Optimization problems, generalized equations, and a multitude of other variational problems invariably lead to the analysis of sets and set-valued mappings as well as their approximations. We review the central concept of set-convergence and explain its role in defining a notion of proximity between sets, especially for epigraphs of functions and graphs of set-valued mappings. The development leads to an approximation theory for optimization problems and generalized equations with profound consequences for the construction of algorithms. We also introduce the role of set-convergence in variational geometry and subdifferentiability with applications to optimality conditions. Examples illustrate the importance of set-convergence in stability analysis, error analysis, construction of algorithms, statistical estimation, and probability theory.



Bio: Dr. Johannes O. Royset is a Professor of Operations Research at the Naval Postgraduate School. His research focuses on formulating and solving stochastic optimization problems arising in data analytics and risk management. He was awarded a National Research Council postdoctoral fellowship in 2003, a Young Investigator Award from the Air Force Office of Scientific Research in 2007, and the Barchi Prize as well as the MOR Journal Award from the Military Operations Research Society in 2009. He received the Carl E. and Jessie W. Menneken Faculty Award for Excellence in Scientific Research in 2010 and the Goodeve Medal from the Operational Research Society in 2019. Dr. Royset was a plenary speaker at the International Conference on Stochastic Programming (2016), the SIAM Conference on Uncertainty Quantification (2018), and the INFORMS Conference on Security (2022). He has a Doctor of Philosophy degree from the University of California at Berkeley (2002). Dr. Royset has been an associate or guest editor of SIAM Journal on Optimization, Operations Research, Mathematical Programming, Journal of Optimization Theory and Applications, Naval Research Logistics, Journal of Convex Analysis, Set-Valued and Variational Analysis, and Computational Optimization and Applications. He has published more than 100 papers and two books.