Convex Optimization Theory Algorithms And Applications
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Convex Optimization
Author: Stephen P. Boyd
language: en
Publisher: Cambridge University Press
Release Date: 2004-03-08
Convex optimization problems arise frequently in many different fields. This book provides a comprehensive introduction to the subject, and shows in detail how such problems can be solved numerically with great efficiency. The book begins with the basic elements of convex sets and functions, and then describes various classes of convex optimization problems. Duality and approximation techniques are then covered, as are statistical estimation techniques. Various geometrical problems are then presented, and there is detailed discussion of unconstrained and constrained minimization problems, and interior-point methods. The focus of the book is on recognizing convex optimization problems and then finding the most appropriate technique for solving them. It 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
Over the past two decades, it has been recognized that advanced image processing techniques provide valuable information to physicians for the diagnosis, image guided therapy and surgery, and monitoring of human diseases. Convex Optimization: Theory, Methods and Applications introduces novel and sophisticated mathematical problems which encourage the development of advanced optimization and computing methods, especially convex optimization.The authors go on to study Steffensen-King-type methods of convergence to approximate a locally unique solution of a nonlinear equation and also in problems of convex optimization. Real-world applications are also provided.The following study is focused on the design and testing of a Matlab code of the Frank-Wolfe algorithm. The Nesterov step is proposed in order to accelerate the algorithm, and the results of some numerical experiments of constraint optimization are also provided.Lagrangian methods for numerical solutions to constrained convex programs are also explored. For enhanced algorithms, the traditional Lagrange multiplier update is modified to take a soft reflection across the zero boundary. This, coupled with a modified drift expression, is shown to yield improved performance.Next, Newton's mesh independence principle was used to solve a certain class of optimal design problems from earlier studies. Motivated by optimization considerations, the authors show that under the same computational cost, a finer mesh independence principle can be given than before.This compilation closes with a presentation on a local convergence analysis for eighth�order variants of Hansen�Patrick�s family for approximating a locally unique solution of a nonlinear equation. The radius of convergence and computable error bounds on the distances involved are also provided.
Convex Optimization—Theory, Algorithms and Applications
Author: Balendu Bhooshan Upadhyay
language: en
Publisher: Springer Nature
Release Date: 2025-04-25
This volume includes chapters on topics presented at the conference on Recent Trends in Convex Optimization: Theory, Algorithms and Applications (RTCOTAA-2020), held at the Department of Mathematics, Indian Institute of Technology Patna, Bihar, India, from 29–31 October 2020. It discusses a comprehensive exploration of the realm of optimization, encompassing both the theoretical underpinnings and the multifaceted real-life implementations of the optimization theory. It meticulously features essential optimization concepts, such as convex analysis, generalized convexity, monotonicity, etc., elucidating their theoretical advancements and significance in the optimization sphere. Multiobjective optimization is a pivotal topic which addresses the inherent difficulties faced in conflicting objectives. The book delves into various theoretical concepts and covers some practical algorithmic approaches to solve multiobjective optimization, such as the line search and the enhanced non-monotone quasi-Newton algorithms. It also deliberates on several other significant topics in optimization, such as the perturbation approach for vector optimization, and solution methods for set-valued optimization. Nonsmooth optimization is extensively covered, with in-depth discussions on various well-known tools of nonsmooth analysis, such as convexificators, limiting subdifferentials, tangential subdifferentials, quasi-differentials, etc. Notable optimization algorithms, such as the interior point algorithm and Lemke’s algorithm, are dissected in detail, offering insights into their applicability and effectiveness. The book explores modern applications of optimization theory, for instance, optimized image encryption, resource allocation, target tracking problems, deep learning, entropy optimization, etc. Ranging from gradient-based optimization algorithms to metaheuristic approaches such as particle swarm optimization, the book navigates through the intersection of optimization theory and deep learning, thereby unravelling new research perspectives in artificial intelligence, machine learning and other fields of modern science. Designed primarily for graduate students and researchers across a variety of disciplines such as mathematics, operations research, electrical and electronics engineering, computer science, robotics, deep learning, image processing and artificial intelligence, this book serves as a comprehensive resource for someone interested in exploring the multifaceted domain of mathematical optimization and its myriad applications.