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

The Nelder--Mead simplex algorithm, first published in 1965, is an enormously popular direct search method for multidimensional unconstrained minimization. Despite its widespread use, essentially no theoretical results have been proved explicitly for the Nelder--Mead algorithm. This paper presents convergence properties of the Nelder--Mead algorithm applied to strictly convex functions in dimensions 1 and 2. We prove convergence to a minimizer for dimension 1, and various limited convergence results for dimension 2. A counterexample of McKinnon gives a family of strictly convex functions in two dimensions and a set of initial conditions for which the Nelder--Mead algorithm converges to a nonminimizer. It is not yet known whether the Nelder--Mead method can be proved to converge to a minimizer for a more specialized class of convex functions in two dimensions.

/pdf/convergence-properties-of-the-nelder-mead-simplex-method-in-1e36h79lmd.pdf

Convergence Properties of the Nelder--Mead Simplex Method in Low Dimensions

Interior methods are an omnipresent, conspicuous feature of the constrained optimization landscape today, but it was not always so. Primarily in the form of barrier methods, interior-point techniques were popular during the 1960s for solving nonlinearly constrained problems. However, their use for linear programming was not even contemplated because of the total dominance of the simplex method. Vague but continuing anxiety about barrier methods eventually led to their abandonment in favor of newly emerging, apparently more efficient alternatives such as augmented Lagrangian and sequential quadratic programming methods. By the early 1980s, barrier methods were almost without exception regarded as a closed chapter in the history of optimization. 
This picture changed dramatically with Karmarkar's widely publicized announcement in 1984 of a fast polynomial-time interior method for linear programming; in 1985, a formal connection was established between his method and classical barrier methods. Since then, interior methods have advanced so far, so fast, that their influence has transformed both the theory and practice of constrained optimization. This article provides a condensed, selective look at classical material and recent research about interior methods for nonlinearly constrained optimization.

/pdf/interior-methods-for-nonlinear-optimization-dm3rhmoym6.pdf

Interior Methods for Nonlinear Optimization

: This report forms the user's guide for Version 4.0 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 and 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 directions 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 Hessian of the Lagrangian function. The steplength at each iteration is required to produce a sufficient decrease 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. (Author)

/pdf/user-s-guide-for-npsol-version-4-0-a-fortran-package-for-3qtrvyutx7.pdf

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

Numerical Linear Algebra and Optimization

Interest in linear programming has been intensified recently by Karmarkar's publication in 1984 of an algorithm that is claimed to be much faster than the simplex method for practical problems. We review classical barrier-function methods for nonlinear programming based on applying a logarithmic transformation to inequality constraints. For the special case of linear programming, the transformed problem can be solved by a "projected Newton barrier" method. This method is shown to be equivalent to Karmarkar's projective method for a particular choice of the barrier parameter. We then present details of a specific barrier algorithm and its practical implementation. Numerical results are given for several non-trivial test problems, and the implications for future developments in linear programming are discussed.

/pdf/on-projected-newton-barrier-methods-for-linear-programming-5602e4xzad.pdf

Margaret H. Wright

Papers

Convergence Properties of the Nelder--Mead Simplex Method in Low Dimensions

Interior Methods for Nonlinear Optimization

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

Numerical Linear Algebra and Optimization

On projected Newton barrier methods for linear programming and an equivalence to Karmarkar's projective method