This Special Issue of Optimization is dedicated to Professor Boris Mordukhovich on the occasion of his 70th birthday.
Professor Mordukhovich is a Distinguished University Professor and Lifetime Scholar of the Academy of Scholars at Wayne State University, USA. He is one of the founders of modern variational analysis, who has made tremendous contributions to the theory of variational analysis with applications to many other fields such as optimization, optimal control, and mathematical economics. Professor Boris Mordukhovich is an AMS Fellow, a SIAM Fellow, and the recipient of many international and national awards including honorary degrees Doctor Honoris Causa by six universities and research institutions. Besides being a great scholar, he is also a great teacher who has educated thirty Ph.D. students from all over the world.
The era of variational and convex analysis started in the 1960s with the pioneering works of Jean Jacques Moreau and R. Tyrrell Rockafellar, who initiated the study of generalized differentiation for convex functions and sets. Convex analysis serves as the mathematical foundation for convex optimization, a field with great impacts in many applied areas nowadays. The 1970s witnessed great effort in developing generalized differentiation theory for nonsmooth functions that are not necessarily convex. Professor Mordukhovich is now considered as one of the founders of modern variational analysis due to his generalized differentiation theory for nonsmooth functions and set-valued mappings that goes beyond convexity and is independent of convexity. Based on a geometric dual approach, in the mid of 1970s, Professor Mordukhovich, who was in the Soviet Union, introduced the concept of limiting/Mordukhovich subgradients for extended-real-valued functions. The generalized differentiation calculus for limiting subgradients relies on another novel notion called the extremal system for nonconvex sets along with the extremal principle, which is as important as the separation theorem in the convex case. In the early of 1980s, Professor Mordukhovich applied his revolutionary idea to the case of set-valued mappings by introducing the concept of coderivatives for set-valued mappings and developed their comprehensive calculus as well as applications to optimization and optimal control.
With his departure to the United States in 1988, he started a new journey in bringing variational analysis to a new level and to the world. In a series of papers written in the 1990s and 2000s, Professor Mordukhovich developed limiting generalized differentiation theory for nonsmooth functions and set-valued mappings in Asplund spaces, an important class of Banach spaces introduced in 1968 by the mathematician Edgar Asplund. By this time, his generalized differentiation theory had reached a very high level of comprehensiveness and mathematical beauty. Nowadays, he continues to explore new findings in variational analysis and bring variational analysis to applications in many fields such as optimization theory, numerical optimization algorithms, optimal control, and mathematical economics. His theory of generalized differentiation stays as an important part of modern variational analysis beyond convexity, while shedding new light on the theory of convex analysis. His important works have attracted and motivated the contributions of many other mathematicians to the field of variational analysis and applications.
This Special Issue of Optimization is a tribute to Professor Mordukhovich for his great achievements in variational analysis and applications to optimization. To all of us, he is not only a great researcher but also a friend, a collaborator, a colleague, and a mentor. The Special Issue consists of seventeen research articles devoted to many areas of variational analysis and applications to optimization. We would like to thank all authors for their scientific contributions.