(2) Something missing: As a grad student
studying the Kuhn-Tucker (KT) conditions
and the constraint qualifications (CQ),
there was interest in implications among
the CQs. Two of the famous CQ were (a)
the Zangwill and the (b) KT, but the
implications between them were "missing".
So, that was a problem, a theorem "need to
prove". My approach was to look for a
counterexample among wildly goofy sets,
e.g., the Mandelbrot set or Brownian
motion. As appropriate for the KT work,
both sets were closed. Hmm .... So,
needed an optimization objective
function to be minimized. So, ..., soon
enough, for each closed set there is a
real valued function zero on the closed
set, strictly positive otherwise, and
infinitely differentiable. Then I had a
counter example. Two weeks of work in a
reading course. Published it.
Was doing some AI for monitoring but
wanted a better approach, the "need".
Used the probabilistic concept tightness
to get another approach, the basis of the
"proof", widely applicable because still
distribution free (i.e., made no
assumptions about probability
distributions, e.g., Gaussian). Published
it.
the FedEx BoD wanted some revenue
projections, wanted so much it nearly
killed FedEx. So, as in that Hacker News
URL, got some "intuition", ..., got a
simple differential equation (the
theorem), solved that (the proof).
Currently my startup needed some progress,
and I formulated a suitable theorem and
proved it.
A concern about such theorems and proofs,
for the published ones, the check has yet
to arrive.
Was eating lunch with some well known
mathematicians, and they asked what I was
working on. I explained, "scheduling the
fleet at FedEx, which airplanes go to what
cities in what order". Immediately one of
the mathematicians with contempt scoffed
and said "the traveling salesman problem"
as if that was the "theorem" to be proved,
i.e., P = NP.
Nope: I was just trying to save FedEx
some money. So, my approach was 0-1
integer linear programming (ILP) set
covering; that this is in
NP-Complete was to me next to irrelevant; I
just wanted feasible solutions that would
save money. Maybe over a year the savings
would be some $millions, but each feasible
solution might be $1000 above an optimal
solution. At 365 days a year, I'd leave
$365,000 on the table. Fine with me!!!
To the mathematician, all that
consideration of money was irrelevant --
he wanted to focus on P = NP and regarded
that as too difficult (it still is) and I
was foolish for working on it (I wasn't
working on it). In short I was counting
the millions to be saved, not the
thousands of saving to be missed.
Later there was a 0-1 ILP with 40,000
constraints and 600,000 variables. I used
the IBM OSL (Optimization Subroutine
Library) and in 900 primal-dual iteration
for Lagrangian relaxation got a feasible
solution within 0.025% of optimality.
Lesson: O-1 ILP can be a good tool in
practice, sometimes can save a lot of
money.
So the well-known mathematician and I
disagreed on your "which theorems you need
to prove"!!!
Of course there is the now famous
Garey and Johnson, Computers, and
Intractability, Bell Labs, 1979.
The authors were trying to find a least
cost design for some Bell network.
On pages 2-3 we see some cartoons with
"I can't find an efficient algorithm, I
guess I'm just too dumb."
and
"I can't find an efficient algorithm, but
neither can all these famous people."
It turns out by "an efficient algorithm"
they meant (a) getting least cost
solutions, least down to the last tiny
fraction of a penny, (b) to worst case
problems, (c) guaranteed, (d) with
computer time growing no faster than some
polynomial in the size of the problem.
I just wanted to save FedEx some $millions
a year.
The famous mathematician insisted on
(a)-(d) or no savings at all.
For research math, it is "immature", incomplete, .... But I was aiming the remark at students plus/minus a few years of being a college freshman.
For what theorems to prove:
(1) "Alexander Grothendieck, who spoke of tackling hard problems by creating a gradually rising sea of ideas around them."
https://www.quantamagazine.org/monumental-proof-settles-geom...
(2) Something missing: As a grad student studying the Kuhn-Tucker (KT) conditions and the constraint qualifications (CQ), there was interest in implications among the CQs. Two of the famous CQ were (a) the Zangwill and the (b) KT, but the implications between them were "missing". So, that was a problem, a theorem "need to prove". My approach was to look for a counterexample among wildly goofy sets, e.g., the Mandelbrot set or Brownian motion. As appropriate for the KT work, both sets were closed. Hmm .... So, needed an optimization objective function to be minimized. So, ..., soon enough, for each closed set there is a real valued function zero on the closed set, strictly positive otherwise, and infinitely differentiable. Then I had a counter example. Two weeks of work in a reading course. Published it.
Was doing some AI for monitoring but wanted a better approach, the "need". Used the probabilistic concept tightness to get another approach, the basis of the "proof", widely applicable because still distribution free (i.e., made no assumptions about probability distributions, e.g., Gaussian). Published it.
(3) A problem from outside math: As in
https://news.ycombinator.com/item?id=40893566
the FedEx BoD wanted some revenue projections, wanted so much it nearly killed FedEx. So, as in that Hacker News URL, got some "intuition", ..., got a simple differential equation (the theorem), solved that (the proof).
Currently my startup needed some progress, and I formulated a suitable theorem and proved it.
A concern about such theorems and proofs, for the published ones, the check has yet to arrive.
Was eating lunch with some well known mathematicians, and they asked what I was working on. I explained, "scheduling the fleet at FedEx, which airplanes go to what cities in what order". Immediately one of the mathematicians with contempt scoffed and said "the traveling salesman problem" as if that was the "theorem" to be proved, i.e., P = NP.
Nope: I was just trying to save FedEx some money. So, my approach was 0-1 integer linear programming (ILP) set covering; that this is in NP-Complete was to me next to irrelevant; I just wanted feasible solutions that would save money. Maybe over a year the savings would be some $millions, but each feasible solution might be $1000 above an optimal solution. At 365 days a year, I'd leave $365,000 on the table. Fine with me!!! To the mathematician, all that consideration of money was irrelevant -- he wanted to focus on P = NP and regarded that as too difficult (it still is) and I was foolish for working on it (I wasn't working on it). In short I was counting the millions to be saved, not the thousands of saving to be missed.
Later there was a 0-1 ILP with 40,000 constraints and 600,000 variables. I used the IBM OSL (Optimization Subroutine Library) and in 900 primal-dual iteration for Lagrangian relaxation got a feasible solution within 0.025% of optimality.
Lesson: O-1 ILP can be a good tool in practice, sometimes can save a lot of money.
So the well-known mathematician and I disagreed on your "which theorems you need to prove"!!!
Of course there is the now famous
Garey and Johnson, Computers, and Intractability, Bell Labs, 1979.
The authors were trying to find a least cost design for some Bell network.
On pages 2-3 we see some cartoons with
"I can't find an efficient algorithm, I guess I'm just too dumb."
and
"I can't find an efficient algorithm, but neither can all these famous people."
It turns out by "an efficient algorithm" they meant (a) getting least cost solutions, least down to the last tiny fraction of a penny, (b) to worst case problems, (c) guaranteed, (d) with computer time growing no faster than some polynomial in the size of the problem.
I just wanted to save FedEx some $millions a year.
The famous mathematician insisted on (a)-(d) or no savings at all.