Numerical Optimization presents a comprehensive and up-to-date description of the most effective methods in continuous optimization. It responds to the growing interest in optimization in engineering, science, and business by focusing on the methods that are best suited to practical problems. For this new edition the book has been thoroughly updated throughout. There are new chapters on nonlinear interior methods and derivative-free methods for optimization, both of which are used widely in practice and the focus of much current research. Because of the emphasis on practical methods, as well as the extensive illustrations and exercises, the book is accessible to a wide audience. It can be used as a graduate text in engineering, operations research, mathematics, computer science, and business. It also serves as a handbook for researchers and practitioners in the field. The authors have strived to produce a text that is pleasant to read, informative, and rigorous - one that reveals both the beautiful nature of the discipline and its practical side.

Numerical Optimization

The model selection literature has been generally poor at reflecting the deep foundations of the Akaike information criterion (AIC) and at making appropriate comparisons to the Bayesian information...

https://faculty.washington.edu/skalski/classes/QERM597/papers_xtra/Burnham%20and%20Anderson.pdf

Multimodel Inference Understanding AIC and BIC in Model Selection

SUMMARY
Over the last three years, a major international effort has been made by the Sub-Commission on Earthquake Algorithms of the International Association of Seismology and the Physics of the Earth's Interior (IASPEI) to generate new global traveltime tables for seismic phases to update the tables of Jeffreys & Bullen (1940). The new tables are specifically designed for convenient computational use, with high-accuracy interpolation in both depth and range. The new iasp91 traveltime tables are derived from a radially stratified velocity model which has been constructed so that the times for the major seismic phases are consistent with the reported times for events in the catalogue of the International Seismological Centre (ISC) for the period 1964–1987. The baseline for the P-wave traveltimes in the iasp91 model has been adjusted to provide only a small bias in origin time for well-constrained events at the main nuclear testing sites around the world.

For P-waves at teleseismic distances, the new tables are about 0.7s slower than the 1968 P-tables (Herrin 1968) and on average about 1.8–1.9 s faster than the Jeffreys & Bullen (1940) tables. For S-waves the teleseismic times lie between those of the JB tables and the results of Randall (1971).

Because the times for all phases are derived from the same velocity model, there is complete consistency between the traveltimes for different phases at different focal depths. The calculation scheme adopted for the new iasp91 tables is that proposed by Buland & Chapman (1983). Tables of delay time as a function of slowness are stored for each traveltime branch, and interpolated using a specially designed tau spline which takes care of square-root singularities in the derivative of the traveltime curve at certain critical slownesses. With this representation, once the source depth is specified, it is straightforward to find the traveltime explicitly for a given epicentral distance. The computational cost is no higher than a conventional look-up table, but there is increased accuracy in constructing the traveltimes for a source at arbitrary depth. A further advantage over standard tables is that exactly the same procedure can be used for each phase. For a given source depth, it is therefore possible to generate very rapidly a comprehensive list of traveltimes and associated derivatives for the main seismic phases which could be observed at a given epicentral distance.

/pdf/traveltimes-for-global-earthquake-location-and-phase-42n3pji8yf.pdf

Traveltimes for global earthquake location and phase identification

We briefly outline the information-theoretic (I-T) approaches to valid inference including a review of some simple methods for making formal inference from all the hypotheses in the model set (multimodel inference). The I-T approaches can replace the usual t tests and ANOVA tables that are so inferentially limited, but still commonly used. The I-T methods are easy to compute and understand and provide formal measures of the strength of evidence for both the null and alternative hypotheses, given the data. We give an example to highlight the importance of deriving alternative hypotheses and representing these as probability models. Fifteen technical issues are addressed to clarify various points that have appeared incorrectly in the recent literature. We offer several remarks regarding the future of empirical science and data analysis under an I-T framework.

/pdf/aic-model-selection-and-multimodel-inference-in-behavioral-192nrc36ut.pdf

AIC model selection and multimodel inference in behavioral ecology: some background, observations, and comparisons

This unique book provides an introduction to a subject whose use has steadily increased over the past 40 years. An update of Ramon Moore s previous books on the topic, it provides broad coverage of the subject as well as the historical perspective of one of the originators of modern interval analysis. The authors provide a hands-on introduction to INTLAB, a high-quality, comprehensive MATLAB toolbox for interval computations, making this the first interval analysis book that does with INTLAB what general numerical analysis texts do with MATLAB. Readers will find the following features of interest: elementary motivating examples and notes that help maximize the reader s chance of success in applying the techniques; exercises and hands-on MATLAB-based examples woven into the text; INTLAB-based examples and explanations integrated into the text, along with a comprehensive set of exercises and solutions, and an appendix with INTLAB commands; an extensive bibliography and appendices that will continue to be valuable resources once the reader is familiar with the subject; and a Web page with links to computational tools and other resources of interest. Audience: Introduction to Interval Analysis will be valuable to engineers and scientists interested in scientific computation, especially in reliability, effects of roundoff error, and automatic verification of results. The introductory material is particularly important for experts in global optimization and constraint solution algorithms. This book is suitable for introducing the subject to students in these areas. Contents: Preface; Chapter 1: Introduction; Chapter 2: The Interval Number System; Chapter 3: First Applications of Interval Arithmetic; Chapter 4: Further Properties of Interval Arithmetic; Chapter 5: Introduction to Interval Functions; Chapter 6: Interval Sequences; Chapter 7: Interval Matrices; Chapter 8: Interval Newton Methods; Chapter 9: Integration of Interval Functions; Chapter 10: Integral and Differential Equations; Chapter 11: Applications; Appendix A: Sets and Functions; Appendix B: Formulary; Appendix C: Hints for Selected Exercises; Appendix D: Internet Resources; Appendix E: INTLAB Commands and Functions; References; Index.

Introduction to Interval Analysis

The iterative method for solving system of linear equations, due to Kaczmarz [2], is investigated. It is shown that the method works well for both singular and non-singular systems and it determines the affine space formed by the solutions if they exist. The method also provides an iterative procedure for computing a generalized inverse of a matrix.

Projection method for solving a singular system of linear equations and its applications

Whole mantle P-wave travel time tomography

Foreword.- A Conversation with Hirotugu Akaike.- List of Publications of Hirotugu Akaike.- Papers.- 1. Precursors.- 1. On a zero-one process and some of its applications.- 2. On a successive transformation of probability distribution and its application to the analysis of the optimum gradient method.- 2. Frequency Domain Time Series Analysis.- 1. Effect of timing-error on the power spectrum of sampled-data.- 2. On a limiting process which asymptotically produces f-2 spectral density.- 3. On the statistical estimation of frequency response function.- 3. Time Domain Time Series Analysis.- 1. On the use of a linear model for the identification of feedback systems.- 2. Fitting autoregressive models for prediction.- 3. Statistical predictor identification.- 4. Autoregressive model fitting for control.- 5. Statistical approach to computer control of cement rotary kilns.- 6. Statistical identification for optimal control of supercritical thermal power plants.- 4. AIC and Parametrization.- 1. Information theory and an extension of the maximum likelihood princilple.- 2. A new look at the statistical model identification.- 3. Markovian representation of stochastic processes and its application to the analysis of autoregressive moving average processes.- 4. Covariance matrix computation of the state variable of a stationary Gaussian process.- 5. Analysis of cross classified data by AIC.- 6. On linear intensity models for mixed doubly stochastic Poisson and self-exciting point processes.- 5. Bayesian Approach.- 1. A Baysian analysis of the minimum AIC procedure.- 2. A new look at the Bayes procedure.- 3. On the likelihood of a time series model.- 4. Likelihood and the Bayes procedure.- 5. Seasonal adjustment by a Bayesian modeling.- 6. A quasi Bayesian approach to outlier detection.- 7. On the fallacy of the likelihood principle.- 8. A Bayesian apporach to the analysis of earth tides.- 9. Factor analysis and AIC.- 6. General Views on Statistics.- 1. Prediction and entropy.- 2. Experiences on the development of time series models.- 3. Implications of informational point of view on the development of statistical science.

Selected papers of Hirotugu Akaike

The purpose of this paper is to introduce a generic class of algorithms for solving a system of nonlinear equations, Linear Programming problems, Quadratic Programming problems, Nonlinear Programming problems and general complementarity problems. The algorithms were obtained by modifying the standard Newton-Raphson method applied to a system of nonlinear equations in the complementarity conditions so that it is biased towards ‘center curve’ passing through the solutions. The search direction of the methods is a positive combination of the Newton direction and a ‘centering’ direction which is also given by applying the Newton method to a projected system of the complementarity equations. These two directions guide the generated sequence of the approximations towards the solution and the center variety respectively. A class of ‘penalized norms’ and ‘guiding cones’ is also introduced for choosing step lengths in bivariate search.

Centered newton method for mathematical programming

A differential geometric approach to the constrained function maximization problem is presented. The continuous analogue of the Newton-Raphson method due to Branin for solving a system of nonlinear equations is extended to the case where the system is under-determined. The method is combined with the continuous analogue of the gradient-projection method to obtain a constrained maximization method with enforced constraint restoration. Detailed analysis of the global behavior of both methods is provided. It is shown that the conjugate-gradient algorithm can take advantage of the sparse structure of the problem in the computation of a vector field, which constitutes the main computational task in the methods.

Kunio Tanabe

Papers

Projection method for solving a singular system of linear equations and its applications

Whole mantle P-wave travel time tomography

Selected papers of Hirotugu Akaike

Centered newton method for mathematical programming

A geometric method in nonlinear programming