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

In this paper we give a new algorithm for determining if a graph is planar. The algorithm is based on Tutte's theory of 3-connected graphs, and provides a structural decomposition of the graph. Results are presented in the algorithmic form, and a computer program is described.

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A new planarity test based on 3-connectivity

We propose a new physical medium for computation: colliding solitons. As particlelike nonlinear waves, solitons can carry and exchange information, thus acting as computing agents. Optical solitons have pulse widths measured in picoseconds or femtoseconds, and this approach could ultimately oﬀer an alternative to electronics. This work grew from an abstract computational model called the particle machine (PM), which uses discrete propagating and colliding particles to compute. We show that this model is universal, and describe several eﬃcient PM algorithms, such as convolution, certain systolic-array computations, and linear-time, arbitrary-precision arithmetic, including an iterative algorithm for division. This leads to the question of physical instantiation of PMs that could do useful computation, and optical solitons provide a natural example. Propagating solitons carry information in their amplitudes, velocities, and phases. We show how colliding solitons can transfer such information in nontrivial ways, thus making computation possible. We characterize the state transformations caused by collisions in a particular system of coupled optical solitons — the Manakov system — and show how to implement several computations, such as a NOT processor and certain other reversible operators. Our results resolve some questions about computational power in diﬀerent soliton systems, and serve as ﬁrst steps towards realizing a practical soliton-based computer.

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Computing with solitons in bulk media

We review work on computing with solitons, from the discovery of solitons in cellular automata, to an abstract model for particle computation, information transfer in collisions of optical solitons, state transformations in collisions of vector solitons, a proof of the universality of blinking spatial solitons, and the demonstration of multistable collision cycles and their application to state-restoring logic. We conclude by discussing open problems and the prospects for practical computing applications using optical soliton collisions in photo-refractive crystals and fibers.

Computing with Classical Soliton Collisions

We show that soliton collisions can reduce propagation-induced quantum phase noise. This effect originates from a negative correlation between self- and cross-phase-modulation-induced phase fluctuations. Furthermore, we show how this effect can be applied directly to improve the quality of soliton-based quantum nondemolition measurements, simply by adjusting the parameter regime in which the measurement is performed. Optimal implementation, which we show to be technologically feasible, favors short propagation distances, small wavelength separation between solitons, and approximately equal soliton amplitudes.

Quantum phase noise reduction in soliton collisions and application to nondemolition measurements

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Kenneth Steiglitz

Papers

A new planarity test based on 3-connectivity

Computing with solitons in bulk media

Computing with Classical Soliton Collisions

Quantum phase noise reduction in soliton collisions and application to nondemolition measurements

A Note on Constant-Gain Digital Resonators