Advertisement
| convex vs non convex optimization: Non-convex Optimization for Machine Learning Prateek Jain, Purushottam Kar, 2017-12-04 Non-convex Optimization for Machine Learning takes an in-depth look at the basics of non-convex optimization with applications to machine learning. It introduces the rich literature in this area, as well as equips the reader with the tools and techniques needed to apply and analyze simple but powerful procedures for non-convex problems. Non-convex Optimization for Machine Learning is as self-contained as possible while not losing focus of the main topic of non-convex optimization techniques. The monograph initiates the discussion with entire chapters devoted to presenting a tutorial-like treatment of basic concepts in convex analysis and optimization, as well as their non-convex counterparts. The monograph concludes with a look at four interesting applications in the areas of machine learning and signal processing, and exploring how the non-convex optimization techniques introduced earlier can be used to solve these problems. The monograph also contains, for each of the topics discussed, exercises and figures designed to engage the reader, as well as extensive bibliographic notes pointing towards classical works and recent advances. Non-convex Optimization for Machine Learning can be used for a semester-length course on the basics of non-convex optimization with applications to machine learning. On the other hand, it is also possible to cherry pick individual portions, such the chapter on sparse recovery, or the EM algorithm, for inclusion in a broader course. Several courses such as those in machine learning, optimization, and signal processing may benefit from the inclusion of such topics. |
| convex vs non convex optimization: Convex Optimization Stephen P. Boyd, Lieven Vandenberghe, 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 vs non convex optimization: Global Optimization with Non-Convex Constraints Roman G. Strongin, Yaroslav D. Sergeyev, 2013-11-09 Everything should be made as simple as possible, but not simpler. (Albert Einstein, Readers Digest, 1977) The modern practice of creating technical systems and technological processes of high effi.ciency besides the employment of new principles, new materials, new physical effects and other new solutions ( which is very traditional and plays the key role in the selection of the general structure of the object to be designed) also includes the choice of the best combination for the set of parameters (geometrical sizes, electrical and strength characteristics, etc.) concretizing this general structure, because the Variation of these parameters ( with the structure or linkage being already set defined) can essentially affect the objective performance indexes. The mathematical tools for choosing these best combinations are exactly what is this book about. With the advent of computers and the computer-aided design the pro bations of the selected variants are usually performed not for the real examples ( this may require some very expensive building of sample op tions and of the special installations to test them ), but by the analysis of the corresponding mathematical models. The sophistication of the mathematical models for the objects to be designed, which is the natu ral consequence of the raising complexity of these objects, greatly com plicates the objective performance analysis. Today, the main (and very often the only) available instrument for such an analysis is computer aided simulation of an object's behavior, based on numerical experiments with its mathematical model. |
| convex vs non convex optimization: Convex Optimization Theory Dimitri Bertsekas, 2009-06-01 An insightful, concise, and rigorous treatment of the basic theory of convex sets and functions in finite dimensions, and the analytical/geometrical foundations of convex optimization and duality theory. Convexity theory is first developed in a simple accessible manner, using easily visualized proofs. Then the focus shifts to a transparent geometrical line of analysis to develop the fundamental duality between descriptions of convex functions in terms of points, and in terms of hyperplanes. Finally, convexity theory and abstract duality are applied to problems of constrained optimization, Fenchel and conic duality, and game theory to develop the sharpest possible duality results within a highly visual geometric framework. This on-line version of the book, includes an extensive set of theoretical problems with detailed high-quality solutions, which significantly extend the range and value of the book. The book may be used as a text for a theoretical convex optimization course; the author has taught several variants of such a course at MIT and elsewhere over the last ten years. It may also be used as a supplementary source for nonlinear programming classes, and as a theoretical foundation for classes focused on convex optimization models (rather than theory). It is an excellent supplement to several of our books: Convex Optimization Algorithms (Athena Scientific, 2015), Nonlinear Programming (Athena Scientific, 2017), Network Optimization(Athena Scientific, 1998), Introduction to Linear Optimization (Athena Scientific, 1997), and Network Flows and Monotropic Optimization (Athena Scientific, 1998). |
| convex vs non convex optimization: Convex Analysis and Nonlinear Optimization Jonathan Borwein, Adrian S. Lewis, 2010-05-05 Optimization is a rich and thriving mathematical discipline, and the underlying theory of current computational optimization techniques grows ever more sophisticated. This book aims to provide a concise, accessible account of convex analysis and its applications and extensions, for a broad audience. Each section concludes with an often extensive set of optional exercises. This new edition adds material on semismooth optimization, as well as several new proofs. |
| convex vs non convex optimization: Algorithms for Convex Optimization Nisheeth K. Vishnoi, 2021-10-07 In the last few years, Algorithms for Convex Optimization have revolutionized algorithm design, both for discrete and continuous optimization problems. For problems like maximum flow, maximum matching, and submodular function minimization, the fastest algorithms involve essential methods such as gradient descent, mirror descent, interior point methods, and ellipsoid methods. The goal of this self-contained book is to enable researchers and professionals in computer science, data science, and machine learning to gain an in-depth understanding of these algorithms. The text emphasizes how to derive key algorithms for convex optimization from first principles and how to establish precise running time bounds. This modern text explains the success of these algorithms in problems of discrete optimization, as well as how these methods have significantly pushed the state of the art of convex optimization itself. |
| convex vs non convex optimization: Convex Optimization Sébastien Bubeck, 2015-11-12 This monograph presents the main complexity theorems in convex optimization and their corresponding algorithms. It begins with the fundamental theory of black-box optimization and proceeds to guide the reader through recent advances in structural optimization and stochastic optimization. The presentation of black-box optimization, strongly influenced by the seminal book by Nesterov, includes the analysis of cutting plane methods, as well as (accelerated) gradient descent schemes. Special attention is also given to non-Euclidean settings (relevant algorithms include Frank-Wolfe, mirror descent, and dual averaging), and discussing their relevance in machine learning. The text provides a gentle introduction to structural optimization with FISTA (to optimize a sum of a smooth and a simple non-smooth term), saddle-point mirror prox (Nemirovski's alternative to Nesterov's smoothing), and a concise description of interior point methods. In stochastic optimization it discusses stochastic gradient descent, mini-batches, random coordinate descent, and sublinear algorithms. It also briefly touches upon convex relaxation of combinatorial problems and the use of randomness to round solutions, as well as random walks based methods. |
| convex vs non convex optimization: Nonsmooth Optimization and Related Topics F.H. Clarke, Vladimir F. Dem'yanov, F. Giannessi, 2013-11-11 This volume contains the edited texts of the lect. nres presented at the International School of Mathematics devoted to Nonsmonth Optimization, held from . June 20 to July I, 1988. The site for the meeting was the Ettore ~Iajorana Centre for Sci entific Culture in Erice, Sicily. In the tradition of these meetings the main purpose was to give the state-of-the-art of an important and growing field of mathematics, and to stimulate interactions between finite-dimensional and infinite-dimensional op timization. The School was attended by approximately 80 people from 23 countries; in particular it was possible to have some distinguished lecturers from the SO\·iet Union, whose research institutions are here gratt-fnlly acknowledged. Besides the lectures, several seminars were delivered; a special s·~ssion was devoted to numerical computing aspects. The result was a broad exposure. gi ·. ring a deep knowledge of the present research tendencies in the field. We wish to express our appreciation to all the participants. Special mention 5hould be made of the Ettorc ;. . Iajorana Centre in Erice, which helped provide a stimulating and rewarding experience, and of its staff which was fundamental for the success of the meeting. j\, loreover, WP want to extend uur deep appreci |
| convex vs non convex optimization: Convex Analysis and Variational Problems Ivar Ekeland, Roger Temam, 1999-12-01 This book contains different developments of infinite dimensional convex programming in the context of convex analysis, including duality, minmax and Lagrangians, and convexification of nonconvex optimization problems in the calculus of variations (infinite dimension). It also includes the theory of convex duality applied to partial differential equations; no other reference presents this in a systematic way. The minmax theorems contained in this book have many useful applications, in particular the robust control of partial differential equations in finite time horizon. First published in English in 1976, this SIAM Classics in Applied Mathematics edition contains the original text along with a new preface and some additional references. |
| convex vs non convex optimization: Convex Analysis and Optimization Dimitri Bertsekas, Angelia Nedic, Asuman Ozdaglar, 2003-03-01 A uniquely pedagogical, insightful, and rigorous treatment of the analytical/geometrical foundations of optimization. The book provides a comprehensive development of convexity theory, and its rich applications in optimization, including duality, minimax/saddle point theory, Lagrange multipliers, and Lagrangian relaxation/nondifferentiable optimization. It is an excellent supplement to several of our books: Convex Optimization Theory (Athena Scientific, 2009), Convex Optimization Algorithms (Athena Scientific, 2015), Nonlinear Programming (Athena Scientific, 2016), Network Optimization (Athena Scientific, 1998), and Introduction to Linear Optimization (Athena Scientific, 1997). Aside from a thorough account of convex analysis and optimization, the book aims to restructure the theory of the subject, by introducing several novel unifying lines of analysis, including: 1) A unified development of minimax theory and constrained optimization duality as special cases of duality between two simple geometrical problems. 2) A unified development of conditions for existence of solutions of convex optimization problems, conditions for the minimax equality to hold, and conditions for the absence of a duality gap in constrained optimization. 3) A unification of the major constraint qualifications allowing the use of Lagrange multipliers for nonconvex constrained optimization, using the notion of constraint pseudonormality and an enhanced form of the Fritz John necessary optimality conditions. Among its features the book: a) Develops rigorously and comprehensively the theory of convex sets and functions, in the classical tradition of Fenchel and Rockafellar b) Provides a geometric, highly visual treatment of convex and nonconvex optimization problems, including existence of solutions, optimality conditions, Lagrange multipliers, and duality c) Includes an insightful and comprehensive presentation of minimax theory and zero sum games, and its connection with duality d) Describes dual optimization, the associated computational methods, including the novel incremental subgradient methods, and applications in linear, quadratic, and integer programming e) Contains many examples, illustrations, and exercises with complete solutions (about 200 pages) posted at the publisher's web site http://www.athenasc.com/convexity.html |
| convex vs non convex optimization: Convex Optimization Algorithms Dimitri Bertsekas, 2015-02-01 This book provides a comprehensive and accessible presentation of algorithms for solving convex optimization problems. It relies on rigorous mathematical analysis, but also aims at an intuitive exposition that makes use of visualization where possible. This is facilitated by the extensive use of analytical and algorithmic concepts of duality, which by nature lend themselves to geometrical interpretation. The book places particular emphasis on modern developments, and their widespread applications in fields such as large-scale resource allocation problems, signal processing, and machine learning. The book is aimed at students, researchers, and practitioners, roughly at the first year graduate level. It is similar in style to the author's 2009Convex Optimization Theory book, but can be read independently. The latter book focuses on convexity theory and optimization duality, while the present book focuses on algorithmic issues. The two books share notation, and together cover the entire finite-dimensional convex optimization methodology. To facilitate readability, the statements of definitions and results of the theory book are reproduced without proofs in Appendix B. |
| convex vs non convex optimization: Convex Optimization & Euclidean Distance Geometry Jon Dattorro, 2005 The study of Euclidean distance matrices (EDMs) fundamentally asks what can be known geometrically given onlydistance information between points in Euclidean space. Each point may represent simply locationor, abstractly, any entity expressible as a vector in finite-dimensional Euclidean space.The answer to the question posed is that very much can be known about the points;the mathematics of this combined study of geometry and optimization is rich and deep.Throughout we cite beacons of historical accomplishment.The application of EDMs has already proven invaluable in discerning biological molecular conformation.The emerging practice of localization in wireless sensor networks, the global positioning system (GPS), and distance-based pattern recognitionwill certainly simplify and benefit from this theory.We study the pervasive convex Euclidean bodies and their various representations.In particular, we make convex polyhedra, cones, and dual cones more visceral through illustration, andwe study the geometric relation of polyhedral cones to nonorthogonal bases biorthogonal expansion.We explain conversion between halfspace- and vertex-descriptions of convex cones,we provide formulae for determining dual cones,and we show how classic alternative systems of linear inequalities or linear matrix inequalities and optimality conditions can be explained by generalized inequalities in terms of convex cones and their duals.The conic analogue to linear independence, called conic independence, is introducedas a new tool in the study of classical cone theory; the logical next step in the progression:linear, affine, conic.Any convex optimization problem has geometric interpretation.This is a powerful attraction: the ability to visualize geometry of an optimization problem.We provide tools to make visualization easier.The concept of faces, extreme points, and extreme directions of convex Euclidean bodiesis explained here, crucial to understanding convex optimization.The convex cone of positive semidefinite matrices, in particular, is studied in depth.We mathematically interpret, for example,its inverse image under affine transformation, and we explainhow higher-rank subsets of its boundary united with its interior are convex.The Chapter on Geometry of convex functions,observes analogies between convex sets and functions:The set of all vector-valued convex functions is a closed convex cone.Included among the examples in this chapter, we show how the real affinefunction relates to convex functions as the hyperplane relates to convex sets.Here, also, pertinent results formultidimensional convex functions are presented that are largely ignored in the literature;tricks and tips for determining their convexityand discerning their geometry, particularly with regard to matrix calculus which remains largely unsystematizedwhen compared with the traditional practice of ordinary calculus.Consequently, we collect some results of matrix differentiation in the appendices.The Euclidean distance matrix (EDM) is studied,its properties and relationship to both positive semidefinite and Gram matrices.We relate the EDM to the four classical axioms of the Euclidean metric;thereby, observing the existence of an infinity of axioms of the Euclidean metric beyondthe triangle inequality. We proceed byderiving the fifth Euclidean axiom and then explain why furthering this endeavoris inefficient because the ensuing criteria (while describing polyhedra)grow linearly in complexity and number.Some geometrical problems solvable via EDMs,EDM problems posed as convex optimization, and methods of solution arepresented;\eg, we generate a recognizable isotonic map of the United States usingonly comparative distance information (no distance information, only distance inequalities).We offer a new proof of the classic Schoenberg criterion, that determines whether a candidate matrix is an EDM. Our proofrelies on fundamental geometry; assuming, any EDM must correspond to a list of points contained in some polyhedron(possibly at its vertices) and vice versa.It is not widely known that the Schoenberg criterion implies nonnegativity of the EDM entries; proved here.We characterize the eigenvalues of an EDM matrix and then devisea polyhedral cone required for determining membership of a candidate matrix(in Cayley-Menger form) to the convex cone of Euclidean distance matrices (EDM cone); \ie,a candidate is an EDM if and only if its eigenspectrum belongs to a spectral cone for EDM^N.We will see spectral cones are not unique.In the chapter EDM cone, we explain the geometric relationship betweenthe EDM cone, two positive semidefinite cones, and the elliptope.We illustrate geometric requirements, in particular, for projection of a candidate matrixon a positive semidefinite cone that establish its membership to the EDM cone. The faces of the EDM cone are described,but still open is the question whether all its faces are exposed as they are for the positive semidefinite cone.The classic Schoenberg criterion, relating EDM and positive semidefinite cones, isrevealed to be a discretized membership relation (a generalized inequality, a new Farkas''''''''-like lemma)between the EDM cone and its ordinary dual. A matrix criterion for membership to the dual EDM cone is derived thatis simpler than the Schoenberg criterion.We derive a new concise expression for the EDM cone and its dual involvingtwo subspaces and a positive semidefinite cone.Semidefinite programming is reviewedwith particular attention to optimality conditionsof prototypical primal and dual conic programs,their interplay, and the perturbation method of rank reduction of optimal solutions(extant but not well-known).We show how to solve a ubiquitous platonic combinatorial optimization problem from linear algebra(the optimal Boolean solution x to Ax=b)via semidefinite program relaxation.A three-dimensional polyhedral analogue for the positive semidefinite cone of 3X3 symmetricmatrices is introduced; a tool for visualizing in 6 dimensions.In EDM proximitywe explore methods of solution to a few fundamental and prevalentEuclidean distance matrix proximity problems; the problem of finding that Euclidean distance matrix closestto a given matrix in the Euclidean sense.We pay particular attention to the problem when compounded with rank minimization.We offer a new geometrical proof of a famous result discovered by Eckart \& Young in 1936 regarding Euclideanprojection of a point on a subset of the positive semidefinite cone comprising all positive semidefinite matriceshaving rank not exceeding a prescribed limit rho.We explain how this problem is transformed to a convex optimization for any rank rho. |
| convex vs non convex optimization: Lectures on Convex Optimization Yurii Nesterov, 2018-11-19 This book provides a comprehensive, modern introduction to convex optimization, a field that is becoming increasingly important in applied mathematics, economics and finance, engineering, and computer science, notably in data science and machine learning. Written by a leading expert in the field, this book includes recent advances in the algorithmic theory of convex optimization, naturally complementing the existing literature. It contains a unified and rigorous presentation of the acceleration techniques for minimization schemes of first- and second-order. It provides readers with a full treatment of the smoothing technique, which has tremendously extended the abilities of gradient-type methods. Several powerful approaches in structural optimization, including optimization in relative scale and polynomial-time interior-point methods, are also discussed in detail. Researchers in theoretical optimization as well as professionals working on optimization problems will find this book very useful. It presents many successful examples of how to develop very fast specialized minimization algorithms. Based on the author’s lectures, it can naturally serve as the basis for introductory and advanced courses in convex optimization for students in engineering, economics, computer science and mathematics. |
| convex vs non convex optimization: Recent Advances in Optimization and its Applications in Engineering Moritz Diehl, Francois Glineur, Elias Jarlebring, Wim Michiels, 2010-09-21 Mathematical optimization encompasses both a rich and rapidly evolving body of fundamental theory, and a variety of exciting applications in science and engineering. The present book contains a careful selection of articles on recent advances in optimization theory, numerical methods, and their applications in engineering. It features in particular new methods and applications in the fields of optimal control, PDE-constrained optimization, nonlinear optimization, and convex optimization. The authors of this volume took part in the 14th Belgian-French-German Conference on Optimization (BFG09) organized in Leuven, Belgium, on September 14-18, 2009. The volume contains a selection of reviewed articles contributed by the conference speakers as well as three survey articles by plenary speakers and two papers authored by the winners of the best talk and best poster prizes awarded at BFG09. Researchers and graduate students in applied mathematics, computer science, and many branches of engineering will find in this book an interesting and useful collection of recent ideas on the methods and applications of optimization. |
| convex vs non convex optimization: Interior-point Polynomial Algorithms in Convex Programming Yurii Nesterov, Arkadii Nemirovskii, 1994-01-01 Specialists working in the areas of optimization, mathematical programming, or control theory will find this book invaluable for studying interior-point methods for linear and quadratic programming, polynomial-time methods for nonlinear convex programming, and efficient computational methods for control problems and variational inequalities. A background in linear algebra and mathematical programming is necessary to understand the book. The detailed proofs and lack of numerical examples might suggest that the book is of limited value to the reader interested in the practical aspects of convex optimization, but nothing could be further from the truth. An entire chapter is devoted to potential reduction methods precisely because of their great efficiency in practice. |
| convex vs non convex optimization: Modern Nonconvex Nondifferentiable Optimization Ying Cui, Jong-Shi Pang, 2022 This monograph serves present and future needs where nonconvexity and nondifferentiability are inevitably present in the faithful modeling of real-world applications of optimization-- |
| convex vs non convex optimization: Introduction to Global Optimization R. Horst, Panos M. Pardalos, Nguyen Van Thoai, 2000-12-31 A textbook for an undergraduate course in mathematical programming for students with a knowledge of elementary real analysis, linear algebra, and classical linear programming (simple techniques). Focuses on the computation and characterization of global optima of nonlinear functions, rather than the locally optimal solutions addressed by most books on optimization. Incorporates the theoretical, algorithmic, and computational advances of the past three decades that help solve globally multi-extreme problems in the mathematical modeling of real world systems. Annotation copyright by Book News, Inc., Portland, OR |
| convex vs non convex optimization: Optimality Conditions in Convex Optimization Anulekha Dhara, Joydeep Dutta, 2011-10-17 Optimality Conditions in Convex Optimization explores an important and central issue in the field of convex optimization: optimality conditions. It brings together the most important and recent results in this area that have been scattered in the literature—notably in the area of convex analysis—essential in developing many of the important results in this book, and not usually found in conventional texts. Unlike other books on convex optimization, which usually discuss algorithms along with some basic theory, the sole focus of this book is on fundamental and advanced convex optimization theory. Although many results presented in the book can also be proved in infinite dimensions, the authors focus on finite dimensions to allow for much deeper results and a better understanding of the structures involved in a convex optimization problem. They address semi-infinite optimization problems; approximate solution concepts of convex optimization problems; and some classes of non-convex problems which can be studied using the tools of convex analysis. They include examples wherever needed, provide details of major results, and discuss proofs of the main results. |
| convex vs non convex optimization: Convex Functions and Optimization Methods on Riemannian Manifolds C. Udriste, 2013-11-11 The object of this book is to present the basic facts of convex functions, standard dynamical systems, descent numerical algorithms and some computer programs on Riemannian manifolds in a form suitable for applied mathematicians, scientists and engineers. It contains mathematical information on these subjects and applications distributed in seven chapters whose topics are close to my own areas of research: Metric properties of Riemannian manifolds, First and second variations of the p-energy of a curve; Convex functions on Riemannian manifolds; Geometric examples of convex functions; Flows, convexity and energies; Semidefinite Hessians and applications; Minimization of functions on Riemannian manifolds. All the numerical algorithms, computer programs and the appendices (Riemannian convexity of functions f:R ~ R, Descent methods on the Poincare plane, Descent methods on the sphere, Completeness and convexity on Finsler manifolds) constitute an attempt to make accesible to all users of this book some basic computational techniques and implementation of geometric structures. To further aid the readers,this book also contains a part of the folklore about Riemannian geometry, convex functions and dynamical systems because it is unfortunately nowhere to be found in the same context; existing textbooks on convex functions on Euclidean spaces or on dynamical systems do not mention what happens in Riemannian geometry, while the papers dealing with Riemannian manifolds usually avoid discussing elementary facts. Usually a convex function on a Riemannian manifold is a real valued function whose restriction to every geodesic arc is convex. |
| convex vs non convex optimization: Finite Dimensional Convexity and Optimization Monique Florenzano, Cuong Le Van, 2012-12-06 This book discusses convex analysis, the basic underlying structure of argumentation in economic theory. Convex analysis is also common to the optimization of problems encountered in many applications. The text is aimed at senior undergraduate students, graduate students, and specialists of mathematical programming who are undertaking research into applied mathematics and economics. The text consists of a systematic development in eight chapters, and contains exercises. The book is appropriate as a class text or for self-study. |
| convex vs non convex optimization: Mixed Integer Nonlinear Programming Jon Lee, Sven Leyffer, 2011-12-02 Many engineering, operations, and scientific applications include a mixture of discrete and continuous decision variables and nonlinear relationships involving the decision variables that have a pronounced effect on the set of feasible and optimal solutions. Mixed-integer nonlinear programming (MINLP) problems combine the numerical difficulties of handling nonlinear functions with the challenge of optimizing in the context of nonconvex functions and discrete variables. MINLP is one of the most flexible modeling paradigms available for optimization; but because its scope is so broad, in the most general cases it is hopelessly intractable. Nonetheless, an expanding body of researchers and practitioners — including chemical engineers, operations researchers, industrial engineers, mechanical engineers, economists, statisticians, computer scientists, operations managers, and mathematical programmers — are interested in solving large-scale MINLP instances. |
| convex vs non convex optimization: Generalized Convexity and Optimization Alberto Cambini, Laura Martein, 2008-10-14 The authors have written a rigorous yet elementary and self-contained book to present, in a unified framework, generalized convex functions. The book also includes numerous exercises and two appendices which list the findings consulted. |
| convex vs non convex optimization: Nonlinear Optimization Andrzej Ruszczynski, 2011-09-19 Optimization is one of the most important areas of modern applied mathematics, with applications in fields from engineering and economics to finance, statistics, management science, and medicine. While many books have addressed its various aspects, Nonlinear Optimization is the first comprehensive treatment that will allow graduate students and researchers to understand its modern ideas, principles, and methods within a reasonable time, but without sacrificing mathematical precision. Andrzej Ruszczynski, a leading expert in the optimization of nonlinear stochastic systems, integrates the theory and the methods of nonlinear optimization in a unified, clear, and mathematically rigorous fashion, with detailed and easy-to-follow proofs illustrated by numerous examples and figures. The book covers convex analysis, the theory of optimality conditions, duality theory, and numerical methods for solving unconstrained and constrained optimization problems. It addresses not only classical material but also modern topics such as optimality conditions and numerical methods for problems involving nondifferentiable functions, semidefinite programming, metric regularity and stability theory of set-constrained systems, and sensitivity analysis of optimization problems. Based on a decade's worth of notes the author compiled in successfully teaching the subject, this book will help readers to understand the mathematical foundations of the modern theory and methods of nonlinear optimization and to analyze new problems, develop optimality theory for them, and choose or construct numerical solution methods. It is a must for anyone seriously interested in optimization. |
| convex vs non convex optimization: Optimization for Machine Learning Suvrit Sra, Sebastian Nowozin, Stephen J. Wright, 2012 An up-to-date account of the interplay between optimization and machine learning, accessible to students and researchers in both communities. The interplay between optimization and machine learning is one of the most important developments in modern computational science. Optimization formulations and methods are proving to be vital in designing algorithms to extract essential knowledge from huge volumes of data. Machine learning, however, is not simply a consumer of optimization technology but a rapidly evolving field that is itself generating new optimization ideas. This book captures the state of the art of the interaction between optimization and machine learning in a way that is accessible to researchers in both fields. Optimization approaches have enjoyed prominence in machine learning because of their wide applicability and attractive theoretical properties. The increasing complexity, size, and variety of today's machine learning models call for the reassessment of existing assumptions. This book starts the process of reassessment. It describes the resurgence in novel contexts of established frameworks such as first-order methods, stochastic approximations, convex relaxations, interior-point methods, and proximal methods. It also devotes attention to newer themes such as regularized optimization, robust optimization, gradient and subgradient methods, splitting techniques, and second-order methods. Many of these techniques draw inspiration from other fields, including operations research, theoretical computer science, and subfields of optimization. The book will enrich the ongoing cross-fertilization between the machine learning community and these other fields, and within the broader optimization community. |
| convex vs non convex optimization: Linear and Convex Optimization Michael H. Veatch, 2020-12-16 Discover the practical impacts of current methods of optimization with this approachable, one-stop resource Linear and Convex Optimization: A Mathematical Approach delivers a concise and unified treatment of optimization with a focus on developing insights in problem structure, modeling, and algorithms. Convex optimization problems are covered in detail because of their many applications and the fast algorithms that have been developed to solve them. Experienced researcher and undergraduate teacher Mike Veatch presents the main algorithms used in linear, integer, and convex optimization in a mathematical style with an emphasis on what makes a class of problems practically solvable and developing insight into algorithms geometrically. Principles of algorithm design and the speed of algorithms are discussed in detail, requiring no background in algorithms. The book offers a breadth of recent applications to demonstrate the many areas in which optimization is successfully and frequently used, while the process of formulating optimization problems is addressed throughout. Linear and Convex Optimization contains a wide variety of features, including: Coverage of current methods in optimization in a style and level that remains appealing and accessible for mathematically trained undergraduates Enhanced insights into a few algorithms, instead of presenting many algorithms in cursory fashion An emphasis on the formulation of large, data-driven optimization problems Inclusion of linear, integer, and convex optimization, covering many practically solvable problems using algorithms that share many of the same concepts Presentation of a broad range of applications to fields like online marketing, disaster response, humanitarian development, public sector planning, health delivery, manufacturing, and supply chain management Ideal for upper level undergraduate mathematics majors with an interest in practical applications of mathematics, this book will also appeal to business, economics, computer science, and operations research majors with at least two years of mathematics training. |
| convex vs non convex optimization: Geometric Algorithms and Combinatorial Optimization Martin Grötschel, Laszlo Lovasz, Alexander Schrijver, 2012-12-06 Historically, there is a close connection between geometry and optImization. This is illustrated by methods like the gradient method and the simplex method, which are associated with clear geometric pictures. In combinatorial optimization, however, many of the strongest and most frequently used algorithms are based on the discrete structure of the problems: the greedy algorithm, shortest path and alternating path methods, branch-and-bound, etc. In the last several years geometric methods, in particular polyhedral combinatorics, have played a more and more profound role in combinatorial optimization as well. Our book discusses two recent geometric algorithms that have turned out to have particularly interesting consequences in combinatorial optimization, at least from a theoretical point of view. These algorithms are able to utilize the rich body of results in polyhedral combinatorics. The first of these algorithms is the ellipsoid method, developed for nonlinear programming by N. Z. Shor, D. B. Yudin, and A. S. NemirovskiI. It was a great surprise when L. G. Khachiyan showed that this method can be adapted to solve linear programs in polynomial time, thus solving an important open theoretical problem. While the ellipsoid method has not proved to be competitive with the simplex method in practice, it does have some features which make it particularly suited for the purposes of combinatorial optimization. The second algorithm we discuss finds its roots in the classical geometry of numbers, developed by Minkowski. This method has had traditionally deep applications in number theory, in particular in diophantine approximation. |
| convex vs non convex optimization: Optimization Models Giuseppe C. Calafiore, Laurent El Ghaoui, 2014-10-31 This accessible textbook demonstrates how to recognize, simplify, model and solve optimization problems - and apply these principles to new projects. |
| convex vs non convex optimization: First-order and Stochastic Optimization Methods for Machine Learning Guanghui Lan, 2020-05-15 This book covers not only foundational materials but also the most recent progresses made during the past few years on the area of machine learning algorithms. In spite of the intensive research and development in this area, there does not exist a systematic treatment to introduce the fundamental concepts and recent progresses on machine learning algorithms, especially on those based on stochastic optimization methods, randomized algorithms, nonconvex optimization, distributed and online learning, and projection free methods. This book will benefit the broad audience in the area of machine learning, artificial intelligence and mathematical programming community by presenting these recent developments in a tutorial style, starting from the basic building blocks to the most carefully designed and complicated algorithms for machine learning. |
| convex vs non convex optimization: Online Learning and Online Convex Optimization Shai Shalev-Shwartz, 2012 Online Learning and Online Convex Optimization is a modern overview of online learning. Its aim is to provide the reader with a sense of some of the interesting ideas and in particular to underscore the centrality of convexity in deriving efficient online learning algorithms. |
| convex vs non convex optimization: Convex Optimization in Signal Processing and Communications Daniel P. Palomar, Yonina C. Eldar, 2010 Leading experts provide the theoretical underpinnings of the subject plus tutorials on a wide range of applications, from automatic code generation to robust broadband beamforming. Emphasis on cutting-edge research and formulating problems in convex form make this an ideal textbook for advanced graduate courses and a useful self-study guide. |
| convex vs non convex optimization: Handbook of Convex Optimization Methods in Imaging Science Vishal Monga, 2017-10-27 This book covers recent advances in image processing and imaging sciences from an optimization viewpoint, especially convex optimization with the goal of designing tractable algorithms. Throughout the handbook, the authors introduce topics on the most key aspects of image acquisition and processing that are based on the formulation and solution of novel optimization problems. The first part includes a review of the mathematical methods and foundations required, and covers topics in image quality optimization and assessment. The second part of the book discusses concepts in image formation and capture from color imaging to radar and multispectral imaging. The third part focuses on sparsity constrained optimization in image processing and vision and includes inverse problems such as image restoration and de-noising, image classification and recognition and learning-based problems pertinent to image understanding. Throughout, convex optimization techniques are shown to be a critically important mathematical tool for imaging science problems and applied extensively. Convex Optimization Methods in Imaging Science is the first book of its kind and will appeal to undergraduate and graduate students, industrial researchers and engineers and those generally interested in computational aspects of modern, real-world imaging and image processing problems. |
| convex vs non convex optimization: Convex Optimization of Power Systems Joshua Adam Taylor, 2015-02-12 A mathematically rigorous guide to convex optimization for power systems engineering. |
| convex vs non convex optimization: Primal-dual Interior-Point Methods Stephen J. Wright, 1997-01-01 In the past decade, primal-dual algorithms have emerged as the most important and useful algorithms from the interior-point class. This book presents the major primal-dual algorithms for linear programming in straightforward terms. A thorough description of the theoretical properties of these methods is given, as are a discussion of practical and computational aspects and a summary of current software. This is an excellent, timely, and well-written work. The major primal-dual algorithms covered in this book are path-following algorithms (short- and long-step, predictor-corrector), potential-reduction algorithms, and infeasible-interior-point algorithms. A unified treatment of superlinear convergence, finite termination, and detection of infeasible problems is presented. Issues relevant to practical implementation are also discussed, including sparse linear algebra and a complete specification of Mehrotra's predictor-corrector algorithm. Also treated are extensions of primal-dual algorithms to more general problems such as monotone complementarity, semidefinite programming, and general convex programming problems. |
| convex vs non convex optimization: Introductory Lectures on Convex Optimization Yurii Nesterov, 2003-12-31 It was in the middle of the 1980s, when the seminal paper by Kar markar opened a new epoch in nonlinear optimization. The importance of this paper, containing a new polynomial-time algorithm for linear op timization problems, was not only in its complexity bound. At that time, the most surprising feature of this algorithm was that the theoretical pre diction of its high efficiency was supported by excellent computational results. This unusual fact dramatically changed the style and direc tions of the research in nonlinear optimization. Thereafter it became more and more common that the new methods were provided with a complexity analysis, which was considered a better justification of their efficiency than computational experiments. In a new rapidly develop ing field, which got the name polynomial-time interior-point methods, such a justification was obligatory. Afteralmost fifteen years of intensive research, the main results of this development started to appear in monographs [12, 14, 16, 17, 18, 19]. Approximately at that time the author was asked to prepare a new course on nonlinear optimization for graduate students. The idea was to create a course which would reflect the new developments in the field. Actually, this was a major challenge. At the time only the theory of interior-point methods for linear optimization was polished enough to be explained to students. The general theory of self-concordant functions had appeared in print only once in the form of research monograph [12]. |
| convex vs non convex optimization: Communications, Computation, Control, and Signal Processing Arogyaswami Paulraj, Vwani Roychowdhury, Charles D. Schaper, 2012-12-06 A. Paulraj*, V. Roychowdhury**, and C. Schaper* * Dept. of Electrical Engineering, Stanford University ** Dept. of Electrical Engineering, UCLA Innumerable conferences are held around the world on the subjects of commu nications, computation, control and signal processing, and on their numerous subdisciplines. Therefore one might not envision a coherent conference encom passing all these areas. However, such an event did take place June 22-26, 1995, at an international symposium held at Stanford University to celebrate Professor Thomas Kailath's sixtieth birthday and to honor the notable con tributions made by him and his students and associates. The depth of these contributions was evident from the participation of so many leading figures in each of these fields. Over the five days of the meeting, there were about 200 at tendees, from eighteen countries, more than twenty government and industrial organizations, and various engineering, mathematics and statistics faculties at nearly 50 different academic institutions. They came not only to celebrate but also to learn and to ponder the threads and the connections that Professor Kailath has discovered and woven among so many apparently disparate areas. The organizers received many comments about the richness of the occasion. A distinguished academic wrote of the conference being the single most rewarding professional event of my life. The program is summarized in Table 1. 1; a letter of reflections by Dr. C. Rohrs appears a little later. |
| convex vs non convex optimization: Beyond the Worst-Case Analysis of Algorithms Tim Roughgarden, 2021-01-14 Introduces exciting new methods for assessing algorithms for problems ranging from clustering to linear programming to neural networks. |
| convex vs non convex optimization: Introductory Lectures on Convex Optimization Y. Nesterov, 2013-12-01 It was in the middle of the 1980s, when the seminal paper by Kar markar opened a new epoch in nonlinear optimization. The importance of this paper, containing a new polynomial-time algorithm for linear op timization problems, was not only in its complexity bound. At that time, the most surprising feature of this algorithm was that the theoretical pre diction of its high efficiency was supported by excellent computational results. This unusual fact dramatically changed the style and direc tions of the research in nonlinear optimization. Thereafter it became more and more common that the new methods were provided with a complexity analysis, which was considered a better justification of their efficiency than computational experiments. In a new rapidly develop ing field, which got the name polynomial-time interior-point methods, such a justification was obligatory. Afteralmost fifteen years of intensive research, the main results of this development started to appear in monographs [12, 14, 16, 17, 18, 19]. Approximately at that time the author was asked to prepare a new course on nonlinear optimization for graduate students. The idea was to create a course which would reflect the new developments in the field. Actually, this was a major challenge. At the time only the theory of interior-point methods for linear optimization was polished enough to be explained to students. The general theory of self-concordant functions had appeared in print only once in the form of research monograph [12]. |
| convex vs non convex optimization: Convex Analysis Ralph Tyrell Rockafellar, 2015-04-29 Available for the first time in paperback, R. Tyrrell Rockafellar's classic study presents readers with a coherent branch of nonlinear mathematical analysis that is especially suited to the study of optimization problems. Rockafellar's theory differs from classical analysis in that differentiability assumptions are replaced by convexity assumptions. The topics treated in this volume include: systems of inequalities, the minimum or maximum of a convex function over a convex set, Lagrange multipliers, minimax theorems and duality, as well as basic results about the structure of convex sets and the continuity and differentiability of convex functions and saddle- functions. This book has firmly established a new and vital area not only for pure mathematics but also for applications to economics and engineering. A sound knowledge of linear algebra and introductory real analysis should provide readers with sufficient background for this book. There is also a guide for the reader who may be using the book as an introduction, indicating which parts are essential and which may be skipped on a first reading. |
| convex vs non convex optimization: Multi-Period Trading Via Convex Optimization Stephen Boyd, Enzo Busseti, Steven Diamond, Ronald N. Kahn, Kwangmoo Koh, Peter Nystrup, Jan Spethmann, 2017-07-28 This monograph collects in one place the basic definitions, a careful description of the model, and discussion of how convex optimization can be used in multi-period trading, all in a common notation and framework. |
| convex vs non convex optimization: Learning with Submodular Functions Francis Bach, Now Publishers, 2013 Submodular functions are relevant to machine learning for at least two reasons: (1) some problems may be expressed directly as the optimization of submodular functions and (2) the Lovász extension of submodular functions provides a useful set of regularization functions for supervised and unsupervised learning. In this monograph, we present the theory of submodular functions from a convex analysis perspective, presenting tight links between certain polyhedra, combinatorial optimization and convex optimization problems. In particular, we show how submodular function minimization is equivalent to solving a wide variety of convex optimization problems. This allows the derivation of new efficient algorithms for approximate and exact submodular function minimization with theoretical guarantees and good practical performance. By listing many examples of submodular functions, we review various applications to machine learning, such as clustering, experimental design, sensor placement, graphical model structure learning or subset selection, as well as a family of structured sparsity-inducing norms that can be derived and used from submodular functions. |
Atlas Sales Intelligence and Pipeline Management | Convex
Convex Atlas combines proprietary data sources into a single property-centric view, enabling teams to find qualified properties and the right decision maker, in the office or on the go.
#1 Sales Intelligence Software for Commercial Services | Convex
Convex gives you up-to-date data on each property, with names, emails, and mobile numbers of key decision makers—giving your sales team more ammo for relevant outreach. Learn More …
About Our Revenue Acceleration Software Team | Convex
Convex provides a secure platform designed specially for the commercial services industry that leverages a powerful combination of property-based intelligence and easy-to-use applications, …
Software for Commercial Service Teams | Convex
Leading commercial service providers trust Convex to uncover new leads and grow existing accounts. We’ll show you how.
HVAC Sales Team Software Solution | Convex
With Convex’s HVAC Sales Software, leading service providers can find and win high-margin business in record time. Request a demo to learn more.
Facilities & Janitorial Sales Intelligence Solutions | Convex
Learn how leading facilities service providers partner with Convex to find and win high-margin business in record time. Request a demo today to learn more.
Cold Call Ratio Calculator for Commercial Sales | Convex
Cold Call Ratio Calculator for Commercial Sales | Convex Calculate the number, value, and more of cold calls based on your strengths. Download our free spreadsheet for commercial services …
Convex Data
Convex offers actionable, reliable intelligence, designed exclusively for commercial services businesses. Learn more about the breadth and quality of our data.
Schedule a Demo | Convex
Schedule a demo today of our Commercial Services Platform and see how Convex can make your team more efficient and increase your margins.
Market Intelligence for Roofing Sales Teams | Convex
See Convex in action. Leading roofing and solar teams trust Convex to uncover new leads and grow existing accounts. We’ll show you how.
Convex Optimization - Stanford University
Nonlinear optimization traditional techniques for general nonconvex problems involve compromises local optimization methods (nonlinear programming) find a point that minimizesf …
Machine learning mathematical optimization techniques for …
This section categorizes optimization techniques into convex vs. non-convex optimization, gradient-based methods, and gradient-free approaches[3]. 2.1. Convex vs. Non-Convex …
Analysis, Convexity, and Optimization - Columbia University
Contents I Introduction 1 1 Some Examples 2 1.1 The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Examples in Several Variables ...
Adaptive Methods and Non-convex Optimization
•Yes, non-convex optimization is at least NP-hard •Can encode most problems as non-convex optimization problems •Example: subset sum problem •Given a set of integers, is there a non …
Disciplined Convex-Concave Programming - Stanford University
The convex optimization modeling systems YALMIP [34], CVX [35], CVXPY [36], and Convex.jl [37] use DCP to verify problem convexity and automatically convert convex programs into …
Nonlinear Optimization: Algorithms and Models - Princeton …
Dec 12, 2005 · 8. Convex vs. Nonconvex Optimization Probs Nonlinear Programming (NLP) minimize f(x) subject to h i(x)= 0, i ∈ E, h i(x)≥ 0, i ∈ I. NLP is convex if • h i ’s in equality …
Non-Smooth Optimization - University of British Columbia
Smoothing Approximations of Non-Smooth Functions Smoothing: replace non-smooth f with smooth f . Apply a fast method for smooth optimization. Smooth approximation to the absolute …
Math 408A: Non-Linear Optimization - University of …
Math 408A: Non-Linear Optimization Introduction Professor James Burke Math Dept, University of Washington ... Outline What is non-linear programming? Problem Types Convex …
Convex vs Non-Convex Estimators for Regression and …
Learning (MKL), boils down to solving convex optimization problems. On the other hand, a Bayesian approach commonly known as Sparse Bayesian Learning (SBL), a version of which …
srihari@cedar.buffalo
Deep Learning Convex vs Non-convex Srihari •Convex methods: –Converge from any initial parameters –Robust--but can encounter numerical problems •SGD with non-convex: …
Nonsmooth, Nonconvex Optimization - New York University
Nonsmooth, Nonconvex Optimization Algorithms and Examples Michael L. Overton Courant Institute of Mathematical Sciences New York University Convex and Nonsmooth Optimization …
COMPUTATIONAL OPTIMIZATION - DNSC 8392 - UMD
capacity allocation problems. These problems can take multiple forms (single- vs. multi-objective; deterministic vs. stochastic; continuous vs. integer; convex vs. non-convex, etc.). Despite their …
Issues in Non-Convex Optimization - MIT OpenCourseWare
Then ¯x is strict local minimum of NLP. q.e.d. 3.1 Algorithm Issues • It is rare that an algorithm for NLP will compute a global minimum. • It is more usual for an algorithm to try to compute a …
1 Theory of convex functions - Princeton University
lecture, we shift our focus to the other important player in convex optimization, namely, convex functions. Here are some of the topics that we will touch upon: Convex, concave, strictly …
5: Convex Optimization - Columbia University
Convex optimization minimize f 0(x) subject to g i(x) ≤0, i= 1,...,m aT j x= b, j= 1,...,p •x∈Rn: decision variables •f 0: Rn→R: convex objective function •g i: Rn→R: convex inequality …
Non-Convex Optimization - Department of Computer Science
Non-convex SGD: A Systems Perspective •It’s exactly the same as the convex case! •The hardware doesn’t care whether our gradients are from a convex function or not •This means …
signSGD: Compressed Optimisation for Non-Convex Problems
on convex problems, where local information in the gradi-ent tells you global information about the direction to the minimum. Whilst elegant, this theory is less relevant for modern problems in …
arXiv:1712.07897v1 [stat.ML] 21 Dec 2017
Dec 22, 2017 · as possible while not losing focus of the main topic of non-convex optimization techniques. Consequently, we have devoted entire sections to present a tutorial-like treatment …
Short Course Robust Optimization and Machine Learning
Global vs. local optima Convex problems Software Non-convex problems Non-convex problems References Optimization problem A standard form An optimization problem is a problem of the …
SequentialConvexProgramming - Stanford University
in variables λ 0, which is a convex optimization problem. While there may be other possible dual problems—which is common in non-convex optimization—solving this problem provides the …
princeton univ. F’13 cos 521: Advanced Algorithm Design
For convex optimization it gives the global optimum under fairly general conditions. For nonconvex optimization it arrives at a local optimum. Figure 1: For nonconvex functions, a …
Non-Convex Optimization in ML - cvut.cz
Saddle point or local optima? 𝒇 ≈𝒇 + ∇𝑥𝑓 , − + − 𝑻∇2𝑓 ( − ) •Local minima: critical point with ∇2𝑓 >0 •Non degenerate saddle point: critical point where ∇2𝑓 has strictly positive and negative eigenvalues …
Statistical machine learning and convex optimization - École …
2. Classical methods for convex optimization • Smooth optimization (gradient descent, Newton method) • Non-smooth optimization (subgradient descent) • Proximal methods 3. Non-smooth …
Statistical machine learning and convex optimization - ENS
2. Classical methods for convex optimization • Smooth optimization (gradient descent, Newton method) • Non-smooth optimization (subgradient descent) • Proximal methods 3. Non-smooth …
Optimization 1: Gradient Descent - University of Washington
2.For non-convex problems, often setting the learning rate based on the insights from the convex case work well. 3.Also, for SGD (for both the convex and non-convex case), if we set the …
A-Z Linear Algebra & Calculus for Machine Learning
Convex vs. Non-Convex Functions Gradient Descent Concepts for Optimization Hessian Matrix & Its Role in Optimization Module 03: Integral Calculus for Machine Learning Indefinite Integral, …
Data Science - Convex optimization and application - univ …
3Data Science - Convex optimization and application Aggregation rules Gradient descent Stochastic Gradient descent Sequential prediction Bandit games, minimax policies Matrix …
Stochastic optimization: Beyond stochastic gradients and …
− Optimization of finite sums − Existing optimization methods for finite sums 2. Convex finite-sum problems – Linearly-convergent stochastic gradient method – SAG, SAGA, SVRG, …
A Brief Overview Why optimization? of Optimization Problems
Global vs. Local Optimization •For general nonlinear functions, most algorithms only guarantee a local optimum –that is, a feasible x o such that f 0(x o) ! f 0(x) for all feasible x within some …
Multi-Robot Trajectory Generation via Consensus ADMM: …
We propose a convex trajectory planning problem by leveraging C-ADMM and Buffered Voronoi Cells (BVCs) to get around the non-convex collision avoidance constraint and compare this …
Convex Optimization - GitHub Pages
Convex Optimization Brennan Gebotys Introduction to Convexity Main Theorems and Definitions PGD Lemmas Convergence Mirror Descent Lemmas Convergence Stochastics Non-Smooth …
Convex Optimization - Stanford University
Unconstrained minimization unconstrained minimization problem minimize f(x) we assume – fconvex, twice continuously differentiable (hencedom open) –optimal value p★ =infx f(x)is …
Convex vs nonconvex approaches for sparse estimation: …
for initializing the non-convex search, which may be seen as an instance of the screening type of approach for variable selection discussed in [15]. An optimization procedure for the HGLasso …
Optimization Methods for Machine Learning - Keerthi S
Optimization Methods for Machine Learning Sathiya Keerthi Microsoft Talks given at UC Santa Cruz February 21-23, 2017 The slides for the talks will be made available at: ... Di erentiable …
Course notes on Optimization for Machine Learning - GitHub …
Figure 4: Convex vs. non-convex functions ; Strictly convex vs. non strictly convex functions. Figure 5: Comparison of convex functions f: Rp→R (for p= 1) and convex sets C⊂Rp(for p= 2). …
Faster Lagrangian-Based Methods in Convex Optimization
Non-ergodic rate of O(1=N2) in the strongly convex case and a Non-ergodic O(1=N) in the convex case (see Theorem 4.1 and Theorem 4.2, respectively). As a by product, we also derive …
A Brief Overview Why optimization? of Optimization Problems
Global vs. Local Optimization" •!For general nonlinear functions, most algorithms only guarantee a local optimum" –!that is, a feasible x o such that f 0(x o) # f 0(x) for all feasible x within some …
Introduction to non-linear optimization - MIT
For convex problems rrf is always positive semi-denite and for strictly convex it is positive denite . What do we want? nd a convex neighborhood of x (be robust against mistakes) apply a …
Evolutionary Gradient Descent for Non-convex …
Non-convex optimization is often involved in arti-ficial intelligence tasks, which may have many sad-dle points, and is NP-hard to solve. Evolutionary algorithms (EAs) are general-purpose …
Monotone, Linear, and Convex Functions - Brendan Cooley
Convex functions generalize some of the properties of linear functions while providing more suitable functional forms. A real-valued function f de ned on a convex set of a linear space X is …
Stationary points, non-convex optimization, and more
Stationary points, non-convex optimization, and more... Instructor: Sham Kakade 1 Terminology stationary point of f(w): a point which has zero gradient. local minima of f(w): a point which …
Convex and non-convex worlds in machine learning - IIT …
Machine learning and optimization- many machine learning problems are formulated as minimization of some loss function on a training set of examples. Loss functions expresses the …
2.1 Review 2.2 Convex Sets - Carnegie Mellon University
y(x) = jjx yjjis convex in xfor any xed y, so by pointwise maximization rule, fis convex. Let Cbe convex, and consider the minimum distance to C: f(x) = min y2C jjx yjj Proof: g(x;y) = jjx yjjis …
Convex Optimization - Stanford University
strong duality holds if there is a non-vertical supporting hyperplane to Aat (0,p★) for convex problem, Ais convex, hence has supporting hyperplane at (0,p★) Slater’s condition: if there …
Efficient and Practical Stochastic Subgradient Descent for …
Problem setup Convex optimization f(X) is any convex loss function, ||X|| is nuclear norm defined to be sum of singular values. Nuclear norm is used to promote low-rank solutions. It is the tight …
Between Discrete and Continuous Optimization: …
convex optimization • ellipsoid method (Grötschel-Lovász-Schrijver 81) • subgradient method (improved: Chakrabarty-Lee-Sidford-Wong 16) combinatorial optimization • network flow …
A Brief Overview of Optimization Problems - MIT …
• Global versus local optimization • Convex vs. non-convex optimization • Unconstrained or box-constrained optimization, and other special-case constraints • Special classes of functions …
Convex and non-convex polygons: Information sheet
A polygon is convex if all the interior angles are less than 180 degrees. If one or more of the interior angles is more than 180 degrees the polygon is non-convex (or concave). All triangles …
Convex Optimization — Boyd & Vandenberghe 1. …
Brief history of convex optimization theory (convex analysis): 1900–1970 algorithms • 1947: simplex algorithm for linear programming (Dantzig) • 1970s: ellipsoid method and other …
Convex Optimization - Brennan Gebotys
Convex Optimization Brennan Gebotys Introduction to Convexity Main Theorems and Definitions PGD Lemmas Convergence Mirror Descent Lemmas Convergence Stochastics Non-Smooth …
