Experimental investigation of an interior search method within a simple framework

A steepest gradient method for solving Linear Programming (LP) problems, followed by a procedure for purifying a non-basic solution to an improved extreme point solution have been embedded within an otherwise simplex based optimiser. The algorithm is designed to be hybrid in nature and exploits many aspects of sparse matrix and revised simplex technology. The interior search step terminates at a boundary point which is usually non-basic. This is then followed by a series of minor pivotal steps which lead to a basic feasible solution with a superior objective function value. It is concluded that the procedures discussed in this paper are likely to have three possible applications, which are (i) improving a non-basic feasible solution to a superior extreme point solution, (iii) an improved starting point for the revised simplex method, and (iii) an efficient implementation of the multiple price strategy of the revised simplex method

Experimental investigations in combining primal dual interior point method and simplex based LP solvers

The use of a primal dual interior point method (PD) based optimizer as a robust linear programming (LP) solver is now well established. Instead of replacing the sparse simplex algorithm (SSX), the PD is increasingly seen as complementing it. The progress of PD iterations is not hindered by the degeneracy or the stalling problem of the SSX, indeed it reaches the 'near optimum' solution very quickly. The SSX algorithm, in contrast, is not affected by the boundary conditions which slow down the convergence of the PD. If the solution to the LP problem is non unique, the PD algorithm converges to an interior point of the solution set while the SSX algorithm finds an extreme point solution. To take advantage of the attractive properties of both the PD and the SSX, we have designed a hybrid framework whereby cross over from PD to SSX can take place at any stage of the PD optimization run. The cross over to SSX involves the partition of the PD solution set to active and dormant variables. In this paper we examine the practical difficulties in partitioning the solution set, we discuss the reliability of predicting the solution set partition before optimality is reached and report the results of combining exact and inexact prediction with SSX basis recovery

Adapting the interior point method for the solution of LPs on serial, coarse grain parallel and massively parallel computers

In this paper we describe a unified scheme for implementing an interior point algorithm (IPM) over a range of computer architectures. In the inner iteration of the IPM a search direction is computed using Newton's method. Computationally this involves solving a sparse symmetric positive definite (SSPD) system of equations. The choice of direct and indirect methods for the solution of this system, and the design of data structures to take advantage of serial, coarse grain parallel and massively parallel computer architectures, are considered in detail. We put forward arguments as to why integration of the system within a sparse simplex solver is important and outline how the system is designed to achieve this integration

Simplicial Multivalued Maps and the Witness Complex for Dynamical Analysis of Time Series

Topology based analysis of time-series data from dynamical systems is powerful: it potentially allows for computer-based proofs of the existence of various classes of regular and chaotic invariant sets for high-dimensional dynamics. Standard methods are based on a cubical discretization of the dynamics and use the time series to construct an outer approximation of the underlying dynamical system. The resulting multivalued map can be used to compute the Conley index of isolated invariant sets of cubes. In this paper we introduce a discretization that uses instead a simplicial complex constructed from a witness-landmark relationship. The goal is to obtain a natural discretization that is more tightly connected with the invariant density of the time series itself. The time-ordering of the data also directly leads to a map on this simplicial complex that we call the witness map. We obtain conditions under which this witness map gives an outer approximation of the dynamics, and thus can be used to compute the Conley index of isolated invariant sets. The method is illustrated by a simple example using data from the classical H\'enon map.Comment: laTeX, 9 figures, 32 page

Regularized Optimal Transport and the Rot Mover's Distance

This paper presents a unified framework for smooth convex regularization of discrete optimal transport problems. In this context, the regularized optimal transport turns out to be equivalent to a matrix nearness problem with respect to Bregman divergences. Our framework thus naturally generalizes a previously proposed regularization based on the Boltzmann-Shannon entropy related to the Kullback-Leibler divergence, and solved with the Sinkhorn-Knopp algorithm. We call the regularized optimal transport distance the rot mover's distance in reference to the classical earth mover's distance. We develop two generic schemes that we respectively call the alternate scaling algorithm and the non-negative alternate scaling algorithm, to compute efficiently the regularized optimal plans depending on whether the domain of the regularizer lies within the non-negative orthant or not. These schemes are based on Dykstra's algorithm with alternate Bregman projections, and further exploit the Newton-Raphson method when applied to separable divergences. We enhance the separable case with a sparse extension to deal with high data dimensions. We also instantiate our proposed framework and discuss the inherent specificities for well-known regularizers and statistical divergences in the machine learning and information geometry communities. Finally, we demonstrate the merits of our methods with experiments using synthetic data to illustrate the effect of different regularizers and penalties on the solutions, as well as real-world data for a pattern recognition application to audio scene classification

OPTIMIZING COMPLEX BIOECONOMIC SIMULATIONS USING AN EFFICIENT SEARCH HEURISTIC

For simulation to be truly useful for investigating many problems in agricultural economics, non-simplifying optimization techniques need to be employed. General methods for simulation optimization that do not inhibit system characterization or analysis are available, and they would appear to provide much of the mathematical and optimizing rigor demanded by economists. This paper describes the theory and algorithm of a robust and efficient simulation optimization approach, the Complex Method. An example of implementing the algorithm is illustrated using a pest management problem.simulation, optimization, Complex Method, hill-climbing, Research Methods/ Statistical Methods,

Proceedings of the ECCS 2005 satellite workshop: embracing complexity in design - Paris 17 November 2005

Embracing complexity in design is one of the critical issues and challenges of the 21st century. As the realization grows that design activities and artefacts display properties associated with complex adaptive systems, so grows the need to use complexity concepts and methods to understand these properties and inform the design of better artifacts. It is a great challenge because complexity science represents an epistemological and methodological swift that promises a holistic approach in the understanding and operational support of design. But design is also a major contributor in complexity research. Design science is concerned with problems that are fundamental in the sciences in general and complexity sciences in particular. For instance, design has been perceived and studied as a ubiquitous activity inherent in every human activity, as the art of generating hypotheses, as a type of experiment, or as a creative co-evolutionary process. Design science and its established approaches and practices can be a great source for advancement and innovation in complexity science. These proceedings are the result of a workshop organized as part of the activities of a UK government AHRB/EPSRC funded research cluster called Embracing Complexity in Design (www.complexityanddesign.net) and the European Conference in Complex Systems (complexsystems.lri.fr). Embracing complexity in design is one of the critical issues and challenges of the 21st century. As the realization grows that design activities and artefacts display properties associated with complex adaptive systems, so grows the need to use complexity concepts and methods to understand these properties and inform the design of better artifacts. It is a great challenge because complexity science represents an epistemological and methodological swift that promises a holistic approach in the understanding and operational support of design. But design is also a major contributor in complexity research. Design science is concerned with problems that are fundamental in the sciences in general and complexity sciences in particular. For instance, design has been perceived and studied as a ubiquitous activity inherent in every human activity, as the art of generating hypotheses, as a type of experiment, or as a creative co-evolutionary process. Design science and its established approaches and practices can be a great source for advancement and innovation in complexity science. These proceedings are the result of a workshop organized as part of the activities of a UK government AHRB/EPSRC funded research cluster called Embracing Complexity in Design (www.complexityanddesign.net) and the European Conference in Complex Systems (complexsystems.lri.fr)