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

This clearly written , mathematically rigorous text includes a novel algorithmic exposition of the simplex method and also discusses the Soviet ellipsoid algorithm for linear programming; efficient algorithms for network flow, matching, spanning trees, and matroids; the theory of NP-complete problems; approximation algorithms, local search heuristics for NPcomplete problems, more All chapters are supplemented by thoughtprovoking problems A useful work for graduate-level students with backgrounds in computer science, operations research, and electrical engineering Mathematicians wishing a self-contained introduction need look no further—American Mathematical Monthly 1982 ed

Combinatorial optimization: algorithms and complexity

An iterative technique is proposed to identify a linear system from samples of its input and output in the presence of noise by minimizing the mean-square error between system and model outputs. The model chosen has a transfer function which is a ratio of polynomials in z-1. Although the regression equations for the optimal set of coefficients are highly nonlinear and intractable, it is shown that the problem can be reduced to the repeated solution of a related linear problem. Computer simulation of a number of typical discrete systems is used to demonstrate the considerable improvement over the Kalman estimate which can be obtained in a few iterations. The procedure is found to be effective at signal-to-noise ratios less than unity, and with as few as 200 samples of the input and output records.

A technique for the identification of linear systems

A quad tree for representing a picture is a tree in which successively deeper levels represent successively finer subdivisions of picture area. An algorithm is given for superposing N quad trees in time proportional to the total number of nodes in the trees. Warnock-type algorithms are then presented for building the quad tree for the picture of the boundary of a polygon, and for coloring the interior of such a polygon. These algorithms take O(v + p + q) time, where v is the number of polygon vertices, p is the polygon perimeter, and q is a resolution parameter. When the resolution q is fixed, these algorithms are asymptotically optimal.

Operations on Images Using Quad Trees

Fixed step size random search for minimization of functions of several parameters is described and compared with the fixed step size gradient method for a particular surface. A theoretical technique, using the optimum step size at each step, is analyzed. A practical adaptive step size random search algorithm is then proposed, and experimental experience is reported that shows the superiority of random search over other methods for sufficiently high dimension.

/pdf/adaptive-step-size-random-search-pjphwtpjb6.pdf

Adaptive step size random search

A method is presented for computing an orthonormal set of eigenvectors for the discrete Fourier transform (DFT). The technique is based on a detailed analysis of the eigenstructure of a special matrix which commutes with the DFT. It is also shown how fractional powers of the DFT can be efficiently computed, and possible applications to multiplexing and transform coding are suggested.

/pdf/eigenvectors-and-functions-of-the-discrete-fourier-transform-2q9wnfgbp2.pdf

Kenneth Steiglitz

Papers

Combinatorial optimization: algorithms and complexity

A technique for the identification of linear systems

Operations on Images Using Quad Trees

Adaptive step size random search

Eigenvectors and functions of the discrete Fourier transform