Optimality and Stability in Optimization

The issues of optimization, i.e., when one seeks to find the most efficient value or approach, broadly speaking, are absolutely fundamental and arise from a vast variety of problems in natural as well as social sciences, engineering, etc. For instance, think of a problem of constructing an object (bridge, building, etc.) with required properties, using minimal resources. Or, a problem of choosing the best strategy with respect to the strategies of other players in game theory. In fact, one can think of any process when, e.g., maximizing the profit or minimizing the time is at stake. In order to obtain precise qualitative answers to these efficiency issues, one typically employs a suitable mathematical model. This project aims to provide a strong analysis of several prominent issues of optimization and mathematical programming as well as underlying theoretical background from variational analysis. More precisely, we plan to investigate the optimization problems from two main perspectives the first being the area of optimality and stationarity conditions and the second being the area of stability and sensitivity of solutions. Several important optimization problems possess certain special structures, which makes them very difficult to handle. In particular, often it is not even clear how to best define the first-order optimality conditions the most basic tool. Simply put, we do not know what exactly to look for. There exist many different stationarity concepts, all of them having some advantages as well as some rather strong shortcomings. Building upon our former work, we plan to further improve our own stationarity notion of Q-stationarity, which has been shown to be very promising. On the other hand, even when simpler and very standard optimization problems (such as the so- called nonlinear programs) are considered, significant questions regarding the stability and sensitivity of solutions remain open. These questions are often closely related to the rather involved issues of variational analysis, in particular the second-order analysis. Having decent knowledge and experience in this area, and taking into account the superb collaboration possibilities, we plan to resolve several prominent tasks in the field of stability and sensitivity of solutions.

The project revolves around challenging mathematical problems from the area of variational analysis, which can be seen as a theoretical foundation for optimization or mathematical programming - a very lively field with countless applications in engineering, economics, natural sciences, data science, etc. While standard mathematical analysis deals mostly with the differential calculus of typical (single-valued) functions or mappings, variational analysis, among other things, aims to generalize this in order to study the so-called set-valued mappings - objects one inevitably encounters when dealing with constrained or nonsmooth optimization problems, which are in turn problems central to many modern applications. Developing calculus (rules on how to compute various generalized derivatives) for set-valued mappings is one of the main topics of variational analysis. Another important area of variational analysis is second-order analysis, which is deeply connected to issues of the stability of solutions of optimization problems and, in turn, to numerical methods (second-order optimality conditions, convergence analysis of algorithms, Newton-type methods). We have contributed to both of these topics. First, we have provided a full understanding of a certain calculus principle, which was heavily underdeveloped. We have identified a completely new assumption - a continuity-type property of set-valued mappings called (fuzzy) inner calmness* - which is both sufficient and necessary for the validity of that calculus rule. This has strong implications for many other calculus rules (which we have derived afterwards), and, particularly, for second-order analysis and stability issues (which we have only begun to work on). Additionally, we have derived second-order optimality conditions for very general optimization problems, using new variational tools as well as new techniques, including calculus for second subderivatives based on inner calmness*. We have also developed various sufficient conditions for inner calmness* in order to enable easier application of this property. Finally, this project was extremely helpful for accumulating new collaborations for the principal investigator and for his career development. Indeed, we have worked intensively with P. Mehlitz and we have established a new collaboration with R. T. Rockafellar, one of the founders of and most important figures in convex and variational analysis. On top of it, we have established many other connections in France, the USA, Canada, Chile, and elsewhere.