Most floating point computations, in particular the additive operations, can have an annoying side-effect: The results can be inaccurate (the relative error can be any large) which may lead to qualitatively wrong answers to a computational problem. For the most prevalent applications there are advanced methods that avoid unreliable computational results in most of the cases. However, these methods usually decrease the performance of computations. In this paper, the most frequently occurring vector-vector addition and dot-product operations are investigated. We propose a faster stable add method which has been made possible by the appropriate utilization of the advanced SIMD architecture. In this work we focus only on the widely used Intel systems that are available for most of users. Based on it, stable vector addition and dot-product implementations are also introduced. Finally, we show the performance of these techniques.

Optimization is a widely used field of science in many applications. Optimization problems are becoming more and more complex and difficult to solve as the new models tend to be very large. To keep up with the growing requirements the solvers need to operate faster and more accurately. An important field of optimization is linear optimization which is very widely used. It is also often the hidden computational engine behind algorithms of other fields of optimization. Since linear optimization solvers use a high amount of special linear algebraic vector operations their performance is greatly influenced by their linear algebraic kernels. These kernels shall exploit the general characteristics of large-scale linear optimization problem models as efficiently as possible. To construct more efficient linear algebraic kernels the critical implementational factors influencing operation performance were identified via performance analysis and are presented in this paper. With the results of this analysis a new kernel has been developed for the open-source linear optimization solver called Pannon Optimizer developed at the Operations Research Laboratory at the University of Pannonia. A novel application of indexed dense vectors is also introduced which is designed specifically for linear optimization solvers. Finally a computational study is performed comparing the performance of vector operations of different linear optimization kernels to validate the high efficiency of our kernel. It shows that in case of large scale operations the indexed dense vector outperforms the state-of-the-art open-source linear optimization kernels.

It is well known that the simplex method is inherently a sequential algorithm with little scope for parallelization. Even so, during the last decades several attempts were made to parallelize it since it is one of the most important algorithms for solving linear optimization problems. Such parallelization ideas mostly rely on iteration parallelism and overlapping. Since the simplex method goes through a series of basic solutions until it finds an optimal solution, each of them must be available before performing the next basis change. This phenomenon imposes a limit on the performance of the parallelized version of the simplex method which uses overlapping iterations. Another approach can be considered if we think about alternative paths on the n-dimensional simplex polyhedron. As the simplex method goes through the edges of this polyhedron it is generally true that the speed of convergence of the algorithm is not smooth. It depends on the actual part of the surface. If a parallel version of the simplex algorithm simultaneously goes on different paths on this surface a highly reliable algorithm can be constructed. There is no known dominating strategy for pivot selection. Therefore, one can try different pivot selection methods in parallel in order to guide the algorithm on different pathways. This approach can be used effectively with periodic synchronization on shared memory multi-core computing environments to speed up the solution algorithm and get around numerically and/or algorithmically difficult situations throughout the computations.

Sparse linear algebraic data structures are widely used during the solution of large scale linear optimization problems. The efficiency of the solver is significantly influenced by the used data structures. The implementations of such data structures are not trivial. A performance analysis of the available data structures can provide valuable information to improve efficiency. In the talk we present our software that supports this task as well as our new, special vector representation. We also report results covering the solution for numerical issues affecting the performance of sparse linear algebraic operations.

Abstract The simplex algorithm is one of the most widely used linear optimization algorithms. Although its history goes back to the 1950’s it is still an evergreen topic because it is usually the hidden computational engine behind other algorithms. Thus its performance heavily affects other optimization systems. The implementation of the simplex algorithm is not a trivial task. An efficient implementation uses very complex methods. Since most of the open-source systems can hardly be used for research purposes the development of a structured and well-documented system is relevant. In this talk we present our own state-of-the-art solver and our development experiences.

The purpose of this work is to investigate the effect of certain numerical factors on the electrocardiographic inverse solution. Since usually the zero-order Tikhonov regularization is applied in order to solve the inverse problem, choosing the right regularization parameter is very important. The used linear equation system solver also plays an important role in the accuracy of the solution. Therefore these two factors were studied in a well defined model environment. Reference epicardial and body surface potential maps were generated in 1003 epicardial and 344 body surface points, respectively, by the Wei-Harumi model, for the QRS interval. The inverse problem has been solved with various regularization parameters and linear equation system solvers. The highest correlations (> 0.9) occurred in the first half of the QRS interval and the correlation fell below 0.6 at the end of the QRS. The result was highly dependent on the chosen regularization parameter and equation system solver, and it was significantly influenced also by the complexity of the epicardial potential distribution. The optimal solving configuration proved to be the application of the Cholesky factorization with the regularization parameter of 10−14.

Using the simplex method (SM) is one of the most effective ways of solving large scale real life linear optimization problems. The efficiency of the solver is crucial. The SM is an iterative procedure, where each iteration is defined by a basis of the constraint set. In order to speed up iterations, proper basis handling procedures must be applied. Two methodologies exist in the state-of-the-art literature, the product form of the inverse (PFI) and lower-upper triangular (LU) factorization. Nowadays the LU method is widely used because 120-150 iterations can be done without the need of refactorization while the PFI can make only about 30-60 iterations without reinversion in order to maintain acceptable numerical accuracy. In this paper we revisit the PFI and present a new version that can make hundreds or sometimes even few thousands of iterations without losing accuracy. The novelty of our approach is in the processing of the non-triangular part of the basis, based on block-triangularization algorithms. The new PFI performs much better than those found in the literature. The results can shed new light on the usefulness of the PFI.

SPECT (Single Photon Emission Computerized Tomography) presents poor performances in term of sensitivity and spatial resolution. The bottleneck of the process is due to the use of thin hole collimators. Collimators with large and long holes have been proposed (CACAO), but they involve a difficult inverse problem. OSEM (Ordered Subset Expectation Maximization) has been extensively used with success in conventional Thin Hole Collimator SPECT (THC-SPECT). Unfortunately this algorithm presents some limitations, when applied to the CACAO problem. We developed an algorithm based on the search for a median solution. The first publication describing this algorithm was limited to unrealistic small size problems (6×6 pixels). We present here a dualisation approach, which allows us to solve larger problems. Results on simulations for a 64×64 pixels size problem are given. Comparisons of the same problems treated by the OSEM approach are given.

Abstract Degeneracy is a phenomenon in linear programming (LP) that has important implications if the simplex method (SM) is used to solve an LP problem. It can happen that some of the vertices of the convex polyhedron of the feasible solutions (basic solutions) are not uniquely defined. In such a case the SM method may move from one representation of a vertex to another representation of the same vertex, while the value of the objective function remains unchanged. This can lead to a sequence of nonimproving solutions and a return to the first basis in the sequence. If it happens the sequence is repeated infinitely resulting in cycling and the SM does not terminate. There are, however, various theoretical and computational approaches that can prevent the SM from cycling. Even if they may not be able to overcome a long series of nonimproving iterations, called stalling , they still can substantially reduce the number of such iterations. This article gives details of all these phenomena and the indicated remedies.

Abstract The simplex method is widely used to solve linear programming problems in diverse application areas. It is used as a stand‐alone solver and also as the computational engine of the solution of other optimization problems. The emerging problems tend to be large to very large. Practical solution of such problems requires sophisticated implementation of the simplex method. This article gives an account of the most important developments that have enabled the simplex method to become a reliable, efficient, and flexible solver.

The GDPO algorithm for phase-1 of the dual simplex method developed by Maros possesses some interesting theoretical features that have potentially huge computational advantages. This paper gives account of a computational analysis of GDPO that has investigated how these features work in practice by exploring the internal operation of the algorithm. Experience of a systematic study involving 48 problems gives an insight how the predicted performance advantages materialize that ultimately make GDPO an indispensable tool for dual phase-1.

Today, sustainable management of the environment and its resources is an issue of vital importance in our contemporary societies and especially in developed societies. The scarcity of resources may well account for this fact. The importance of this problem underlies the necessity for adapting measures aiming at its solution. Many research papers and scientific conferences have been devoted to this direction. The challenge is to combine the high quality of life with a continuing economic growth in a sustainable way over the long term. However, this goal requires multidisciplinary and integrated research effort. Identification and evaluation of the likely environmental, economic and

1. Decision making and operations research. (a) The need for optimization. (b) Typical models of industry and finance. 2. Advanced linear programming (LP) (a) Need for linear models. (b) General form of LP problems. (c) The sparse simplex method; primal and dual algorithms; computational implications. (d) Economic interpretation of the optimality conditions: Sensitivity analysis. (e) Interior point algorithms. (f) State-of-the-art in solving LP problems. 3. Integer programming (IP) (a) Situations requiring integer valued models. (b) IP problem formulation. (c) Solution methods; Implicit enumeration, branch and bound (B&B) method. (d) Network programming (optimization in networks). (e) Logic constraints and IP. (f) State-of-the-art in solving IP problems. 4. Modeling (a) Basics of modeling. (b) Model generation and management. (c) Modeling languages, modeling systems.

The challenge of automating the formulation of optimization models is to produce, from a problem description, a well-formed model, which is a mathematically accurate representation of the real-world decision-making problem being considered, and which is suitable for computational pur poses. This can be stated more formally as the automation problem, which is the problem of providing intelligent (i.e. automated) assistance during the formulation stage of the mathematical programming process. In this paper, we explore the need to autom ate model formulation, thus providing a background on the automation problem. We also detail a solution to the problem which is based on evolutionary search techniques.

Optimization software e.g. solvers and modelling systems, expose software and vendor specific interfacing mechanisms to client applications e.g. decision support systems, thus introducing close coupling. The ‘Optimization Service Connectivity Protocol’ (OSCP) is an abstraction of the interfaces to optimization software, which is aimed primarily at simplifying the process of integrating optimization systems into software solutions by providing an abstracted, uniform and easy to use interface to such systems, regardless of system or vendor specific requirements. This paper presents a high-level overview of OSCP including descriptions of its main interfaces, and illustrates its use via