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Optimality and stability are two important notions in applied mathematics. This book is a study of these notions and their relationship in linear and convex parametric programming models. It begins with a survey of basic optimality conditions in nonlinear programming. Then new results in convex programming, using LFS functions, for single-objective, multi-objective, differentiable and non-smooth programs are introduced. Parametric programming models are studied using basic tools of point-to-set topology. Stability of the models is introduced, essentially, as continuity of the feasible set of decision variables under continuous perturbations of the parameters. Perturbations that preserve this continuity are regions of stability. It is shown how these regions can be identified. The main results on stability are characterizations of locally and globally optimal parameters for stable and also for unstable perturbations. The results are straightened for linear models and bi-level programs. Some of the results are extended to abstract spaces after considering parameters as `controls'. Illustrations from diverse fields, such as data envelopment analysis, management, von Stackelberg games of market economy, and navigation problems are given and several case studies are solved by finding optimal parameters. The book has been written in an analytic spirit. Many results appear here for the first time in book form. Audience: The book is written at the level of a first-year graduate course in optimization for students with varied backgrounds interested in modeling of real-life problems. It is expected that the reader has been exposed to a prior elementary course in optimization, such as linear or non-linear programming. The last section of the book requires some knowledge of functional analysis.
The goal of the Encyclopedia of Optimization is to introduce the reader to a complete set of topics that show the spectrum of research, the richness of ideas, and the breadth of applications that has come from this field. The second edition builds on the success of the former edition with more than 150 completely new entries, designed to ensure that the reference addresses recent areas where optimization theories and techniques have advanced. Particularly heavy attention resulted in health science and transportation, with entries such as "Algorithms for Genomics", "Optimization and Radiotherapy Treatment Design", and "Crew Scheduling".
Introduction to Sensitivity and Stability Analysis in Nonlinear Programming
This book introduces models and methodologies that can be employed towards making the Industry 4.0 vision a reality within the process industries, and at the same time investigates the impact of uncertainties in such highly integrated settings. Advances in computing power along with the widespread availability of data have led process industries to consider a new paradigm for automated and more efficient operations. The book presents a theoretically proven optimal solution to multi-parametric linear and mixed-integer linear programs and efficient solutions to problems such as process scheduling and design under global uncertainty. It also proposes a systematic framework for the uncertainty-aware integration of planning, scheduling and control, based on the judicious coupling of reactive and proactive methods. Using these developments, the book demonstrates how the integration of different decision-making layers and their simultaneous optimisation can enhance industrial process operations and their economic resilience in the face of uncertainty.
Researchers working with nonlinear programming often claim "the word is non linear" indicating that real applications require nonlinear modeling. The same is true for other areas such as multi-objective programming (there are always several goals in a real application), stochastic programming (all data is uncer tain and therefore stochastic models should be used), and so forth. In this spirit we claim: The word is multilevel. In many decision processes there is a hierarchy of decision makers, and decisions are made at different levels in this hierarchy. One way to handle such hierar chies is to focus on one level and include other levels' behaviors as assumptions. Multilevel programming is the research area that focuses on the whole hierar chy structure. In terms of modeling, the constraint domain associated with a multilevel programming problem is implicitly determined by a series of opti mization problems which must be solved in a predetermined sequence. If only two levels are considered, we have one leader (associated with the upper level) and one follower (associated with the lower level).
Information and communication technology has helped to provide a more effective network infrastructure and development platform for logistics and service operations. In order to meet the needs of consumers, and particularly to promote low-carbon development processes, new types of services will also emerge. LISS 2012 is a prime international forum for both researchers and industry practitioners to exchange the latest fundamental advances in the state of the art and practice of logistics, informatics, service operations and service science. Experts and researchers from related fields will discuss current issues and future development opportunities, discuss and analyze developing trends and exchange the latest research and academic thought. The theme of the conference is Logistics and Service Science based on the Internet of Things.
A general method is presented for solving the problem of heat-shield response in the stagnation region of a charring type ablator. The analysis is actually for the stagnation point of an axisymmetric blunt body, but it is a valid approximate method for calculations in the stagnation region of any arbitrary blunt body. The analysis is applicable to windtunnel or flight conditions, and the heat loadings are either arbitrarily assigned or they are calculated concurrently with the heat-shield response. Surface heating (or cooling) mechanisms accounted for are those due to convection, radiation, homogeneous combustion, heterogeneous combustion, surface material removal by means other than combustion (includes erosion) , and sublimation. Physical and thermodynamic properties of the ablating material are arbitrarily assigned so that calculations can be made for various materials. A typical application of the analysis is given as an illustration. The analysis is machine programmed for numerical solutions usinga finite difference scheme, and a family of computing programs is used. These programs are described and instructions are provided for using them. The programs can be obtained from COSMIC, University of Georgia, Athens, Georgia, 30601.