Scaling Sparse Matrices for Optimization Algorithms

To iteratively solve large scale optimization problems in various contexts like planning, operations, design etc., we need to generate descent directions that are based on linear system solutions. Irrespective of the optimization algorithm or the solution method employed for the linear systems, ill conditioning introduced by problem characteristics or the algorithm or both need to be addressed. In [GL01] we used an intuitive heuristic approach in scaling linear systems that improved performance of a large scale interior point algorithm significantly. We saw a factor of 10*3* improvements in condition number estimates. In this paper, given our experience with optimization problems from a variety of application backgrounds like economics, finance, engineering, planning etc., we examine the theoretical basis for scaling while solving the linear systems. Our goal is to develop reasonably "good" scaling schemes with sound theoretical basis. We introduce concepts and define "good" scaling schemes in section (1), as well as explain related work in this area. Scaling has been studied extensively and though there is a broad agreement on its importance, the same cannot be said about what constitutes good scaling. A theoretical framework to scale an m x n real matrix is established in section (2). We use the first order conditions associated with the Euclidean metric to develop iterative schemes in section (2.3) that approximate solution in O(mn) time for real matrice. We discuss symmetry preserving scale factors for an n x n symmetric matrix in (3). The importance of symmetry preservation is discussed in section (3.1). An algorithm to directly compute symmetry preserving scale factors in O(n2) time based on Euclidean metric is presented in section (3.4) We also suggest scaling schemes based on rectilinear norm in section (2.4). Though all p-norms are theoretically equivalent, the importance of outliers increases as p increases. For barrier methods, due to large diagnal corrections, we believe that the taxicab metric (p = 1) may be more appropriate. We develop a linear programming model for it and look at a "reduced" dual that can be formulated as a minimum cost flow problem on networks. We are investigating algorithms to solve it in O(mn) time that we require for an efficient scaling procedure. We hope that in future special structure of the "reduced" dual could be exploited to solve it quickly. The dual information can then be used to compute the required scale factors. We discuss Manhattan metric for symmetric matrices in section (3.5) and as in the case of real matrices, we are unable to propose an efficient computational scheme for this metric. We look at a linearized ideal penalty function that only uses deviations out of the desired range in section (2.5). If we could use such a metric to generate an efficient solution, then we would like to see impact of changing the range on the numerical behavior.

Scaling Sparse Constrained Nonlinear Problems for Iterative Solvers

We look at scaling a nonlinear optimization problem for iterative solvers that use at least first derivatives. These derivatives are either computed analytically or by differncing. We ignore iterative methods that are based on function evaluations only and that do not use any derivative information. We also exclude methods where the full problem structure is unknown like variants of delayed column generation. We look at related work in section (1). Despite its importance as evidenced in widely used implementations of nonlinear programming algorithms, scaling has not received enough attention from a theoretical point of view. What do we mean by scaling a nonlinear problem itself is not very clear. In this paper we attempt a scaling framework definition. We start with a description of a nonlinear problem in section (2). Various authors prefer different forms, but all forms can be converted to the form we show. We then describe our scaling framework in section (3). We show the equivalence between the original problem and the scaled problem. The correctness results of section (3.3) play an important role in the dynamic scaling scheme suggested. In section (4), we develop a prototypical algorithm that can be used to represent a variety of iterative solution methods. Using this we examine the impact of scaling in section (5). In the last section (6), we look at what the goal should be for an ideal scaling scheme and make some implementation suggestions for nonlinear solvers.

Identification of Top Performing Economies

Using seven indicators of the economic performance of 187 countries, the paper identifies the top 50 performers during the decades of 1981-90 and 1991-2000. Five of these indicators are the trend rates of growth over a decade in imports, FDI, capital formation, per capita income and forex reserves. Average inflation rate and HDI are the remaining indicators. Comparison of top performers of the 1980s and the 1990s suggest that high performance in inflation and HDI are the precondition for consistency of high overall performance over time. The paper also examines the interrelationship among the indicators over time.

Program Transformations for Asynchronous and Batched Query Submission

The performance of database/Web-service backed applications can be significantly improved by asynchronous submission of queries/requests well ahead of the point where the results are needed, so that results are likely to have been fetched already when they are actually needed. However, manually writing applications to exploit asynchronous query submission is tedious and error-prone. In this paper we address the issue of automatically transforming a program written assuming synchronous query submission, to one that exploits asynchronous query submission. Our program transformation method is based on data flow analysis and is framed as a set of transformation rules. Our rules can handle query executions within loops, unlike some of the earlier work in this area. We also present a novel approach that, at runtime, can combine multiple asynchronous requests into batches, thereby achieving the benefits of batching in addition to that of asynchronous submission. We have built a tool that implements our transformation techniques on Java programs that use JDBC calls; our tool can be extended to handle Web service calls. We have carried out a detailed experimental study on several real-life applications, which shows the effectiveness of the proposed rewrite techniques, both in terms of their applicability and the performance gains achieved.Comment: 14 page

Implementing Tabu Search to Exploit Sparsity in ATSP Instances

Real life traveling salesman problem (TSP) instances are often large,sparse, and asymmetric. Conventional tabu search implementations for the TSP that have been reported in the literature, almost always deals with small, dense and symmetric instances. In this paper, we outline data structures and a tabu search implementation that takes advantage of such data structures, which can exploit sparsity of a TSP instances, and hence can solve relatively large TSP instances (with up to 3000 nodes) much faster than conventional implementations. We also provide computational experiences with this implementation.