Rockafellar's research is motivated by the goal of organizing mathematical ideas and concepts into robust frameworks that yield new insights and relations. This approach is most salient in his seminal book "Variational Analysis" (1998, with
Roger J-B Wets), where numerous threads developed in the areas of convex analysis, nonlinear analysis, calculus of variation, mathematical optimization, equilibrium theory, and control systems were brought together to produce a unified approach to variational problems in finite dimensions. These various fields of study are now referred to as
variational analysis. In particular, the text dispenses of differentiability as a necessary property in many areas of analysis and embraces nonsmoothness, set-valuedness, and extended real-valuedness, while still developing far-reaching calculus rules.
Contributions to Mathematics The approach of extending the real line with the values
infinity and negative infinity and then allowing (convex) functions to take these values can be traced back to Rockafellar's dissertation and, independently, the work by
Jean-Jacques Moreau around the same time. The central role of
set-valued mappings (also called multivalued functions) was also recognized in Rockafellar's dissertation and, in fact, the standard notation ∂
f(
x) for the set of
subgradients of a function
f at
x originated there. Rockafellar contributed to nonsmooth analysis by extending the rule of
Fermat, which characterizes solutions of
optimization problems, to composite problems using subgradient calculus and variational geometry and thereby bypassing the
implicit function theorem. The approach broadens the notion of
Lagrange multipliers to settings beyond smooth equality and inequality systems. In his doctoral dissertation and numerous later publications, Rockafellar developed a general duality theory based on
convex conjugate functions that centers on embedding a problem within a family of problems obtained by a perturbation of parameters. This encapsulates
linear programming duality and Lagrangian duality, and extends to general convex problems as well as nonconvex ones, especially when combined with an augmentation.
Contributions to Applications Rockafellar also worked on applied problems and computational aspects. In the 1970s, he contributed to the development of the proximal point method, which underpins several successful algorithms including the
proximal gradient method often used in statistical applications. He placed the analysis of expectation functions in
stochastic programming on solid footing by defining and analyzing normal integrands. Rockafellar also contributed to the analysis of
control systems and
general equilibrium theory in economics. Since the late 1990s, Rockafellar has been actively involved with organizing and expanding the mathematical concepts for risk assessment and decision making in
financial engineering and
reliability engineering. This includes examining the mathematical properties of
risk measures and coining the terms "conditional value-at-risk", in 2000 as well as "superquantile" and "buffered failure probability" in 2010, which either coincide with or are closely related to
expected shortfall. ==Selected publications==