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

Today's Web-enabled deluge of electronic data calls for automated methods of data analysis. Machine learning provides these, developing methods that can automatically detect patterns in data and then use the uncovered patterns to predict future data. This textbook offers a comprehensive and self-contained introduction to the field of machine learning, based on a unified, probabilistic approach. The coverage combines breadth and depth, offering necessary background material on such topics as probability, optimization, and linear algebra as well as discussion of recent developments in the field, including conditional random fields, L1 regularization, and deep learning. The book is written in an informal, accessible style, complete with pseudo-code for the most important algorithms. All topics are copiously illustrated with color images and worked examples drawn from such application domains as biology, text processing, computer vision, and robotics. Rather than providing a cookbook of different heuristic methods, the book stresses a principled model-based approach, often using the language of graphical models to specify models in a concise and intuitive way. Almost all the models described have been implemented in a MATLAB software package--PMTK (probabilistic modeling toolkit)--that is freely available online. The book is suitable for upper-level undergraduates with an introductory-level college math background and beginning graduate students.

Machine Learning : A Probabilistic Perspective

A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems which are generally thought to be hard on a classical computer and which have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, e.g., the number of digits of the integer to be factored.

Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer

We present a novel approach to measuring similarity between shapes and exploit it for object recognition. In our framework, the measurement of similarity is preceded by: (1) solving for correspondences between points on the two shapes; (2) using the correspondences to estimate an aligning transform. In order to solve the correspondence problem, we attach a descriptor, the shape context, to each point. The shape context at a reference point captures the distribution of the remaining points relative to it, thus offering a globally discriminative characterization. Corresponding points on two similar shapes will have similar shape contexts, enabling us to solve for correspondences as an optimal assignment problem. Given the point correspondences, we estimate the transformation that best aligns the two shapes; regularized thin-plate splines provide a flexible class of transformation maps for this purpose. The dissimilarity between the two shapes is computed as a sum of matching errors between corresponding points, together with a term measuring the magnitude of the aligning transform. We treat recognition in a nearest-neighbor classification framework as the problem of finding the stored prototype shape that is maximally similar to that in the image. Results are presented for silhouettes, trademarks, handwritten digits, and the COIL data set.

Shape matching and object recognition using shape contexts

Planning algorithms are impacting technical disciplines and industries around the world, including robotics, computer-aided design, manufacturing, computer graphics, aerospace applications, drug design, and protein folding. This coherent and comprehensive book unifies material from several sources, including robotics, control theory, artificial intelligence, and algorithms. The treatment is centered on robot motion planning but integrates material on planning in discrete spaces. A major part of the book is devoted to planning under uncertainty, including decision theory, Markov decision processes, and information spaces, which are the “configuration spaces” of all sensor-based planning problems. The last part of the book delves into planning under differential constraints that arise when automating the motions of virtually any mechanical system. Developed from courses taught by the author, the book is intended for students, engineers, and researchers in robotics, artificial intelligence, and control theory as well as computer graphics, algorithms, and computational biology.

Planning Algorithms: Introductory Material

Stochastic method for calculating wave functions.

Serious roadblocks have been encountered in several areas of computer application, for example in the solution of intractable (NP-complete) combinatorial problems, or in the simulation of fluid flow. In this talk we will explore two alternatives to the usual kinds of computers, and ask if they provide some hope of ultimately by-passing what appear to be essential difficulties.

Two non-standard paradigms for computation: analog machines and cellular automata

This investigation is concerned with the optimal control of unknown, time-invariant. linear discrete systems. Of particular interest is the relationship among the identification problem, the specification of the control law assuming the system is known and the overall optimalizing control. For single-stage control with noiseless observations, conditions under which the use of separate identification and control procedures results in overall optimal control are established. The complexity inherent in the control problem is illustrated with a simple single-stage example wherein optimal control calls for filtering of the observed data by a time-varying, data-dependent operator, for which no simple recursive implementation exists.

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The identification and control of unknown linear discrete systems

The minimum bandwidth pulse for a given rms value and peak amplitude is found using the Pontryagin Maximum Principle for bounded phase space. The solution is obtained by actually solving the equations of the maximum principle rather than by a verification of a solution arrived at by other means. The resulting optimal pulse shape can be used for comparisons with more easily generated pulses, and may be considered to be the optimal modification of a rectangular pulse from the point of view of minimum bandwidth. Pulses of this sort are useful in pulse communication systems.

Application of the Maximum Principle to the Design of Minimum Bandwidth Pulses

We describe the design and testing of an 18- processor chip that implements the update rule for a fixed one-dimensional binary-valued cellular automaton. The VLSI design was done in the high level, procedural language ALLENDE [5,6]. The processors are bit-serial, completely pipelined, and cascaded, so that one chip performs 18 updates per major clock cycle. In principle, any number of chips can be cascaded, with a corresponding linear speedup. Sixteen chips were fabricated in 4 µ nMOS using the MOSIS facility, of which 10 were fully functional. When tested, 9 of the fully functional chips operated at maximum clock rates between 6 and 8 Mhz. This implies a maximum speed of about 108bit updates per second per chip, and opens the way for experimentation that is too time-consuming using general purpose devices.

Kenneth Steiglitz

Papers

Stochastic method for calculating wave functions.

Two non-standard paradigms for computation: analog machines and cellular automata

The identification and control of unknown linear discrete systems

Application of the Maximum Principle to the Design of Minimum Bandwidth Pulses

A multi-processor cellular automaton chip