The mathematical tools for building optimization models and algorithms grow out of linear algebra, differential calculus and real analysis. However, the needs of applications have led to a new area of mathematics that can handle systems of inequalities and functions that are neither smooth nor well-defined in a traditional sense. Variational analysis is the broad term for this area of mathematics. In this presentation, we show its crucial role in the development of optimization models and algorithms in finite dimensions. First, we examine variational geometry and definitions of normal and tangent vectors that extend the classical notions for smooth manifolds. This in turn leads to subdifferentiability, a wide range of calculus rules and optimality conditions for arbitrary functions. Second, we develop an approximation theory for optimization problems that leads to consistent approximations, error bounds and rates of convergence even in the nonconvex and nonsmooth setting.
Dr. Johannes O. Royset is Professor of Operations Research at the Naval Postgraduate School. Dr. Royset's research focuses on formulating and solving stochastic and deterministic optimization problems arising in data analytics, sensor management, and reliability engineering. 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 in 2016 and at the SIAM Conference on Uncertainty Quantification in 2018. 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 Operations Research, Mathematical Programming, Journal of Optimization Theory and Applications, Journal of Convex Analysis, Set-Valued and Variational Analysis, Naval Research Logistics, and Computational Optimization and Applications.