Difference equations in Banach spaces

Difference equations which discretely approximate boundary value problems for second-order ordinary differential equations are analysed. It is well known that the existence of solutions to the continuous problem does not necessarily imply existence of solutions to the discrete problem and, even if solutions to the discrete problem are guaranteed, they may be unrelated and inapplicable to the continuous problem.Analogues to theorems for the continuous problem regarding a priori bounds and existence of solutions are formulated for the discrete problem. Solutions to the discrete problem are shown to converge to solutions of the continuous problem in an aggregate sense.An example which arises in the study of the finite deflections of an elastic string under a transverse load is investigated. The earlier results are applied to show the existence of a solution; the sufficient estimates on the step size are presented.     

Completion time variance minimisation on two identical parallel processors

The problem of scheduling jobs to minimise completion time variance (CTV) is a well-known problem in scheduling research. CTV is categorized as a non-regular performance measure and its value may decrease by increasing the job completion times. This objective is relevant in situations where providing uniform service to customers is important, and is in-line with just-in-time philosophy. The problem concerned in this paper is to schedule n jobs on two identical parallel machines to minimise CTV. We consider the unrestricted version of the problem. The problem is said to be restricted when a machine is not allowed to remain idle when jobs are available for processing. It may be necessary to delay the start of job processing on a machine in order to reduce the completion time deviations. This gives rise to the unrestricted version of the problem. We discuss several properties of an optimal schedule to the problem. In this paper, we develop a lower bound on CTV for a known partial schedule and propose a branch and bound algorithm to solve the problem. Optimal solutions are obtained and results are reported.     

Numerical solution for the Cauchy problem in nonlinear 1-D-thermoelasticity

We approximate the Cauchy problem by a problem in a bounded domain Ω _R =(?R,R) withR>0 sufficiently large, and the boundary conditions on ?Ω _R are imposed in terms of the far field behavior of solutions to the Cauchy problem. Then we solve this approximate problem by the finite element method for the spatial variable and the difference method for the time variable. Moreover a coupled numerical scheme for the Cauchy problem is presented. The error estimates are established.     

Dynamic Control with Indistinguishable Events

We consider the problem of synthesizing controllers for dicrete event systems with indistinguishable events. The specification for the supervised system as well as the constraints for the controllers are defined by an extension of the logic of the μ-calculus that generalize the framework introduced by Ramadge and Wonham. More precisely, in order to express that some events are indistinguishable, one can specify with this extension that, from a node of a graph, two edges reach the same node. As for the μ-calculus, the model cheking problem and also the emptiness problem for this extension amount to computing winning strategies in parity games. We show that the centralized control problem with indistinguishable events is decidable by reducing it to the satisfiability problem of a formula of this extension of the μ-calculus. We prove also that, unfortunately, the decentralized control problem is undecidable by a reduction of the halting problem of a Turing machine. Moreover, we exhibit a decidable case of decentralized control by restricting the specifications to those which we called here “conic”.     

Approximate solutions to a weighted mixed-sensitivity H∞-control design for irrational transfer matrices

Approximate solutions to a weighted mixed-sensitivity H^∞-control problem for an irrational transfer matrix are obtained by solving the same problem for a reduced-order (rational) transfer matrix. Upper and lower bounds are given in terms of the solution to the reduced-order problem and the approximation error. A delay example and a pde example are discussed.     

Single-plant sourcing decisions with a multidimensional knapsack model

This paper addresses sourcing decisions and how those decisions can affect the management of a company’s assets. The study begins with a single-plant insourcing problem in which the facility chooses, from a list of parts, which parts to bring in-house. This model is extended to include an outsourcing element where, in addition to the decision of which parts to insource, parts currently in-house are considered for outsourcing. The selection is based on maximizing the value of the selected parts while remaining within the plant’s capacity. This is defined as the sourcing problem and modeled as a multidimensional knapsack problem (MKP). Characteristics of the sourcing problem are identified and compared to standard MKP test problems, then utilized to provide guidelines for generating industry-motivated data sets. Experiences with modeling and solving the sourcing problem for a sheet metal fabrication facility are reported.     

On the efficient and accurate solution of the skew-symmetric eigenvalue problem: An arrangement of new and already known algorithmic formulations

Algorithms are presented to solve the special eigenvalue problem $\text{AZ}\text{=}\text{Zλ}$, where $\text{A}$ is skew-symmetric. The effective use of Householder's method, the bisection method and inverse iteration for solving the complete eigen-value problem are described in some detail. Simultaneous vector iteration is formulated for skew-symmetric matrices. The amount of work for the skew-symmetric Jacobi algorithm and the simultaneous vector iteration may be reduced by using the solution of a simplified eigenvalue problem. For Hermitian matrices also quadratic eigenvalue bounds for groups of eigenvalues and linear bounds for groups of eigenvectors are derived. The case where the set of calculated eigenvectors is not orthonormal is considered in some detail. In principle, the skew-symmetric eigenvalue problem may be easily transformed into a symmetric eigenvalue problem; but such a procedure has the following disadvantages: first, the results are in general less accurate, and, second, the eigenvectors which belong to well separated eigenvalues are not uniquely determined.     

Maximizing bichromatic reverse nearest neighbor for <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-norm in two- and three-dimensional spaces

Bichromatic reverse nearest neighbor (BRNN) has been extensively studied in spatial database literature. In this paper, we study a related problem called MaxBRNN: find an optimal region that maximizes the size of BRNNs for L _p -norm in two- and three- dimensional spaces. Such a problem has many real-life applications, including the problem of finding a new server point that attracts as many customers as possible by proximity. A straightforward approach is to determine the BRNNs for all possible points that are not feasible since there are a large (or infinite) number of possible points. To the best of our knowledge, there are no existing algorithms which solve MaxBRNN for any L _p -norm space of two- and three-dimensionality. Based on some interesting properties of the problem, we come up with an efficient algorithm called MaxOverlap for to solve this problem. Extensive experiments are conducted to show that our algorithm is efficient.     

Network Model for the Problem of Integer Balancing of a Four-Dimensional Matrix

The problem of integer balancing of a four-dimensional matrix is studied. The elements of the inner part (all four indices are greater than zero) of the given real matrix are summed in each direction and each two- and three-dimensional section of the matrix; the total sum is also found. These sums are placed into the elements where one or more indices are equal to zero (according to the summing directions). The problem is to find an integer matrix of the same structure, which can be produced from the initial one by replacing the elements with the largest previous or the smallest following integer. At the same time, the element with four zero indices should be produced with standard rules of rounding-off. In the article the problem of finding the maximum multiple flow in the network of any natural multiplicity k is also studied. There are arcs of three types: ordinary arcs, multiple arcs and multi-arcs. Each multiple and multi-arc is a union of k linked arcs, which are adjusted with each other. The network constructing rules are described. The definitions of a divisible network and some associated subjects are stated. There are defined the basic principles for reducing the integer balancing problem of an l-dimensional matrix (l ≥ 3) to the problem of finding the maximum flow in a divisible multiple network of multiplicity k. There are stated the rules for reducing the four-dimensional balancing problem to the maximum flow problem in the network of multiplicity 5. The algorithm of finding the maximum flow, which meets the solvability conditions for the integer balancing problem, is formulated for such a network.     

Symmetric cryptographic protocols for extended millionaires’ problem

Yao’s millionaires’ problem is a fundamental problem in secure multiparty computation, and its solutions have become building blocks of many secure multiparty computation solutions. Unfortunately, most protocols for millionaires’ problem are constructed based on public cryptography, and thus are inefficient. Furthermore, all protocols are designed to solve the basic millionaires’ problem, that is, to privately determine which of two natural numbers is greater. If the numbers are real, existing solutions do ...     

Optimal switch location in mobile communication networks using hybrid genetic algorithms

The optimal positioning of switches in a mobile communication network is an important task, which can save costs and improve the performance of the network. In this paper we propose a model for establishing which are the best nodes of the network for allocating the available switches, and several hybrid genetic algorithms to solve the problem. The proposed model is based on the so-called capacitated p-median problem, which have been previously tackled in the literature. This problem can be split in two subproblems: the selection of the best set of switches, and a terminal assignment problem to evaluate each selection of switches. The hybrid genetic algorithms for solving the problem are formed by a conventional genetic algorithm, with a restricted search, and several local search heuristics. In this work we also develop novel heuristics for solving the terminal assignment problem in a fast and accurate way. Finally, we show that our novel approaches, hybridized with the genetic algorithm, outperform existing algorithms in the literature for the p-median problem.     

Multivariate control theory in linear sequential machines

This paper presents a unified survey of some modern multivariate control theory aspects and techniques applied to linear sequential machines over a Galois field GF(p), Utilizing the concepts of controllability and observability for a Mealy-type machine the canonical decomposition problem, the state minimization problem, the identification problem, the transformation to canonical form problem, and the controllability/observability problem of combined machines are studied. Then the problem of designing a linear feedback controller for driving any state of a linear machine to the zero state after a minimum number of time steps A(clock periods), as well as the dual problem of designing a time-optimal state reconstructor for the same machine are solved. Finally, the problems of inverting a linear sequential machine and decoupling its inputs and outputs by using state feedforward and feedback are examined. Several examples illustrate the theoretical results.     

Two-machine group scheduling problems in discrete parts manufacturing with sequence-dependent setups

This paper focuses on minimizing the total completion time in two-machine group scheduling problems with sequence-dependent setups that are typically found in discrete parts manufacturing. As the problem is characterized as strongly NP-hard, three search algorithms based on tabu search are developed for solving industry-size scheduling problems. Four different lower bounding mechanisms are developed to identify a lower bound for all problems attempted, and the largest of the four is aptly used in the evaluation of the percentage deviation of the search algorithms to assess their efficacy. The problem sizes are classified as small, medium and large, and to accommodate the variability that might exist in the sequence-dependent setup times on both machines, three different scenarios are considered. Such finer levels of classification have resulted in the generation of nine different categories of problem instances, thus facilitating the performance of a very detailed statistical experimental design to assess the efficacy and efficiency of the three search algorithms. The search algorithm based on long-term memory with maximal frequencies either recorded a statistically better makespan or one that is indifferent when compared with the other two with all three scenarios and problem sizes. Hence, it is recommended for solving the research problem. Under the three scenarios, the average percentage deviation for all sizes of problem instances solved has been remarkably low. In particular, a mathematical programming based lower bounding mechanism, which focuses on converting (reducing) the original sequence-dependent group scheduling problem with several jobs in each group to a sequence-dependent job scheduling problem, has served well in identifying a high quality lower bound for the original problem, making it possible to evaluate a lower average percentage deviation for the search algorithm. Also, a 16–17-fold reduction in average computation time for solving a large problem instance with the recommended search algorithm compared with identifying just the lower bound of (not solving) the same instance by the mathematical programming based mechanism speaks strongly in favor of the search algorithm for solving industry-size group scheduling problems.     

An integrated approach to automated innovization for discovering useful design principles: Case studies from engineering

Computational optimization methods are most often used to find a single or multiple optimal or near-optimal solutions to the underlying optimization problem describing the problem at hand. In this paper, we elevate the use of optimization to a higher level in arriving at useful problem knowledge associated with the optimal or near-optimal solutions to a problem. In the proposed innovization process, first a set of trade-off optimal or near-optimal solutions are found using an evolutionary algorithm. Thereafter, the trade-off solutions are analyzed to decipher useful relationships among problem entities automatically so as to provide a better understanding of the problem to a designer or a practitioner. We provide an integrated algorithm for the innovization process and demonstrate the usefulness of the procedure to three real-world engineering design problems. New and innovative design principles obtained in each case should clearly motivate engineers and practitioners for its further application to more complex problems and its further development as a more efficient data analysis procedure.     

A reconstruction algorithm for a subclass of instances of the 2-color problem

In the field of Discrete Tomography, the 2-color problem consists in reconstructing a matrix whose elements are of two different types, starting from its horizontal and vertical projections. It is known that the 1-color problem admits a polynomial time reconstruction algorithm, while the c-color problem, with c≥2, is NP-hard. Thus, the 2-color problem constitutes an interesting example of a problem just in the frontier between hard and easy problems.In this paper we define a linear time algorithm (in the size of the output matrix) to solve a subclass of its instances, where some values of the horizontal and vertical projections are constant, while the others are upper bounded by a positive number proportional to the dimension of the problem. Our algorithm relies on classical studies for the solution of the 1-color problem.     

Modelling and solving grid resource allocation problem with network resources for workflow applications

A problem of allocating resources of a grid to workflow applications is considered. The problem consists, generally, in allocating distributed grid resources to tasks of a workflow in such a way that the resource demands of each task are satisfied. Grid resources are divided into computational resources and network resources. Computational tasks and transmission tasks of a workflow are distinguished. We present a model of the problem, and an algorithm for finding feasible resource allocations. A numerical example is included, showing the importance of the resource allocation phase on a grid. Some conclusions and directions for future research are given.     

Time and output warping of control systems: Comparing and imitating motions

The problem of making one system “mimic” a motion generated by another is studied. To address this, an optimal tracking problem is introduced that, in addition to tracking, optimizes over functions that deform or “warp” the time axis and the output space. Parametric and nonparametric versions of the time-warped tracking problem are presented and reduced to standard Bolza problems. The output warping problem is treated for affine and piecewise affine output warping functions. Example applications are given, including that of controlling a marionette to mimic a human dancer.     

An approach for minimizing a quadratically constrained fractional quadratic problem with application to the communications over wireless channels

Studies for the cognitive model are relatively new in the literature; however there is a growing interest in the communication field nowadays. This paper considers the cognitive model in the communication field as the problem of minimizing a fractional quadratic problem, subject to two or more quadratic constraints in complex field. Although both denominator and numerator in the fractional problem are convex, this problem is not so simple since the quotient of convex functions is not convex in most cases. We first change the fractional problem into a non-fractional one. Second, we consider the semi-definite programming (SDP) method. For the problem with m (m≤2) constraints, we use the SDP relaxation and obtain the exact optimal solution. However, for the problem with m (m>2) constraints, we choose the randomization method to gain an approximation solution in the complex case. At last, we apply this method to practical communications over wireless channels with good results.     

Lagrangean relaxation heuristics for the p-cable-trench problem

We address the p-cable-trench problem. In this problem, p facilities are located, a trench network is dug and cables are laid in the trenches, so that every customer - or demand - in the region is connected to a facility through a cable. The digging cost of the trenches, as well as the sum of the cable lengths between the customers and their assigned facilities, are minimized. We formulate an integer programming model of the problem using multicommodity flows that allows finding the solution for instances of up to 200 nodes. We also propose two Lagrangean Relaxation-based heuristics to solve larger instances of the problem. Computational experience is provided for instances of up to 300 nodes.     

The ripple minimization problems in sample data systems

The problem of minimizing the state value between the sampling times of a sampled data regulator is considered in this paper and is termed the ripple minimization problem. It is found that this problem is equivalent to the discrete problem of minimizing the state at the sampling times with a cost functional that includes in addition to the quadratic terms others which are bilinear in x and u. The results of the solution of the discrete optimal problem are used to solve the ripple minimization problem for both time-variant and time-invariant sampled data system. It is found that an additional condition on the rank of the controller matrices is necessary which is to some extent analogous to the condition necessary for preservation of controllability of sampled data system (Kalman et at., Bar-Ness 1975).