Horner's Rule for Interval Evaluation Revisited

Interval arithmetic can be used to enclose the range of a real function over a domain. However, due to some weak properties of interval arithmetic, a computed interval can be much larger than the exact range. This phenomenon is called dependency problem. In this paper, Horner's rule for polynomial interval evaluation is revisited. We introduce a new factorization scheme based on well-known symbolic identities in order to handle the dependency problem of interval arithmetic. The experimental results show an improvement of 25% of the width of computed intervals with respect to Horner's rule. Received December 14, 2001; revised March 27, 2002 Published online: July 8, 2002     

Semi On-Line Scheduling on Two Identical Machines

This paper investigates two different semi on-line scheduling problems on a two-machine system. In the first case, we assume that all jobs have their processing times in between p and rp . In the second case, we assume that the largest processing time is known in advance. We show that one has a best possible algorithm with worst case ratio 4/3 while LS is still the best possible for the other problem with ratio which is still in the worst case . Received: February 23, 1998; revised August 5, 1998     

A New Well-Solvable Class of PNS Problems

A manufacturing system consists of operating units which convert their input materials into their output materials. In the problem of designing a process network, we have to find a suitable network of operating units which produces the desired products from the given raw materials. If we consider this process network design problem from a structural point of view, then we obtain a combinatorial optimization problem called the Process Network Synthesis or (PNS) problem. It is known that the PNS problem is NP-complete. Here, a new method is presented to reduce the solution of some more difficult PNS problems to the solution of simpler ones, and using this method, a new well-solvable class of PNS problems is established. Received February 12, 1999; revised October 24, 2000     

Global Convergence of a Class of Collinear Scaling Algorithms with Inexact Line Searches on Convex Functions

Ariyawansa [2] has presented a class of collinear scaling algorithms for unconstrained minimization. A certain family of algorithms contained in this class may be considered as an extension of quasi-Newton methods with the Broyden family [11] of approximants of the objective function Hessian. Byrd, Nocedal and Yuan [7] have shown that all members except the DFP [11] method of the Broyden convex family of quasi-Newton methods with Armijo [1] and Goldstein [12] line search termination criteria are globally and q-superlinearly convergent on uniformly convex functions. Extension of this result to the above class of collinear scaling algorithms of Ariyawansa [2] has been impossible because line search termination criteria for collinear scaling algorithms were not known until recently. Ariyawansa [4] has recently proposed such line search termination criteria. In this paper, we prove an analogue of the result of Byrd, Nocedal and Yuan [7] for the family of collinear scaling algorithms of Ariyawansa [2] with the line search termination criteria of Ariyawansa [4]. Received: May 8, 1996; revised January 27, 1999     

Scheduling a Single Server in a Two-machine Flow Shop

We study the problem of scheduling a single server that processes n jobs in a two-machine flow shop environment. A machine dependent setup time is needed whenever the server switches from one machine to the other. The problem with a given job sequence is shown to be reducible to a single machine batching problem. This result enables several cases of the server scheduling problem to be solved in O(n log n) by known algorithms, namely, finding a schedule feasible with respect to a given set of deadlines, minimizing the maximum lateness and, if the job processing times are agreeable, minimizing the total completion time. Minimizing the total weighted completion time is shown to be NP-hard in the strong sense. Two pseudopolynomial dynamic programming algorithms are presented for minimizing the weighted number of late jobs. Minimizing the number of late jobs is proved to be NP-hard even if setup times are equal and there are two distinct due dates. This problem is solved in O(n ³) time when all job processing times on the first machine are equal, and it is solved in O(n ⁴) time when all processing times on the second machine are equal. Received November 20, 2001; revised October 18, 2002 Published online: January 16, 2003     

Complexity Results for Single-Machine Problems with Positive Finish-Start Time-Lags

In a single-machine problem with time-lags a set of jobs has to be processed on a single machine in such a way that certain timing restrictions between the finishing and starting times of the jobs are satisfied and a given objective function is minimized. We consider the case of positive finish-start time-lags which mean that between the finishing time of job i and the starting time of job j the minimal distance has to be respected. New complexity results are derived for single-machine problems with constant positive time-lags which also lead to new results for flow-shop problems with unit processing times and job precedences. Received: May 13, 1998; revised November 23, 1998     

An Adaptive Routing Algorithm for WK-Recursive Topologies

This paper presents an easy and straightforward routing algorithm for WK-recursive topologies. The algorithm, based on adaptive routing, takes advantage of the geometric properties of such topologies. Once a source node S and destination node D have been determined for a message communication, they characterize, at some level l, two virtual nodes and that respectively contain S but not D and D but not S. Such virtual nodes characterize other (where is the node degree for a fixed topology) virtual nodes of the same level that contain neither S nor D. Consequently, it is possible to locate triangles whose vertices are these virtual nodes with property to share the same path, called the self-routing path, directly connecting to . When the self-routing path is unavailable to transmit a message from S to D because of deadlock, fault, and congestion conditions, the routing strategy can follow what we call the triangle rule to deliver it. The proposed communication scheme has the advantage that 1) it is the same for all three conditions; 2) each node of a WK-recursive network, to transmit messages, does not require any information about their presence or location. Furthermore, this routing algorithm is able to tolerate up to faulty links. Received: July 19, 1998; revised May 17, 1999     

A Dual Framework for Lower Bounds of the Quadratic Assignment Problem Based on Linearization

A dual framework allowing the comparison of various bounds for the quadratic assignment problem (QAP) based on linearization, e.g. the bounds of Adams and Johnson, Carraresi and Malucelli, and Hahn and Grant, is presented. We discuss the differences of these bounds and propose a new and more general bounding procedure based on the dual of the linearization of Adams and Johnson. The new procedure has been applied to problems of dimension up to , and the computational results indicate that the new bound competes well with existing linearization bounds and yields a good trade off between computation time and bound quality. Received: February 5, 1999; revised August 24, 1999     

Acquisition of Complex Model Knowledge by Domain Theory-Controlled Generalization

Nearly all three-dimensional reconstruction methods lack proper model knowledge that reflects the scene. Model knowledge is required in order to reduce ambiguities which occur during the reconstruction process. It must comprise the scene and is therefore complex, and additionally difficult to acquire. In this paper we present an approach for the learning of complex model knowledge. A (large) sample set of three-dimensionally acquired buildings represented as graphs is generalized by the use of background knowledge. The background knowledge entails domain-specific knowledge and is utilized for the search guidance during the generalization process of EXRES. The generalization result is a distribution of relevant patterns which reduces ambiguities occurring in 3D object reconstruction (here: buildings). Three different applications for the 3D reconstruction of buildings from aerial images are executed whereas binary relations of so-called building atoms, namely tertiary nodes and faces, and building models are learned. These applications are evaluated based on (a) the estimated empirical generalization error and (b) the use of information coding theory and statistics by comparing the learned knowledge with non-available a priori knowledge. Received: June 3, 1998; revised November 5, 1998     

Computation of Lagrange Multipliers for Linear Least Squares Problems with Equality Constraints

Received December 21, 2000; revised June 7, 2001     

A New Multisection Technique in Interval Methods for Global Optimization

A new multisection technique in interval methods for global optimization is investigated, and numerical tests demonstrate that the efficiency of the underlying global optimization method can be improved substantially. The heuristic rule is based on experiences that suggest the subdivision of the current subinterval into a larger number of pieces only if it is located in the neighbourhood of a minimizer point. An estimator of the proximity of a subinterval to the region of attraction to a minimizer point is utilized. According to the numerical study made, the new multisection strategies seem to be indispensable, and can improve both the computational and the memory complexity substantially. Received May 31, 1999; revised January 20, 2000     

Descent Iterations for Improving Approximate Eigenpairs of Polynomial Eigenvalue Problems with General Complex Matrices

In this paper, we consider descent iterations with line search for improving an approximate eigenvalue and a corresponding approximate eigenvector of polynomial eigenvalue problems with general complex matrices, where an approximate eigenpair was obtained by some method. The polynomial eigenvalue problem is written as a system of complex nonlinear equations with nondifferentiable normalized condition. Convergence theorems for iterations are established. Finally, some numerical examples are presented to demonstrate the effectiveness of the iterative methods. Received April 9, 2001; revised October 2, 2001 Published online February 18, 2002     

Adaptive Predictor for Lossless Image Compression

A new method for lossless image compression of grey-level images is proposed. The image is treated as a set of stacked bit planes. The compressed version of the image is represented by residuals of a non-linear local predictor spanning the current bit plane as well as a few neighbouring ones. Predictor configurations are grouped in pairs differing in one bit of the representative point only. The frequency of predictor configurations is obtained from the input image. The predictor adapts automatically to the image, it is able to estimate the influence of neighbouring cells and thus copes even with complicated structure or fine texture. The residuals between the original and the predicted image are those that correspond to the less frequent predictor configurations. Efficiently coded residuals constitute the output image. To our knowledge, the performance of the proposed compression algorithm is comparable to the current state of the art. Especially good results were obtained for binary images, grey-level cartoons and man-made drawings. Received: June 29, 1998; revised November 2, 1998     

A Modification of Newton's Method for Analytic Mappings Having Multiple Zeros

We propose a modification of Newton's method for computing multiple roots of systems of analytic equations. Under mild assumptions the iteration converges quadratically. It involves certain constants whose product is a lower bound for the multiplicity of the root. As these constants are usually not known in advance, we devise an iteration in which not only an approximation for the root is refined, but also approximations for these constants. Numerical examples illustrate the effectiveness of our approach. Received: August 17, 1998     

On Solving Periodically Perturbed Nonconservative Systems with a One-Parameter Embedding

The paper is concerned with solving the periodically perturbed nonconservative systems, which will be differentiably imbedded into an one-parameter family of operators. The solution of the systems is then found by continuing the solution curve of operator homotopy. When the Newton-Kantorovich's procedure is applied to the corresponding operator equations, an efficient algorithm is derived. Furthermore, the suitable condition on the optimum step size of the parameter is provided for assuring that the approximation solution will converge to the unique solution of the nonlinear periodically boundary value problem. Finally, the theoretical results are in excellent agreement with the numerical examples. Received August 14, 2001; revised December 19, 2001 Published online: November 18, 2002     

Simulated Annealing for Fitting Linear Combinations of Gaussians to Data

 It is difficult to find a good fit of a combination of Gaussians to arbitrary empirical data. The surface defined by the objective function contains many local minima, which trap gradient descent algorithms and cause stochastic methods to tarry unreasonably in the vicinity. A number of techniques for accelerating convergence when using simulated annealing are presented. These are tested on a sample of known Gaussian combinations and are compared for accuracy and resource consumption. A single `best' set of techniques is found which gives good results on the test samples and on empirical data. Received September 27, 1999; revised March 13, 2000     

Analysis of Some Elementary Algorithms for Prime Factorization

For factoring a positive integer n into primes, four variants of the elementary algorithm are analysed. The worst-case time complexities vary from Θ() up to Θ(). The average time complexities vary from Θ() up to Θ(). Received August 21, 1998; revised September 14, 2000     

From Methods to Images

This paper presents a system for automatically selecting images in an image database to be used as illustrations of an image method or analysis process. This problem is related to the teleteaching of image processing which uses already implemented libraries of algorithms. This is in the context of a teleteaching European project. We first give a Bayesian approach to this problem, by using an image basis. Then we show how to use the Haar transform for this purpose. Finally we give examples and discuss our approach. Received: June 8, 1998; revised December 17, 1998     

Tighter Bounds on Preemptive Job Shop Scheduling with Two Machines

We consider the preemptive job shop scheduling problem with two machines, with the objective to minimize the makespan. We present an algorithm that finds a schedule of length at most P _max/2 greater than the optimal schedule length, where P _max is the length of the longest job. Received June 13, 2000