## Calculus for Functions That Don't Have Derivatives

###### Associated People:

Heinz H. Bauschke (University of British Columba Okanagan)

Warren L. Hare (University of British Columbia Okanagan)

Yves Lucet (University of British Columbia Okanagan)

Xianfu Wang (University of British Columbia Okanagan)

###### Associated Sites:

PIMS University of British Columbia###### Associated PIMS Programs:

One of the most successful mathematical discoveries of all time is the differential calculus by Newton and Leibniz. It is an indispensable tool in applied mathematics. For example, in optimization - which aims to solve mathematically optimal allocation of resources - one of the key techniques is finding critical points via derivatives.

Unfortunately, many important functions encountered in optimization do not possess classical derivatives. In this case, one creates generalized derivatives that capture the "first order calculus" of the function without requiring traditional differentiability. Most widely studied in this field is the notion of convex functions, which leads to generalized derivatives that are monotone operators [1]. Another important example is the case when the function represents inclusion in a set, which leads to normal cones and set-valued calculus.

At the Centre for Optimization, Convex and Nonsmooth Analysis (COCANA) at the University of British Columbia, researchers are particularly interested in generalized derivatives and their applications. Recent research topics include exploring connections between generalized derivatives and the method of alternating projections [2], building sophisticated computer-aided convex analysis tools to increase researchers' productivity [3], and designing new techniques to approximate normal cones without calculus [4]. These advancements of knowledge are paving the way for new applications in Optimization worldwide.

###### References

- H.H. BAUSCHKE and P.L. COMBETTES, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer (2011).
- H.H. BAUSCHKE, D.R. LUKE, H.M. PHAN and X. WANG, Restricted normal cones and the method of alternating projections, preprint, May 2012. (arxiv.org:1205.0318)
- Y. LUCET, What shape is your conjugate? A survey of computational convex analysis and its applications, SIAM Review 52 (2010), 505-542.
- C. DAVIS and W. HARE, Exploiting Known Structures to Approximate Normal Cones, Mathematics of Operations Research, (to appear 2013).