"Variational analysis of monotone operators and beyond: theory, algorithms and applications"

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My research will apply variational analysis to monotone operators. Variational analysis is an extension of convex analysis that encompasses a variety of nondifferential functions (convex or nonconvex) and mappings. Variational analysis has more powerful tools such as coderivative, proto-derivative, prox-regularity, variational principle and many modern techniques. Monotone operators play key roles in Optimization, Convex Analysis, Fixed Point Theory, and Partial Differential Equations. Up to now, the second order theory of nonsmooth analysis is not satisfactory; to make good progress, it is worthwhile to systematically analyze coderivatives and graphical derivatives of monotone operators. Up to now, almost all splitting algorithms need both operators to be monotone. What happens if one operator is not monotone? Even for alternating projections, little progress has been made in this direction. While monotone operator field provides an excellent test ground for variational analysis, to go beyond monotonicity (or convexity) one has to rely on modern techniques in variational analysis. The goal of this proposal is to study monotone operators by variational (nonconvex) analysis, develop local convergence theory and algorithms for solving inclusions in which one operator is monotone and the other not necessarily monotone, and exploit some applications in image processing and mathematical economics. The q-positive sets and SSDB spaces by Simons will also be explored. Amazingly, while post-doctoral 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 the analytical properties and structures of monotone (n-monotone) operators, gain new insights on SSDB spaces and bring splitting algorithms to a new level. Practical impacts are efficient algorithms for solving for zeros of the sum of a monotone operator and a Lipschitz mapping. These could then be applied to modelling in engineering and cancer research, and economic equilibrium. The success of my research will benefit Canada and have global impacts.