RESUME for Bruce CRAVEN

D. Sc. (Melbourne) (1973)
Department of Mathematics and Statistics, University of Melbourne..
Victoria 3010, Australia.
craven@ms.unimelb.edu.au

What do I do?

For many years, a mathematician at Melbourne University, with an industrial background before that. Published five books (the latest on Control and Optimization) and 180 articles. Main professional area is Operations Research and Optimization, but interested in everything else mathematical as well. Involved in computing, on and off, for 40 years. (Fond memories of dear old CSIRAC, the first digital computer built in Australia.) Heavily involved in desktop computing since the TRS80 (I bought the Dick Smith clone), and then the Macintosh. Now retired but still researching mathematics, in association with Melbourne University and Victoria University.

Professional information

See the following links.
Curriculum Vitae
Books and Research Papers
Notes on Research and Graduate Students

Research

Some of the ideas behind my research are briefly as follows. The minimization of a function f(x), subject to constraints g(x) <= 0, can be described using a Lagrangian function f(x) + vg(x), where v is a Lagrange multiplier. If f is vector valued, then "minimum" at a point p must be redefined, replacing f(x) >= f(p) by NOT [f(x) < f(p)], with some ordering of the vectors. The Lagrangian L is replaced by tf(x) + vg(x), with an additional multiplier t. If the functions are differentiable, then L has zero gradient at the minimum, for some t and v. There are analogs of this result when the functions are not differentiable, or their values are sets instead of points. One approach is to smooth f and g by averaging their values over nearby points; this gives a smooth problem, approximating the given one.
When do the Lagrangian conditions, in turn, imply a minimum? This is true if the vector function F = (f,g) is convex, thus if
F(x) - F(p) >= F'(p)(x-p).
But we can replace x-p by some "scale function" of x-p, and it still works. This makes F an "invex" vector function. This extends to functions F that are not differentiable, in two ways. One can replace derivatives by tangent cones to the graph of F. Or F may satisfy an inequality
tF(x) + (1-t)F(p) >= F(z)
for some suitable z, depending on x,p,t. We still get sufficient conditions.
If there is an equality constraint h(x) = 0, then this makes x fall on some curved surface. More generally, what happens to "invex" when x lies on a manifold? A somewhat restricted "invex" corresponds to "convexifiable", thus F can be made convex by transforming the underlying space. This idea works for manifolds, if we assume singular points are well behaved, and then "invex" corresponds to a topological property ("zero index") at singular points. There are extensions to invexity for vector functions. For a characterization of invex that does not require the "scale function" to be known, see the Global Invexity paper cited below.
Lagrange multipliers, like shadow prices in linear programming, measure the sensitivity of a minimum value to small perturbations of the problem. It is harder to find what happens to a minimum point when the problem is perturbed. This requires the problem to have a stability property, and this happens when "invex" is strengthened a bit. All this works also with optimal control problems, where x is replaced by a "state function" and a "control function".
More recently, optimal control has been applied to various economic and financial models.

What else?

In Ogden Nash's lines:

... got no chillun !
Got no use for penicillun !

Put time and effort into Christian church activities, modern
languages, travel, mathematical links with several Asian countries,
Macintosh computer, and giving and getting information on the internet.
Other interests include reading, classical music, theoretical physics,
more reading. What computing things do I care about? Technical word
processing (MarinerWrite preferred, Word disliked), a few selected
Web sites, mathematical number-crunching and programs to do it,
desktop computers (rather than mainframes).

A few of my research papers may be downloaded (as zip or pdf files)

Global Invexity
Optimal control on an infinite domain
Pontryagin principle for a partial differential equation
Characterizing invex
Generalized invexity
MATLAB files for SCOM Optimal Control package

Electronic mail:

craven@ms.unimelb.edu.au