In contrast to other disciplines in mathematics, problems in
optimization are usually quite easy to state and to understand-even for
those with limited mathematical sophistication. Writing circa 200 BC, the
Greek mathematician Zenodorus considered the so-called isoperimetric
problem: Determine, from all simple closed planar curves of the same
perimeter, the one that encloses the greatest area. Even the mathematically
uninitiated guess correctly that the solution is the circle.
The isoperimetric problem has been the most influential mathematics problem
of all time. It played a major role in motivating the calculus of
variations activity credited to the Bernoullis, Newton, Euler, and Lagrange
in the late 1600's and early 1700's. Yet a complete solution of the
isoperimetric problem eluded these early pioneers. Indeed, it was
Weierstrass who first gave a complete proof more than a century later. In
this talk the speaker will argue that Euler and later Lagrange were one
direct observation away from deriving a sufficiency condition that would
have given a straightforward resolution of the isoperimetric problem. The
missing ingredient was function convexity. We then ask rhetorically: was
convexity of functions not known to the great mathematicians of that time.
Set convexity was known to the early Greeks.
Moreover, the derivation of the Euler-Lagrange equation presented by Euler
and Lagrange is well known to be flawed. A correct derivation was given by
du Bois-Reymond some 150 years later. We argue quite surprisingly that the
du Bois-Reymond's derivation can be viewed as presenting the Euler-Lagrange
equation as a multiplier rule. As such, it would be the world's first
multiplier rule and would precede the very notion of Lagrange multiplier
rules.
