This book investigates several duality approaches for vector optimization problems, while also comparing them. Special attention is paid to duality for linear vector optimization problems, for which a vector dual that avoids the shortcomings of the classical ones is proposed. Moreover, the book addresses different efficiency concepts for vector optimization problems. Among the problems that appear when the framework is generalized by considering set-valued functions, an increasing interest is generated by those involving monotone operators, especially now that new methods for approaching them by means of convex analysis have been developed. Following this path, the book provides several results on different properties of sums of monotone operators.

This book brings together research articles and state-of-the-art surveys in broad areas of optimization and numerical analysis with particular emphasis on algorithms. The discussion also focuses on advances in monotone operator theory and other topics from variational analysis and nonsmooth optimization, especially as they pertain to algorithms and concrete, implementable methods. The theory of monotone operators is a central framework for understanding and analyzing splitting algorithms. Topics discussed in the volume were presented at the interdisciplinary workshop titled Splitting Algorithms, Modern Operator Theory, and Applications held in Oaxaca, Mexico in September, 2017. Dedicated to Jonathan M. Borwein, one of the most versatile mathematicians in contemporary history, this compilation brings theory together with applications in novel and insightful ways.

This book contains a selection of refereed papers presented at the "International Conference on Operations Research (OR 2014)", which took place at RWTH Aachen University, Germany, September 2-5, 2014. More than 800 scientists and students from 47 countries attended OR 2014 and presented more than 500 papers in parallel topical streams, as well as special award sessions. The theme of the conference and its proceedings is "Business Analytics and Optimization".

Oleg Wilfer presents a new conjugate duality concept for geometric and cone constrained optimization problems whose objective functions are a composition of finitely many functions. As an application, the author derives results for single minmax location problems formulated by means of extended perturbed minimal time functions as well as for multi-facility minmax location problems defined by gauges. In addition, he provides formulae of projections onto the epigraphs of gauges to solve these kinds of location problems numerically by using parallel splitting algorithms. Numerical comparisons of recent methods show the excellent performance of the proposed solving technique. ​About the Author: Dr. Oleg Wilfer received his PhD at the Faculty of Mathematics of Chemnitz University of Technology, Germany. He is currently working as a development engineer in the automotive industry.

This book presents a unified theory of convex functions, sets, and set-valued mappings in topological vector spaces with its specifications to locally convex, Banach and finite-dimensional settings. These developments and expositions are based on the powerful geometric approach of variational analysis, which resides on set extremality with its characterizations and specifications in the presence of convexity. Using this approach, the text consolidates the device of fundamental facts of generalized differential calculus to obtain novel results for convex sets, functions, and set-valued mappings in finite and infinite dimensions. It also explores topics beyond convexity using the fundamental machinery of convex analysis to develop nonconvex generalized differentiation and its applications. The text utilizes an adaptable framework designed with researchers as well as multiple levels of students in mind. It includes many exercises and figures suited to graduate classes in mathematical sciences that are also accessible to advanced students in economics, engineering, and other applications. In addition, it includes chapters on convex analysis and optimization in finite-dimensional spaces that will be useful to upper undergraduate students, whereas the work as a whole provides an ample resource to mathematicians and applied scientists, particularly experts in convex and variational analysis, optimization, and their applications.

This book serves as an introductory text to optimization theory in normed spaces and covers all areas of nonlinear optimization. It presents fundamentals with particular emphasis on the application to problems in the calculus of variations, approximation and optimal control theory. The reader is expected to have a basic knowledge of linear functional analysis.

The aim of this work is to present several new results concerning duality in scalar convex optimization, the formulation of sequential optimality conditions and some applications of the duality to the theory of maximal monotone operators. After recalling some properties of the classical generalized interiority notions which exist in the literature, we give some properties of the quasi interior and quasi-relative interior, respectively. By means of these notions we introduce several generalized interior-point regularity conditions which guarantee Fenchel duality. By using an approach due to Magnanti, we derive corresponding regularity conditions expressed via the quasi interior and quasi-relative interior which ensure Lagrange duality. These conditions have the advantage to be applicable in situations when other classical regularity conditions fail. Moreover, we notice that several duality results given in the literature on this topic have either superfluous or contradictory assumptions, the investigations we make offering in this sense an alternative. Necessary and sufficient sequential optimality conditions for a general convex optimization problem are established via perturbation theory. These results are applicable even in the absence of regularity conditions. In particular, we show that several results from the literature dealing with sequential optimality conditions are rediscovered and even improved. The second part of the thesis is devoted to applications of the duality theory to enlargements of maximal monotone operators in Banach spaces. After establishing a necessary and sufficient condition for a bivariate infimal convolution formula, by employing it we equivalently characterize the $\varepsilon$-enlargement of the sum of two maximal monotone operators. We generalize in this way a classical result concerning the formula for the $\varepsilon$-subdifferential of the sum of two proper, convex and lower semicontinuous functions. A characterization of fully en.

The results presented in this book originate from the last decade research work of the author in the ?eld of duality theory in convex optimization. The reputation of duality in the optimization theory comes mainly from the major role that it plays in formulating necessary and suf?cient optimality conditions and, consequently, in generatingdifferent algorithmic approachesfor solving mathematical programming problems. The investigations made in this work prove the importance of the duality theory beyond these aspects and emphasize its strong connections with different topics in convex analysis, nonlinear analysis, functional analysis and in the theory of monotone operators. The ?rst part of the book brings to the attention of the reader the perturbation approach as a fundamental tool for developing the so-called conjugate duality t- ory. The classical Lagrange and Fenchel duality approaches are particular instances of this general concept. More than that, the generalized interior point regularity conditions stated in the past for the two mentioned situations turn out to be p- ticularizations of the ones given in this general setting. In our investigations, the perturbationapproachrepresentsthestartingpointforderivingnewdualityconcepts for several classes of convex optimization problems. Moreover, via this approach, generalized Moreau–Rockafellar formulae are provided and, in connection with them, a new class of regularity conditions, called closedness-type conditions, for both stable strong duality and strong duality is introduced. By stable strong duality we understand the situation in which strong duality still holds whenever perturbing the objective function of the primal problem with a linear continuous functional.

Building on fundamental results in variational analysis, this monograph presents new and recent developments in the field as well as selected applications. Accessible to a broad spectrum of potential readers, the main material is presented in finite-dimensional spaces. Infinite-dimensional developments are discussed at the end of each chapter with comprehensive commentaries which emphasize the essence of major results, track the genesis of ideas, provide historical comments, and illuminate challenging open questions and directions for future research. The first half of the book (Chapters 1–6) gives a systematic exposition of key concepts and facts, containing basic material as well as some recent and new developments. These first chapters are particularly accessible to masters/doctoral students taking courses in modern optimization, variational analysis, applied analysis, variational inequalities, and variational methods. The reader’s development of skills will be facilitated as they work through each, or a portion of, the multitude of exercises of varying levels. Additionally, the reader may find hints and references to more difficult exercises and are encouraged to receive further inspiration from the gems in chapter commentaries. Chapters 7–10 focus on recent results and applications of variational analysis to advanced problems in modern optimization theory, including its hierarchical and multiobjective aspects, as well as microeconomics, and related areas. It will be of great use to researchers and professionals in applied and behavioral sciences and engineering.