Perturbation Analysis of Optimization Problems

Perturbation Analysis of Optimization Problems
Author: J.Frederic Bonnans
Publisher: Springer Science & Business Media
Total Pages: 618
Release: 2013-11-22
Genre: Mathematics
ISBN: 1461213940

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A presentation of general results for discussing local optimality and computation of the expansion of value function and approximate solution of optimization problems, followed by their application to various fields, from physics to economics. The book is thus an opportunity for popularizing these techniques among researchers involved in other sciences, including users of optimization in a wide sense, in mechanics, physics, statistics, finance and economics. Of use to research professionals, including graduate students at an advanced level.

Stochastic Simulation Optimization For Discrete Event Systems: Perturbation Analysis, Ordinal Optimization And Beyond

Stochastic Simulation Optimization For Discrete Event Systems: Perturbation Analysis, Ordinal Optimization And Beyond
Author: Chun-hung Chen
Publisher: World Scientific
Total Pages: 274
Release: 2013-07-03
Genre: Technology & Engineering
ISBN: 9814513024

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Discrete event systems (DES) have become pervasive in our daily lives. Examples include (but are not restricted to) manufacturing and supply chains, transportation, healthcare, call centers, and financial engineering. However, due to their complexities that often involve millions or even billions of events with many variables and constraints, modeling these stochastic simulations has long been a “hard nut to crack”. The advance in available computer technology, especially of cluster and cloud computing, has paved the way for the realization of a number of stochastic simulation optimization for complex discrete event systems. This book will introduce two important techniques initially proposed and developed by Professor Y C Ho and his team; namely perturbation analysis and ordinal optimization for stochastic simulation optimization, and present the state-of-the-art technology, and their future research directions.

Perturbations, Optimization, and Statistics

Perturbations, Optimization, and Statistics
Author: Tamir Hazan
Publisher: MIT Press
Total Pages: 412
Release: 2017-09-22
Genre: Computers
ISBN: 0262337940

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A description of perturbation-based methods developed in machine learning to augment novel optimization methods with strong statistical guarantees. In nearly all machine learning, decisions must be made given current knowledge. Surprisingly, making what is believed to be the best decision is not always the best strategy, even when learning in a supervised learning setting. An emerging body of work on learning under different rules applies perturbations to decision and learning procedures. These methods provide simple and highly efficient learning rules with improved theoretical guarantees. This book describes perturbation-based methods developed in machine learning to augment novel optimization methods with strong statistical guarantees, offering readers a state-of-the-art overview. Chapters address recent modeling ideas that have arisen within the perturbations framework, including Perturb & MAP, herding, and the use of neural networks to map generic noise to distribution over highly structured data. They describe new learning procedures for perturbation models, including an improved EM algorithm and a learning algorithm that aims to match moments of model samples to moments of data. They discuss understanding the relation of perturbation models to their traditional counterparts, with one chapter showing that the perturbations viewpoint can lead to new algorithms in the traditional setting. And they consider perturbation-based regularization in neural networks, offering a more complete understanding of dropout and studying perturbations in the context of deep neural networks.

Constructive Nonsmooth Analysis and Related Topics

Constructive Nonsmooth Analysis and Related Topics
Author: Vladimir F. Demyanov
Publisher: Springer Science & Business Media
Total Pages: 258
Release: 2013-11-12
Genre: Mathematics
ISBN: 1461486157

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This volume contains a collection of papers based on lectures and presentations delivered at the International Conference on Constructive Nonsmooth Analysis (CNSA) held in St. Petersburg (Russia) from June 18-23, 2012. This conference was organized to mark the 50th anniversary of the birth of nonsmooth analysis and nondifferentiable optimization and was dedicated to J.-J. Moreau and the late B.N. Pshenichnyi, A.M. Rubinov, and N.Z. Shor, whose contributions to NSA and NDO remain invaluable. The first four chapters of the book are devoted to the theory of nonsmooth analysis. Chapters 5-8 contain new results in nonsmooth mechanics and calculus of variations. Chapters 9-13 are related to nondifferentiable optimization, and the volume concludes with four chapters containing interesting and important historical chapters, including tributes to three giants of nonsmooth analysis, convexity, and optimization: Alexandr Alexandrov, Leonid Kantorovich, and Alex Rubinov. The last chapter provides an overview and important snapshots of the 50-year history of convex analysis and optimization.

Multivalued Analysis and Nonlinear Programming Problems with Perturbations

Multivalued Analysis and Nonlinear Programming Problems with Perturbations
Author: B. Luderer
Publisher: Springer Science & Business Media
Total Pages: 218
Release: 2013-03-09
Genre: Mathematics
ISBN: 1475734689

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The book presents a treatment of topological and differential properties of multivalued mappings and marginal functions. In addition, applications to sensitivity analysis of nonlinear programming problems under perturbations are studied. Properties of marginal functions associated with optimization problems are analyzed under quite general constraints defined by means of multivalued mappings. A unified approach to directional differentiability of functions and multifunctions forms the base of the volume. Nonlinear programming problems involving quasidifferentiable functions are considered as well. A significant part of the results are based on theories and concepts of two former Soviet Union researchers, Demyanov and Rubinov, and have never been published in English before. It contains all the necessary information from multivalued analysis and does not require special knowledge, but assumes basic knowledge of calculus at an undergraduate level.

Convex Optimization

Convex Optimization
Author: Stephen P. Boyd
Publisher: Cambridge University Press
Total Pages: 744
Release: 2004-03-08
Genre: Business & Economics
ISBN: 9780521833783

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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.

Perturbation Analysis, Optimization and Resource Contention Games in Stochastic Hybrid Systems

Perturbation Analysis, Optimization and Resource Contention Games in Stochastic Hybrid Systems
Author: Chen Yao
Publisher:
Total Pages: 342
Release: 2011
Genre:
ISBN:

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Abstract:Stochastic Hybrid Systems (SHS) are systems that combine event-driven and time-driven dynamics, and include elements to model uncertainties in the system. There have been several different types of stochastic hybrid system models proposed. In this dissertation, a unified framework is presented for carrying out perturbation analysis for general SHS with arbitrary structures, in particular, the Infinitesimal Perturbation Analysis (IPA) methodology originally developed for Discrete Event Systems. Some properties are also established, which apply to this framework and justify its effectiveness in recovering useful performance sensitivity estimates. Then, this dissertation concentrates on Stochastic Flow Models (SFMs), which are one type of SHS and are used to abstract the dynamics of many complex discrete event systems to provide the basis for their control and optimization. SFMs have been used to date to study systems with a single user class or some multiclass settings in which performance metrics are not. class-dependent. However, little work has been done for multiclass systems that fully differentiate among classes, where classes contend for single or multiple system resources, and with class-dependent performance metrics. This is partly due to the complexities in modeling SFMs for such systems, and partly clue to the difficulties in applying IPA in this context. In this dissertation, a general framework is built based on multiclass SFMs, to model stochastic resource contention systems, where multiple classes (users) compete for shared resources. The general IPA framework is then applied to stick systems to obtain performance gradient estimates for various user-specific objectives, which enables the study of a new " user centric " optimization perspective, in addition to the usual "system-centric " viewpoint. Following the "user-centric " optimization, each class (user) seeks to optimize its own performance by adjusting its own controls, which leads to resource contention games between classes. A simple instance of such systems is studied to illustrate how the general IPA is applied to specific systems, and the difference between solutions of the two perspectives, which is commonly referred to as the "price of anarchy". Two specific resource contention problems are studied in this dissertation. One is the admission control problem for the multiclass queueing system under a First Come First Served (FCFS) policy, where the buffer capacity thresholds of all classes are determined to optimize system performance; the other problem is the multiclass lot-sizing problem arising in the manufacturing production planning setting, where the objective is to obtain optimal lot sizes for all classes. For both problems, the general IPA framework is applied to the multiclass SFM abstractions to derive sensitivity estimates of performance metrics with respect to control parameters of interest, which are all proven to be unbiased, hence, reliable for control and optimization purposes. These estimates arc then used to drive the on-line optimization of these parameters, and simulation results are provided to contrast the solutions obtained through the " system-centric " and "user-centric " perspectives.

Singular Perturbation Analysis of Aotv-Related Trajectory Optimization Problems

Singular Perturbation Analysis of Aotv-Related Trajectory Optimization Problems
Author: National Aeronautics and Space Administration (NASA)
Publisher: Createspace Independent Publishing Platform
Total Pages: 54
Release: 2018-07-10
Genre:
ISBN: 9781722741037

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The problem of real time guidance and optimal control of Aeroassisted Orbit Transfer Vehicles (AOTV's) was addressed using singular perturbation theory as an underlying method of analysis. Trajectories were optimized with the objective of minimum energy expenditure in the atmospheric phase of the maneuver. Two major problem areas were addressed: optimal reentry, and synergetic plane change with aeroglide. For the reentry problem, several reduced order models were analyzed with the objective of optimal changes in heading with minimum energy loss. It was demonstrated that a further model order reduction to a single state model is possible through the application of singular perturbation theory. The optimal solution for the reduced problem defines an optimal altitude profile dependent on the current energy level of the vehicle. A separate boundary layer analysis is used to account for altitude and flight path angle dynamics, and to obtain lift and bank angle control solutions. By considering alternative approximations to solve the boundary layer problem, three guidance laws were derived, each having an analytic feedback form. The guidance laws were evaluated using a Maneuvering Reentry Research Vehicle model and all three laws were found to be near optimal. For the problem of synergetic plane change with aeroglide, a difficult terminal boundary layer control problem arises which to date is found to be analytically intractable. Thus a predictive/corrective solution was developed to satisfy the terminal constraints on altitude and flight path angle. A composite guidance solution was obtained by combining the optimal reentry solution with the predictive/corrective guidance method. Numerical comparisons with the corresponding optimal trajectory solutions show that the resulting performance is very close to optimal. An attempt was made to obtain numerically optimized trajectories for the case where heating rate is constrained. A first order state variable inequality constra...