguishability in a logic with counting quantifiers and a bounded number of variables. This tight connection has quite striking consequences. For example, it follows immediately from a deep result of Grohe in the context of logics with counting quantifiers that a fixed number of levels of SA suffice to determine isomorphism of planar and minor-free graphs. We also offer applications in both finite model theory and polyhedral combinatorics. First, we show that certain properties of graphs, such as that of having a flow circulation of a prescribed value, are definable in the infinitary logic with counting with a bounded number of variables. Second, we exploit a lower bound construction due to Cai, Fürer, and Immerman in the context of counting logics to give simple explicit instances that show that the SA relaxations of the vertex-cover and cut polytopes do not reach their integer hulls for up to $\Omega(n)$ levels, where $n$ is the number of vertices in the graph.

Structural and operational optimisation : applications in energy systems

In this work mathematical programming models for structural and operational optimisation of energy systems are developed and applied to a selection of energy technology problems. The studied cases are taken from industrial processes and from large regional energy distribution systems. The models are based on Mixed Integer Linear Programming (MILP), Mixed Integer Non-Linear Programming (MINLP) and on a hybrid approach of a combination of Non-Linear Programming (NLP) and Genetic Algorithms (GA). The optimisation of the structure and operation of energy systems in urban regions is treated in the work. Firstly, distributed energy systems (DES) with different energy conversion units and annual variations of consumer heating and electricity demands are considered. Secondly, district cooling systems (DCS) with cooling demands for a large number of consumers are studied, with respect to a long term planning perspective regarding to given predictions of the consumer cooling demand development in a region. The work comprises also the development of applications for heat recovery systems (HRS), where paper machine dryer section HRS is taken as an illustrative example. The heat sources in these systems are moist air streams. Models are developed for different types of equipment price functions. The approach is based on partitioning of the overall temperature range of the system into a number of temperature intervals in order to take into account the strong nonlinearities due to condensation in the heat recovery exchangers. The influence of parameter variations on the solutions of heat recovery systems is analysed firstly by varying cost factors and secondly by varying process parameters. Point-optimal solutions by a fixed parameter approach are compared to robust solutions with given parameter variation ranges. In the work enhanced utilisation of excess heat in heat recovery systems with impingement drying, electricity generation with low grade excess heat and the use of absorption heat transformers to elevate a stream temperature above the excess heat temperature are also studied.

Lagrangian-informed mixed integer programming reformulations

La programmation linéaire en nombres entiers est une approche robuste qui permet de résoudre rapidement de grandes instances de problèmes d'optimisation discrète. Toutefois, les problèmes gagnent constamment en complexité et imposent parfois de fortes limites sur le temps de calcul. Il devient alors nécessaire de développer des méthodes spécialisées afin de résoudre approximativement ces problèmes, tout en calculant des bornes sur leurs valeurs optimales afin de prouver la qualité des solutions obtenues. Nous proposons d'explorer une approche de reformulation en nombres entiers guidée par la relaxation lagrangienne. Après l'identification d'une forte relaxation lagrangienne, un processus systématique permet d'obtenir une seconde formulation en nombres entiers. Cette reformulation, plus compacte que celle de Dantzig et Wolfe, comporte exactement les mêmes solutions entières que la formulation initiale, mais en améliore la borne linéaire: elle devient égale à la borne lagrangienne. L'approche de reformulation permet d'unifier et de généraliser des formulations et des méthodes de borne connues. De plus, elle offre une manière simple d'obtenir des reformulations de moins grandes tailles en contrepartie de bornes plus faibles. Ces reformulations demeurent de grandes tailles. C'est pourquoi nous décrivons aussi des méthodes spécialisées pour en résoudre les relaxations linéaires. Finalement, nous appliquons l'approche de reformulation à deux problèmes de localisation. Cela nous mène à de nouvelles formulations pour ces problèmes; certaines sont de très grandes tailles, mais nos méthodes de résolution spécialisées les rendent pratiques.

Domain/Multi-Domain Protection and Provisioning in Optical Networks

L’évolution récente des commutateurs de sélection de longueurs d’onde (WSS -Wavelength Selective Switch) favorise le développement du multiplexeur optique d’insertionextraction reconfigurable (ROADM - Reconfigurable Optical Add/Drop Multiplexers) à plusieurs degrés sans orientation ni coloration, considéré comme un équipement fort prometteur pour les réseaux maillés du futur relativement au multiplexage en longueur d’onde (WDM -Wavelength Division Multiplexing ). Cependant, leur propriété de commutation asymétrique complique la question de l’acheminement et de l’attribution des longueur d’ondes (RWA - Routing andWavelength Assignment). Or la plupart des algorithmes de RWA existants ne tiennent pas compte de cette propriété d’asymétrie. L’interruption des services causée par des défauts d’équipements sur les chemins optiques (résultat provenant de la résolution du problème RWA) a pour conséquence la perte d’une grande quantité de données. Les recherches deviennent ainsi incontournables afin d’assurer la survie fonctionnelle des réseaux optiques, à savoir, le maintien des services, en particulier en cas de pannes d’équipement. La plupart des publications antérieures portaient particulièrement sur l’utilisation d’un système de protection permettant de garantir le reroutage du trafic en cas d’un défaut d’un lien. Cependant, la conception de la protection contre le défaut d’un lien ne s’avère pas toujours suffisante en termes de survie des réseaux WDM à partir de nombreux cas des autres types de pannes devenant courant de nos jours, tels que les bris d’équipements, les pannes de deux ou trois liens, etc. En outre, il y a des défis considérables pour protéger les grands réseaux optiques multidomaines composés de réseaux associés à un domaine simple, interconnectés par des liens interdomaines, où les détails topologiques internes d’un domaine ne sont généralement pas partagés à l’extérieur. La présente thèse a pour objectif de proposer des modèles d’optimisation de grande taille et des solutions aux problèmes mentionnés ci-dessus. Ces modèles-ci permettent de générer des solutions optimales ou quasi-optimales avec des écarts d’optimalité mathématiquement prouvée. Pour ce faire, nous avons recours à la technique de génération de colonnes afin de résoudre les problèmes inhérents à la programmation linéaire de grande envergure. Concernant la question de l’approvisionnement dans les réseaux optiques, nous proposons un nouveau modèle de programmation linéaire en nombres entiers (ILP - Integer Linear Programming) au problème RWA afin de maximiser le nombre de requêtes acceptées (GoS - Grade of Service). Le modèle résultant constitue celui de l’optimisation d’un ILP de grande taille, ce qui permet d’obtenir la solution exacte des instances RWA assez grandes, en supposant que tous les noeuds soient asymétriques et accompagnés d’une matrice de connectivité de commutation donnée. Ensuite, nous modifions le modèle et proposons une solution au problème RWA afin de trouver la meilleure matrice de commutation pour un nombre donné de ports et de connexions de commutation, tout en satisfaisant/maximisant la qualité d’écoulement du trafic GoS. Relativement à la protection des réseaux d’un domaine simple, nous proposons des solutions favorisant la protection contre les pannes multiples. En effet, nous développons la protection d’un réseau d’un domaine simple contre des pannes multiples, en utilisant les p-cycles de protection avec un chemin indépendant des pannes (FIPP - Failure Independent Path Protecting) et de la protection avec un chemin dépendant des pannes (FDPP - Failure Dependent Path-Protecting). Nous proposons ensuite une nouvelle formulation en termes de modèles de flots pour les p-cycles FDPP soumis à des pannes multiples. Le nouveau modèle soulève un problème de taille, qui a un nombre exponentiel de contraintes en raison de certaines contraintes d’élimination de sous-tour. Par conséquent, afin de résoudre efficacement ce problème, on examine : (i) une décomposition hiérarchique du problème auxiliaire dans le modèle de décomposition, (ii) des heuristiques pour gérer efficacement le grand nombre de contraintes. À propos de la protection dans les réseaux multidomaines, nous proposons des systèmes de protection contre les pannes d’un lien. Tout d’abord, un modèle d’optimisation est proposé pour un système de protection centralisée, en supposant que la gestion du réseau soit au courant de tous les détails des topologies physiques des domaines. Nous proposons ensuite un modèle distribué de l’optimisation de la protection dans les réseaux optiques multidomaines, une formulation beaucoup plus réaliste car elle est basée sur l’hypothèse d’une gestion de réseau distribué. Ensuite, nous ajoutons une bande pasiv sante partagée afin de réduire le coût de la protection. Plus précisément, la bande passante de chaque lien intra-domaine est partagée entre les p-cycles FIPP et les p-cycles dans une première étude, puis entre les chemins pour lien/chemin de protection dans une deuxième étude. Enfin, nous recommandons des stratégies parallèles aux solutions de grands réseaux optiques multidomaines. Les résultats de l’étude permettent d’élaborer une conception efficace d’un système de protection pour un très large réseau multidomaine (45 domaines), le plus large examiné dans la littérature, avec un système à la fois centralisé et distribué.

Optimal Methodology for Synchronized Scheduling of Parallel Station Assembly with Air Transportation

We present an optimal methodology for synchronized scheduling of production assembly with air transportation to achieve accurate delivery with minimized cost in consumer electronics supply chain (CESC). This problem was motivated by a major PC manufacturer in consumer electronics industry, where it is required to schedule the delivery requirements to meet the customer needs in different parts of South East Asia. The overall problem is decomposed into two sub-problems which consist of an air transportation allocation problem and an assembly scheduling problem. The air transportation allocation problem is formulated as a Linear Programming Problem with earliness tardiness penalties for job orders. For the assembly scheduling problem, it is basically required to sequence the job orders on the assembly stations to minimize their waiting times before they are shipped by flights to their destinations. Hence the second sub-problem is modelled as a scheduling problem with earliness penalties. The earliness penalties are assumed to be independent of the job orders.

Power allocation strategies for distributed space-time codes in two-way relay networks

We study a two-way relay network (TWRN), where distributed space-time codes are constructed across multiple relay terminals in an amplify-and-forward mode. Each relay transmits a scaled linear combination of its received symbols and their conjugates,with the scaling factor chosen based on automatic gain control. We consider equal power allocation (EPA) across the relays, as well as the optimal power allocation (OPA) strategy given access to instantaneous channel state information (CSI). For EPA, we derive an upper bound on the pairwise-error-probability (PEP), from which we prove that full diversity is achieved in TWRNs. This result is in contrast to one-way relay networks, in which case a maximum diversity order of only unity can be obtained. When instantaneous CSI is available at the relays, we show that the OPA which minimizes the conditional PEP of the worse link can be cast as a generalized linear fractional program, which can be solved efficiently using the Dinkelback-type procedure.We also prove that, if the sum-power of the relay terminals is constrained, then the OPA will activate at most two relays.

Optimised placement of control and protective devices in electric distribution systems through reactive tabu search algorithm

Optimised placement of control and protective devices in distribution networks allows for a better operation and improvement of the reliability indices of the system. Control devices (used to reconfigure the feeders) are placed in distribution networks to obtain an optimal operation strategy to facilitate power supply restoration in the case of a contingency. Protective devices (used to isolate faults) are placed in distribution systems to improve the reliability and continuity of the power supply, significantly reducing the impacts that a fault can have in terms of customer outages, and the time needed for fault location and system restoration. This paper presents a novel technique to optimally place both control and protective devices in the same optimisation process on radial distribution feeders. The problem is modelled through mixed integer non-linear programming (MINLP) with real and binary variables. The reactive tabu search algorithm (RTS) is proposed to solve this problem. Results and optimised strategies for placing control and protective devices considering a practical feeder are presented. (c) 2007 Elsevier B.V. All rights reserved.

Planning and Projects of Secondary Electric Power Distribution Systems

This paper proposes a methodology to achieve integrated planning and projects for secondary distribution circuits. The planning model is formulated as a mixed integer nonlinear programming problem (MINLP). In order to resolve this problem, a tabu search (TS) algorithm is used, with a neighborhood structure developed to explore the physical characteristics of specific geographies included in the planning and expansion of secondary networks, thus obtaining effective solutions as well as low operating costs and investments. The project stage of secondary circuits consists of calculating the mechanical efforts to determine the support structures of the primary and secondary distribution systems and determining the types of structures that should be used in the system according to topological and electrical parameters of the network and, therefore, accurately assessing the costs involved in the construction and/or reform of secondary systems. A constructive heuristic based on information of the electrical and topological conditions between the medium voltage and low voltage systems is used to connect the primary systems and secondary circuits. The results obtained from planning and design simulations of a real secondary system of electric energy distribution are presented.

A metaheuristic to solve the transmission expansion planning

In this letter, a genetic algorithm (GA) is applied to solve - the static and multistage transmission expansion planning (TEP) problem. The characteristics of the proposed GA to solve the TEP problem are presented. Results using some known systems show that the proposed GA solves a smaller number of linear programming problems in order to find the optimal solutions and obtains a better solution for the multistage TEP problem.

Linear relationship between potential and concentration in flow potentiometry

In potentiometric-flow systems, linear-potential responses for logarithmic concentrations can be attained for first-(or pseudo-first-) order reactions in which the monitored chemical species react with the analyte during a fixed time interval. To demonstrate this property, the determination of glycerol based on its oxidation by periodate and potentiometric monitoring of the remaining periodate was selected. Influence of reagent concentration and timing on the linearity of the analytical curve were investigated. A mathematical treatment was derived, and potentialities/limitations of the approach were outlined. The system was applied to analysis of soap and lixivia samples. The analytical curve within 200 and 2000 mg L-1 (r = 0.99975; n = 5) was described as E = 8.166 + 0.0478 (glycerol). The sample throughput was 100 h(-1), and a measurement repeatability within 0.5 mV was always observed. By applying a t-test, there was no statistical difference between the results obtained by the proposed procedure and by iodimetric titration at the 95% confidence level. (C) 2000 John Wiley & Sons, Inc. Lab Robotics and Automation 12:41-45, 2000.

TIME-OPTIMAL GEOMAGNETIC ATTITUDE MANEUVERS OF AN AXISYMMETRICAL SPINNING SATELLITE

A formulation used to determine the time-optimal geomagnetic attitude maneuvers subject to dynamic and geometric constraints is proposed in this paper. This was obtained by a direct search procedure based on a control function parametrization method, using linear programming to obtain numerical suboptimal solutions by linear perturbation. Due to its characteristics it can be used in small computers and to generate computer programs of general application. The dynamic modeling, the magnetic torque model and the suboptimal control procedure are presented. Simulation runs have verified the feasibility of the formulation thus derived and have shown a notable improvement in performance.

Optimal Contract Pricing of Distributed Generation in Distribution Networks

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Optimized Allocation of Control and Protective Devices in Electric Distribution Systems

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Defects location in power systems components through a dedicated genetic algorithm

In this work the problem of defects location in power systems is formulated through a binary linear programming (BLP) model based on alarms historical database of control and protection devices from the system control center, sets theory of minimal coverage (AI) and protection philosophy adopted by the electric utility. In this model, circuit breaker operations are compared to their expected states in a strictly mathematical manner. For solving this BLP problem, which presents a great number of decision variables, a dedicated Genetic Algorithm (GA), is proposed. Control parameters of the GA, such as crossing over and mutation rates, population size, iterations number and population diversification, are calibrated in order to obtain efficiency and robustness. Results for a test system found in literature, are presented and discussed. © 2004 IEEE.

Influence of the supply of reactive power in the calculation of the transfer capability

This paper presents some initial concepts for including reactive power in linear methods for computing Available Transfer Capability (ATC). It is proposed an approximation for the reactive power flows computation that uses the exact circle equations for the transmission line complex flow, and then it is determined the ATC using active power distribution factors. The transfer capability can be increased using the sensitivities of flow that show the best group of buses which can have their reactive power injection modified in order to remove the overload in the transmission lines. The results of the ATC computation and of the use of the sensitivities of flow are presented using the Cigré 32-bus system. © 2004 IEEE.