The Ultimate Solution Approach to Intractable Problems

There is now strong belief that P ? NP. This means that some very common problems cannot be solved efficiently under current and so called Von Neumann type computer architectures including parallel configurations. And, this will remain the case even in relatively low dimensions. What one may hope to achieve is the best possible solution given the available facilities within the allowed time. This makes the current definition of the optimum redundant for practical purposes. Therefore, a new definition of the optimum is required as well as appropriate approaches to find it. This paper will put forward a definition for the practical or sensible optimum, the s-optimum, consider its consequences and suggest what can be the ultimate approach to finding it. Although this approach is generic and can be applied in any context, optimisation and search are the specific contexts we will be concerned with here

New recurrence relationships between orthogonal polynomials which lead to new Lanczos-type algorithms

Lanczos methods for solving Ax = b consist in constructing a sequence of vectors (Xk),k = 1,... such that rk = b-AXk= Pk(A)r0, where Pk is the orthogonal polynomial of degree at most k with respect to the linear functional c defined as c(εi) = (y, Air0). Let P(1)k be the regular monic polynomial of degree k belonging to the family of formal orthogonal polynomials (FOP) with respect to c(1) defined as c(1)(εi) = c(εi+1). All Lanczos-type algorithms are characterized by the choice of one or two recurrence relationships, one for Pk and one for P(1)k. We shall study some new recurrence relations involving these two polynomials and their possible combinations to obtain new Lanczos-type algorithms. We will show that some recurrence relations exist, but cannot be used to derive Lanczos-type algorithms, while others do not exist at all

Explaining the Mpemba effect

The Mpemba effect goes back a long way; apparently, philosophers like Aristotle already knew about it, [1, 2] in ancient times. Nearer to us, Bacon and Descartes also knew of it and tried to explain it. It has been rediscovered and brought back to prominence in the modern era by a schoolboy in Tanzania, in the 60?s, [3]. It is the observation that warmer water may freeze quicker that colder water. Two things make it interesting and, to many rather irresistible: 1. It is counter-intuitive since cold water should be closer in state to frozen water (ice), than hot water; it, therefore, should freeze quicker. 2. Experiments do not show the effect every time. So, those who observed it want to explain it and those who did not, despite experimenting, think it is a hoax and want to debunk it

A game theory framework for clustering

The Game Theory-based Multi-Agent System (GTMAS) of Toreyen and Salhi, [10] and [12], implements a loosely coupled hybrid algorithm that may involve any number of algorithms suitable, a priori, for the solution of a given optimisation problem. The system allows the available algorithms to co-operate toward the solution of the problem in hand as well as compete for the computing facilities they require to run. This co-operative/competitive aspect is captured through the implementation of the Prisoners? Dilemma paradigm of game theory. Here, we apply GTMAS to the problem of clustering European Union (EU) economies, including Turkey, to find out whether the latter, based on a number of criteria, can fit in the EU and find out which countries, if any, it has strong similaries with. This clustering problem is first converted into an optimisation problem, namely the Travelling Salesman Problem (TSP) before being solved with GTMAS involving two players (agents) each implementing a standard combinatorial optimisation algorithm. Computational results are included

Direct recovering of surface structure characterized by an Nth degree polynomial equation using the UOFF approach

There are two different approaches for estimation of structure and/or motion of objects in the computer vision community today. One is the feature correspondence method, and the other is the optical flow method [1]. There are many difficulties and limitations encountered with the feature correspondence method, while the optical flow method is more feasible, but requires a substantial amount of extra calculations if the optical flow is to be computed as an intermediate step. Direct methods have been developed [2-4], that use the optical flow approach, but avoid computing the full optical flow field as an intermediate step for recovering structure and motion. The unified optical flow field theory was recently established in [5]. It is an extension of the optical flow (UOFF) [1] to stereo imagery. Based on the UOFF, a direct method is developed to reconstruct an Alpha shape surface structure characterized by an third degree polynomial equation, and a Sphere surface characterized by a second degree polynomial [6]. This thesis work uses the methods developed in [5,6], to reconstruct the third degree polynomial describing a surface. The main difference from the simulation results obtained in [6], is that in this case, one of the two surfaces tested is a third order, unbounded surface, and that tbe image gradients are computed directly from the image data, with no prior knowledge of the surface gray function distribution. Another important difference is that the gray levels of the surface are quantized in this work; i.e., the computations are done using integer image data, not the continuous gray levels as in [6]. These differences contribute to proving that the UOFF technique can be used in a practical manner, and with good results. Further discussions of the contributions of this work are included in the last chapter

Nature-Inspired Optimisation Approaches and the New Plant Propagation Algorithm

Nature-inspired algorithms are proving to be very successful on complex optimisation problems. A new algorithm, inspired by the way plants, and in particular the strawberry plant, propagate is presented. The algorithm is explained, tested on standard test functions, and compared with the well known Nelder-Mead algorithm. The new approach is then applied to a complex process design problem that arises in Chlorobenzene purification, a problem that exhibits strong nonlinear behaviour and has a small feasible region

A New Lanczos-Type Algorithm for Systems of Linear Equations

Lanczos-type algorithms are efficient and easy to implement. Unfortunately they breakdown frequently and well before convergence has been achieved. These algorithms are typically based on recurrence relations which involve formal orthogonal polynomials of low degree. In this paper, we consider a recurrence relation that has not been studied before and which involves a relatively higher degree polynomial. Interestingly, it leads to an algorithm that shows superior stability when compared to existing Lanczos-type algorithms. This new algorithm is derived and described. It is then compared to the best known algorithms of this type, namely A5/B10, A8/B10, as well as Arnoldi's algorithm, on a set of standard test problems. Numerical results are included

A quadtree-based allocation method for a class of large discrete Euclidean location problems: large location problems

A special data compression approach using a quadtree-based method is proposed for allocating very large demand points to their nearest facilities while eliminating aggregation error. This allocation procedure is shown to be extremely effective when solving very large facility location problems in the Euclidian space. Our method basically aggregates demand points where it eliminates aggregation-based allocation error, and disaggregates them if necessary. The method is assessed first on the allocation problems and then embedded into the search for solving a class of discrete facility location problems namely the p-median and the vertex p-centre problems. We use randomly generated and TSP datasets for testing our method. The results of the experiments show that the quadtree-based approach is very effective in reducing the computing time for this class of location problems

Erratum: Is it possible to infer the equation of state of a mixture of hard discs from that of the one-component system?

The numerical values in the sixth and seventh columns of table 1 of the paper Molec. Phys., 1999, 96, 1185-1188 are not correct. Consequently, some of the comments made in the paper are wrong. The corrected version of table 1 is reprinted here and the results are briefly discussed.Comment: 2 pages; Erratum to Molec. Phys., 1999, 96, 1185-1188; to be published in Molec. Phy