Applied Variational Analysis: Theory, Algorithms, and Applications

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My research is in applied variational analysis. Variational analysis is an extension of convex analysis and classical analysis to encompass a variety of nondifferential functions and mappings. With its tools well-developed, seeking their applications in different areas, practical problems, and computation algorithms are essential. One important result for set-valued mappings is the Attouch-Thera duality, which assumes that the primal has a solution. If the primal has no solution, it is not clear whether an analogue of the Fenchel-Rockafellar duality for maximal monotone mappings exists. Epi-convergence is essential for studying convergence of extended-real-valued functions. In this topology, what can we say about minimization behaviors of nonconvex functions? Proximal average of convex functions is a powerful tool in modern convex analysis, but its generalizations to nonconvex functions are much less explored. Up to now, almost all splitting algorithms need both operators to be monotone. What happens if at least one operator is not monotone? Prox-regularity of functions and metric regularity of mappings have been systematically studied by Poliquin, Rockafellar, Thibault, Mordukhovich, Ioffe, Lewis, et al., but their algorithmic consequences await a comprehensive study. From convex functions to nonconvex functions, monotone mappings to nonmonotone mappings, these generalizations are intrinsically hard, often they require new methodologies. Although some progress has been made, it is not satisfactory at all. In this proposal, I plan to (1) study Attouch-Thera's duality, generic minimization properties of nonconvex functions by envelopes, proximal average applications and extensions, and the second order nonsmooth analysis of envelopes by Mordukhovich's coderivative analysis and Rockafellar's proto-derivative analysis; (2) develop algorithms and a local convergence theory for solving zeros of a sum of nonmonotone mappings, and for minimizing a sum of nonconvex functions. Works by Chen and Rockafellar, Bauschke, Combettes, Noll, and Thera will be examined. Generalized local nonexpansive mappings are at the heart of the convergence theory. Metric regularity of mappings and prox-regularity of functions will be used extensively to study convergence rates of splitting algorithms; and (3) exploit some applications in signal processing and financial mathematics. Seeing the success of my HQPs brings me joy. Amazingly, while postdoctoral fellows and graduate students can concentrate on theory developments, undergraduates can do some numerical experiments on algorithms and computations. I believe that this research will significantly advance our knowledge about applied variational analysis in theory, algorithms, and applications. People optimize. These much needed research results will have both local influence and global impact on the optimization community.