Variational Perspectives on Mathematical Optimization

CRM Applied Mathematics Seminars (26 oct. 2020 / 26 Oct. 2020) https://dms.umontreal.ca/~mathapp/ Johannes Royset (Naval Postgraduate School, California, USA) Variational Perspectives on Mathematical Optimization Abstract: 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.