Convex optimization problems arise frequently in many different fields. A comprehensive introduction to the subject, this book shows in detail how such problems can be solved numerically with great efficiency. The focus is on recognizing convex optimization problems and then finding the most appropriate technique for solving them. The text contains many worked examples and homework exercises and will appeal to students, researchers and practitioners in fields such as engineering, computer science, mathematics, statistics, finance, and economics.

Convex Optimization

Introduction to a Transient World. Fourier Kingdom. Discrete Revolution. Time Meets Frequency. Frames. Wavelet Zoom. Wavelet Bases. Wavelet Packet and Local Cosine Bases. An Approximation Tour. Estimations are Approximations. Transform Coding. Appendix A: Mathematical Complements. Appendix B: Software Toolboxes.

http://ahvazdownload.persiangig.com/document/article/39.pdf

A wavelet tour of signal processing

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

SciPy is an open source scientific computing library for the Python programming language. SciPy 1.0 was released in late 2017, about 16 years after the original version 0.1 release. SciPy has become a de facto standard for leveraging scientific algorithms in the Python programming language, with more than 600 unique code contributors, thousands of dependent packages, over 100,000 dependent repositories, and millions of downloads per year. This includes usage of SciPy in almost half of all machine learning projects on GitHub, and usage by high profile projects including LIGO gravitational wave analysis and creation of the first-ever image of a black hole (M87). The library includes functionality spanning clustering, Fourier transforms, integration, interpolation, file I/O, linear algebra, image processing, orthogonal distance regression, minimization algorithms, signal processing, sparse matrix handling, computational geometry, and statistics. In this work, we provide an overview of the capabilities and development practices of the SciPy library and highlight some recent technical developments.

/pdf/scipy-1-0-fundamental-algorithms-for-scientific-computing-in-111lr790ol.pdf

SciPy 1.0--Fundamental Algorithms for Scientific Computing in Python

Preface 1. Introduction Overview A Brief History of LMIs in Control Theory Notes on the Style of the Book Origin of the Book 2. Some Standard Problems Involving LMIs. Linear Matrix Inequalities Some Standard Problems Ellipsoid Algorithm Interior-Point Methods Strict and Nonstrict LMIs Miscellaneous Results on Matrix Inequalities Some LMI Problems with Analytic Solutions 3. Some Matrix Problems. Minimizing Condition Number by Scaling Minimizing Condition Number of a Positive-Definite Matrix Minimizing Norm by Scaling Rescaling a Matrix Positive-Definite Matrix Completion Problems Quadratic Approximation of a Polytopic Norm Ellipsoidal Approximation 4. Linear Differential Inclusions. Differential Inclusions Some Specific LDIs Nonlinear System Analysis via LDIs 5. Analysis of LDIs: State Properties. Quadratic Stability Invariant Ellipsoids 6. Analysis of LDIs: Input/Output Properties. Input-to-State Properties State-to-Output Properties Input-to-Output Properties 7. State-Feedback Synthesis for LDIs. Static State-Feedback Controllers State Properties Input-to-State Properties State-to-Output Properties Input-to-Output Properties Observer-Based Controllers for Nonlinear Systems 8. Lure and Multiplier Methods. Analysis of Lure Systems Integral Quadratic Constraints Multipliers for Systems with Unknown Parameters 9. Systems with Multiplicative Noise. Analysis of Systems with Multiplicative Noise State-Feedback Synthesis 10. Miscellaneous Problems. Optimization over an Affine Family of Linear Systems Analysis of Systems with LTI Perturbations Positive Orthant Stabilizability Linear Systems with Delays Interpolation Problems The Inverse Problem of Optimal Control System Realization Problems Multi-Criterion LQG Nonconvex Multi-Criterion Quadratic Problems Notation List of Acronyms Bibliography Index.

/pdf/linear-matrix-inequalities-in-system-and-control-theory-2q5cf0pqm2.pdf

Linear Matrix Inequalities in System and Control Theory

Linear least-squares problems involving very large, completely dense matrices with a highly special structure arise in two applications—system identification and separable nonlinear least-squares with multiple data sets. Generic dense methods applied to such problems would be inordinately costly in terms of both speed and storage. Various techniques for exploiting the structure are described that differ in operation count, storage requirements, accuracy, and suitability for parallel computation. The system identification problem is presented first, to characterize the structure and analyze the algorithmic alternatives. Some indication is then given of how to apply the same techniques to separable nonlinear least-squares with multiple data sets.

Structured Linear Least-Squares Problems in System Identification and Separable Nonlinear Data Fitting

In this note, we consider two of the major issues that have arisen in implementing a sequential quadratic programming (SQP) method for nonlinearly constrained optimization problems (the code NPSOL; Gill et al., 1983). The problem of concern is assumed to be of the form[EQUATION]where F(x) is a smooth nonlinear function, AL is a constant matrix, and c(x) is a vector of smooth nonlinear constraint functions. The matrix AL and the vector c(x) may be empty. Note that upper and lower bounds are specified for all the variables and for all the constraints. This from allows full generality in constraint specification. In particular, the i-th constraint may be defined as an equality by setting li = ui. If certain bounds are not present, the associated elements of l or u can be set to special values that will be treated as - ∞ or +∞.

Some issues in implementing a sequential quadratic programming algorithm

This paper describes some of the important issues of numerical analysis in implementing a sequential quadratic programming method for nonlinearly constrained optimization We consider the separate treatment of linear constraints, design of a specialized quadratic programming algorithm, and control of ill-conditioning The results of applying the method to two specific examples are analyzed in detail

Considerations of numerical analysis in a sequential quadratic programming method

Linearly constrained optimization

: This report forms the user's guide for Version 40 of NPSOL, a set of Fortran subroutines designed to minimize a smooth function subject to constraints, which may include simple bounds on the variables, linear constraints and smooth nonlinear constraints (NPSOL may also be used for unconstrained, bound-constrained and linearly constrained optimization) The user must provide subroutines that define the objective ans constraint functions and (optionally) their gradients All matrices are treated as dense, and hence NPSOL is not intended for large sparse problems NPSOL uses a sequential quadratic programming (SQP) algorithm, in which the search direction is the solution of a quadratic programming (QP) subproblem The algorithm treats bounds, linear constraints and nonlinear constraints separately The Hessian of each QP subproblem is a positive-definite quasi-Newton approximation to the Hessiau of the Lagrangian function The steplength at each iteration is required to produce a sufficient decrease in an augmented Lagrangian merit function Each QP subproblem is solved using a quadratic programming package with several features that improve the efficiency of an SQP algorithm Keywords: Mathematical software; Nonlinear programming; and Finite difference (Author)

Margaret H. Wright

Papers

Structured Linear Least-Squares Problems in System Identification and Separable Nonlinear Data Fitting

Some issues in implementing a sequential quadratic programming algorithm

Considerations of numerical analysis in a sequential quadratic programming method

Linearly constrained optimization

NPSOL (Version 4.0): A Fortran Package for Nonlinear Programming. User's Guide.