The Proper Generalized Decomposition With Application In Problems Of Dynamics

In this paper we address the new reduction method called Proper Generalized Decomposition (PGD) which is a discretization technique based on the use of separated representation of the unknown fields, specially well suited for solving multidimensional parametric equations. In this case, it is applied to the solution of dynamics problems. We will focus on the dynamic analysis of an one-dimensional rod with a unit harmonic load of frequency (ω) applied at a point of interest. In what follows, we will present the application of the methodology PGD to the problem in order to approximate the displacement field as the sum of the separated functions. We will consider as new variables of the problem, parameters models associated with the characteristic of the materials, in addition to the frequency. Finally, the quality of the results will be assessed based on an example.

Regularity and relaxed problems of minimizing biharmonic maps into spheres

For n >= 5 and k >= 4, we show that any minimizing biharmonic map from Omega subset of R-n to S-k is smooth off a closed set whose Hausdorff dimension is at most n - 5. When n = 5 and k = 4, for a parameter lambda is an element of [0, 1] we introduce lambda-relaxed energy H-lambda of the Hessian energy for maps in W-2,W-2 (Omega; S-4) so that each minimizer u(lambda) of H-lambda is also a biharmonic map. We also establish the existence and partial regularity of a minimizer of H-lambda for lambda is an element of [0, 1).

Using min-sum loopy belief propagation for decentralised supply chain formation

Modern business trends such as agile manufacturing and virtual corporations require high levels of flexibility and responsiveness to consumer demand, and require the ability to quickly and efficiently select trading partners. Automated computational techniques for supply chain formation have the potential to provide significant advantages in terms of speed and efficiency over the traditional manual approach to partner selection. Automated supply chain formation is the process of determining the participants within a supply chain and the terms of the exchanges made between these participants. In this thesis we present an automated technique for supply chain formation based upon the min-sum loopy belief propagation algorithm (LBP). LBP is a decentralised and distributed message-passing algorithm which allows participants to share their beliefs about the optimal structure of the supply chain based upon their costs, capabilities and requirements. We propose a novel framework for the application of LBP to the existing state-of-the-art case of the decentralised supply chain formation problem, and extend this framework to allow for application to further novel and established problem cases. Specifically, the contributions made by this thesis are: • A novel framework to allow for the application of LBP to the decentralised supply chain formation scenario investigated using the current state-of-the-art approach. Our experimental analysis indicates that LBP is able to match or outperform this approach for the vast majority of problem instances tested. • A new solution goal for supply chain formation in which economically motivated producers aim to maximise their profits by intelligently altering their profit margins. We propose a rational pricing strategy that allows producers to earn significantly greater profits than a comparable LBP-based profitmaking approach. • An LBP-based framework which allows the algorithm to be used to solve supply chain formation problems in which goods are exchanged in multiple units, a first for a fully decentralised technique. As well as multiple-unit exchanges, we also model in this scenario realistic constraints such as factory capacities and input-to-output ratios. LBP continues to be able to match or outperform an extended version of the existing state-of-the-art approach in this scenario. • Introduction of a dynamic supply chain formation scenario in which participants are able to alter their properties or to enter or leave the process at any time. Our results suggest that LBP is able to deal easily with individual occurences of these alterations and that performance degrades gracefully when they occur in larger numbers.

Using the max-sum algorithm for supply chain emergence in dynamic multiunit environments

Supply chain formation (SCF) is the process of determining the set of participants and exchange relationships within a network with the goal of setting up a supply chain that meets some predefined social objective. Many proposed solutions for the SCF problem rely on centralized computation, which presents a single point of failure and can also lead to problems with scalability. Decentralized techniques that aid supply chain emergence offer a more robust and scalable approach by allowing participants to deliberate between themselves about the structure of the optimal supply chain. Current decentralized supply chain emergence mechanisms are only able to deal with simplistic scenarios in which goods are produced and traded in single units only and without taking into account production capacities or input-output ratios other than 1:1. In this paper, we demonstrate the performance of a graphical inference technique, max-sum loopy belief propagation (LBP), in a complex multiunit unit supply chain emergence scenario which models additional constraints such as production capacities and input-to-output ratios. We also provide results demonstrating the performance of LBP in dynamic environments, where the properties and composition of participants are altered as the algorithm is running. Our results suggest that max-sum LBP produces consistently strong solutions on a variety of network structures in a multiunit problem scenario, and that performance tends not to be affected by on-the-fly changes to the properties or composition of participants.

Multiple criteria optimization of contemporary logistics distribution network problems

In the contemporary customer-driven supply chain, maximization of customer service plays an equally important role as minimization of costs for a company to retain and increase its competitiveness. This article develops a multiple-criteria optimization approach, combining the analytic hierarchy process (AHP) and an integer linear programming (ILP) model, to aid the design of an optimal logistics distribution network. The proposed approach outperforms traditional cost-based optimization techniques because it considers both quantitative and qualitative factors and also aims at maximizing the benefits of deliverer and customers. In the approach, the AHP is used to determine the relative importance weightings or priorities of alternative warehouses with respect to some critical customer-oriented criteria. The results of AHP prioritization are utilized as the input of the ILP model, the objective of which is to select the best warehouses at the lowest possible cost. In this article, two commercial packages are used: including Expert Choice and LINDO.

Isomorphism Problems for the Baire Function Spaces of Topological Spaces

Let a compact Hausdorff space X contain a non-empty perfect subset. If α < β and β is a countable ordinal, then the Banach space Bα (X) of all bounded real-valued functions of Baire class α on X is a proper subspace of the Banach space Bβ (X). In this paper it is shown that: 1. Bα (X) has a representation as C(bα X), where bα X is a compactification of the space P X – the underlying set of X in the Baire topology generated by the Gδ -sets in X. 2. If 1 ≤ α < β ≤ Ω, where Ω is the first uncountable ordinal number, then Bα (X) is uncomplemented as a closed subspace of Bβ (X). These assertions for X = [0, 1] were proved by W. G. Bade [4] and in the case when X contains an uncountable compact metrizable space – by F.K.Dashiell [9]. Our argumentation is one non-metrizable modification of both Bade’s and Dashiell’s methods.

The Boundary Descriptors of the n-dimensional Unit Cube Subset Partitioning

* The research is supported partly by INTAS: 04-77-7173 project, http://www.intas.be

A Method for Solving a Class of Multiple-Criteria Analysis Problems

AMS subject classification: 90C29.

Relationship between Extremal and Sum Processes Generated by the same Point Process

2000 Mathematics Subject Classification: Primary 60G51, secondary 60G70, 60F17.

Distributed Feature Selection in Large n and Large p Regression Problems

Fitting statistical models is computationally challenging when the sample size or the dimension of the dataset is huge. An attractive approach for down-scaling the problem size is to first partition the dataset into subsets and then fit using distributed algorithms. The dataset can be partitioned either horizontally (in the sample space) or vertically (in the feature space), and the challenge arise in defining an algorithm with low communication, theoretical guarantees and excellent practical performance in general settings. For sample space partitioning, I propose a MEdian Selection Subset AGgregation Estimator ({\em message}) algorithm for solving these issues. The algorithm applies feature selection in parallel for each subset using regularized regression or Bayesian variable selection method, calculates the `median' feature inclusion index, estimates coefficients for the selected features in parallel for each subset, and then averages these estimates. The algorithm is simple, involves very minimal communication, scales efficiently in sample size, and has theoretical guarantees. I provide extensive experiments to show excellent performance in feature selection, estimation, prediction, and computation time relative to usual competitors.

While sample space partitioning is useful in handling datasets with large sample size, feature space partitioning is more effective when the data dimension is high. Existing methods for partitioning features, however, are either vulnerable to high correlations or inefficient in reducing the model dimension. In the thesis, I propose a new embarrassingly parallel framework named {\em DECO} for distributed variable selection and parameter estimation. In {\em DECO}, variables are first partitioned and allocated to m distributed workers. The decorrelated subset data within each worker are then fitted via any algorithm designed for high-dimensional problems. We show that by incorporating the decorrelation step, DECO can achieve consistent variable selection and parameter estimation on each subset with (almost) no assumptions. In addition, the convergence rate is nearly minimax optimal for both sparse and weakly sparse models and does NOT depend on the partition number m. Extensive numerical experiments are provided to illustrate the performance of the new framework.

For datasets with both large sample sizes and high dimensionality, I propose a new "divided-and-conquer" framework {\em DEME} (DECO-message) by leveraging both the {\em DECO} and the {\em message} algorithm. The new framework first partitions the dataset in the sample space into row cubes using {\em message} and then partition the feature space of the cubes using {\em DECO}. This procedure is equivalent to partitioning the original data matrix into multiple small blocks, each with a feasible size that can be stored and fitted in a computer in parallel. The results are then synthezied via the {\em DECO} and {\em message} algorithm in a reverse order to produce the final output. The whole framework is extremely scalable.

Exact Solution of Bayes and Minimax Change-Detection Problems

The challenge of detecting a change in the distribution of data is a sequential decision problem that is relevant to many engineering solutions, including quality control and machine and process monitoring. This dissertation develops techniques for exact solution of change-detection problems with discrete time and discrete observations. Change-detection problems are classified as Bayes or minimax based on the availability of information on the change-time distribution. A Bayes optimal solution uses prior information about the distribution of the change time to minimize the expected cost, whereas a minimax optimal solution minimizes the cost under the worst-case change-time distribution. Both types of problems are addressed. The most important result of the dissertation is the development of a polynomial-time algorithm for the solution of important classes of Markov Bayes change-detection problems. Existing techniques for epsilon-exact solution of partially observable Markov decision processes have complexity exponential in the number of observation symbols. A new algorithm, called constellation induction, exploits the concavity and Lipschitz continuity of the value function, and has complexity polynomial in the number of observation symbols. It is shown that change-detection problems with a geometric change-time distribution and identically- and independently-distributed observations before and after the change are solvable in polynomial time. Also, change-detection problems on hidden Markov models with a fixed number of recurrent states are solvable in polynomial time. A detailed implementation and analysis of the constellation-induction algorithm are provided. Exact solution methods are also established for several types of minimax change-detection problems. Finite-horizon problems with arbitrary observation distributions are modeled as extensive-form games and solved using linear programs. Infinite-horizon problems with linear penalty for detection delay and identically- and independently-distributed observations can be solved in polynomial time via epsilon-optimal parameterization of a cumulative-sum procedure. Finally, the properties of policies for change-detection problems are described and analyzed. Simple classes of formal languages are shown to be sufficient for epsilon-exact solution of change-detection problems, and methods for finding minimally sized policy representations are described.

Similarity Measures for Exam Timetabling Problems

A large number of heuristic algorithms have been developed over the years which have been aimed at solving examination timetabling problems. However, many of these algorithms have been developed specifically to solve one particular problem instance or a small subset of instances related to a given real-life problem. Our aim is to develop a more general system which, when given any exam timetabling problem, will produce results which are comparative to those of a specially designed heuristic for that problem. We are investigating a Case based reasoning (CBR) technique to select from a set of algorithms which have been applied successfully to similar problem instances in the past. The assumption in CBR is that similar problems have similar solutions. For our system, the assumption is that an algorithm used to find a good solution to one problem will also produce a good result for a similar problem. The key to the success of the system will be our definition of similarity between two exam timetabling problems. The study will be carried out by running a series of tests using a simple Simulated Annealing Algorithm on a range of problems with differing levels of similarity and examining the data sets in detail. In this paper an initial investigation of the key factors which will be involved in this measure is presented with a discussion of how the definition of good impacts on this.

Similarity Measures for Exam Timetabling Problems

A large number of heuristic algorithms have been developed over the years which have been aimed at solving examination timetabling problems. However, many of these algorithms have been developed specifically to solve one particular problem instance or a small subset of instances related to a given real-life problem. Our aim is to develop a more general system which, when given any exam timetabling problem, will produce results which are comparative to those of a specially designed heuristic for that problem. We are investigating a Case based reasoning (CBR) technique to select from a set of algorithms which have been applied successfully to similar problem instances in the past. The assumption in CBR is that similar problems have similar solutions. For our system, the assumption is that an algorithm used to find a good solution to one problem will also produce a good result for a similar problem. The key to the success of the system will be our definition of similarity between two exam timetabling problems. The study will be carried out by running a series of tests using a simple Simulated Annealing Algorithm on a range of problems with differing levels of similarity and examining the data sets in detail. In this paper an initial investigation of the key factors which will be involved in this measure is presented with a discussion of how the definition of good impacts on this.

Hierarchical Reconstruction Method for Solving Ill-posed Linear Inverse Problems

We present a detailed analysis of the application of a multi-scale Hierarchical Reconstruction method for solving a family of ill-posed linear inverse problems. When the observations on the unknown quantity of interest and the observation operators are known, these inverse problems are concerned with the recovery of the unknown from its observations. Although the observation operators we consider are linear, they are inevitably ill-posed in various ways. We recall in this context the classical Tikhonov regularization method with a stabilizing function which targets the specific ill-posedness from the observation operators and preserves desired features of the unknown. Having studied the mechanism of the Tikhonov regularization, we propose a multi-scale generalization to the Tikhonov regularization method, so-called the Hierarchical Reconstruction (HR) method. First introduction of the HR method can be traced back to the Hierarchical Decomposition method in Image Processing. The HR method successively extracts information from the previous hierarchical residual to the current hierarchical term at a finer hierarchical scale. As the sum of all the hierarchical terms, the hierarchical sum from the HR method provides an reasonable approximate solution to the unknown, when the observation matrix satisfies certain conditions with specific stabilizing functions. When compared to the Tikhonov regularization method on solving the same inverse problems, the HR method is shown to be able to decrease the total number of iterations, reduce the approximation error, and offer self control of the approximation distance between the hierarchical sum and the unknown, thanks to using a ladder of finitely many hierarchical scales. We report numerical experiments supporting our claims on these advantages the HR method has over the Tikhonov regularization method.

Vehicle Routing Problems that Minimize the Completion Time: Heuristics, Worst-Case Analyses, and Computational Results

In the standard Vehicle Routing Problem (VRP), we route a fleet of vehicles to deliver the demands of all customers such that the total distance traveled by the fleet is minimized. In this dissertation, we study variants of the VRP that minimize the completion time, i.e., we minimize the distance of the longest route. We call it the min-max objective function. In applications such as disaster relief efforts and military operations, the objective is often to finish the delivery or the task as soon as possible, not to plan routes with the minimum total distance. Even in commercial package delivery nowadays, companies are investing in new technologies to speed up delivery instead of focusing merely on the min-sum objective. In this dissertation, we compare the min-max and the standard (min-sum) objective functions in a worst-case analysis to show that the optimal solution with respect to one objective function can be very poor with respect to the other. The results motivate the design of algorithms specifically for the min-max objective. We study variants of min-max VRPs including one problem from the literature (the min-max Multi-Depot VRP) and two new problems (the min-max Split Delivery Multi-Depot VRP with Minimum Service Requirement and the min-max Close-Enough VRP). We develop heuristics to solve these three problems. We compare the results produced by our heuristics to the best-known solutions in the literature and find that our algorithms are effective. In the case where benchmark instances are not available, we generate instances whose near-optimal solutions can be estimated based on geometry. We formulate the Vehicle Routing Problem with Drones and carry out a theoretical analysis to show the maximum benefit from using drones in addition to trucks to reduce delivery time. The speed-up ratio depends on the number of drones loaded onto one truck and the speed of the drone relative to the speed of the truck.