The main purpose of this book is to provide a systematic study of asymptotic cones and asymptotic functions in finite dimensional normed spaces. Many results are extracted from articles by the authors and their collaborators, but there are also well-known theorems in convex analysis and variational inequalities which are deduced using asymptotic cones and functions. The book is divided into 6 chapters: Ch. 1: Convex analysis and set-valued maps: a review; Ch. 2: Asymptotic cones and functions; Ch. 3: Existence and stability in optimization problems; Ch. 4: Minimizing and stationary sequences; Ch. 5: Duality in optimization problems; Ch. 6: Maximal monotone operators and variational inequalities.
The first chapter has a preliminary character and the results are stated without proof. The heart of the book is represented by the second chapter where the notions of asymptotic cone and asymptotic function are defined and their properties are studied. An important use of asymptotic cones is for obtaining closedness criteria for images of closed sets by linear mappings; a necessary and sufficient condition is furnished for this. A class of sets whose images by linear operators are automatically closed is that of continuous convex sets; this class of sets is also studied in this chapter. An analytic formula for the asymptotic function is given; this formula turns to be very useful. The chapter ends with applications to semidefinite optimization problems and to modelling and smoothing optimization problems. In the third chapter one introduces the classes of weakly coercive convex functions and of asymptotically level stable functions, respectively. For these classes of functions one obtains the existence of optimal solutions. In this chapter one studies also the stability of constrained optimization problems using perturbation functions and gives applications to subdifferential calculus. The forth chapter is dedicated mainly to the study of asymptotically well-behaved functions, a notion introduced in [A. Auslender and J.-P. Crouzeix, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 6, No. Suppl., 101-121 (1989; Zbl 0675.90070)]. One also studies global error bounds for convex inequality systems and stationary sequences in constrained minimization. Using a perturbation function approach, in the fifth chapter the authors state several general duality results; then Fenchel and Lagrangian dualities are studied. In the last chapter one proves the Minty theorem, the maximal monotonicity of the subdifferential and the fact that the closure and the relative interior of the domain and range of a maximal monotone map are convex. Also, one associates an asymptotic function to a maximal monotone map which is used for studying the range of the sum of two monotone maps. Then one studies the existence of solutions and duality of variational inequalities. Every chapter ends with bibliographical notes. The book ends with a list of 135 references, an index and a list of notations.
The book is addressed to graduate students at an advanced level and to researchers and practitioners in the fields of optimization theory, nonlinear programming and applied mathematical sciences. To our knowledge, this is the first book in the literature which uses the asymptotic analysis as the main tool in studying optimization problems and variational inequalities. Traditionally, asymptotic cones were used for obtaining closedness criteria while asymptotic functions for obtaining existence results. The present book proves that asymptotic analysis became a powerful and versatile field of mathematical analysis. For example the well-known material of Chapters 5 and 6 differ in many respects from the classical literature, on both the way it is presented but also on several proofs, which emphasize the role of asymptotic analysis. We recommend this book to all those who are interested in asymptotic analysis and its use. We also consider that after reading this book most people will have a new perspective on known results. Unfortunately there are some misprints and omissions, especially in the introductory chapter, which ask readers’ care; probably the authors will mention them in an errata.