It has only week on Linear Programming which is nicely done but I think the real value is that it starts with the much more playful Constraint programming and focuses on intuitions and keeping you both entertained and trained which is really hard to do.
The course comes with assignments and the whole thing sort of has a Advent of Code flavor (I kind of have a half baked plan on make this year discrete optimization my AOC theme).
Not strong on the modeling part/business motivation.
On the Optimization Ph.D. qualifying exam, got a "High Pass" and the best score in the class.
In optimization answered a question in the Kuhn-Tucker constraint qualifications and had the paper accepted quickly in Mathematical Programming.
Taught linear programming in a well-known business school for 5 years.
NYC had a few users, in a loose group, of linear programming but were not very good at it. Somehow I stumbled into the group and was by far the best, but there was no chance of a career or job there.
Net, saw next to nothing in "applied". Believing "applied" would have been homeless living on the street, exactly, no exaggeration.
Was not able to help my wife -- she died. Had two cats. They got sick, needed medical care I couldn't afford, and died.
Thousands of resume copies posted -- no meaningful responses.
The Sheriff showed up with several workers with guns, dumped all my clothes, furniture, music materials, e.g., piano, large professional library on the front -- I had to drive off with a few items.
At FedEx I saw a good application, got a memo from the Founder, COB, CEO to pursue it, pleased two crucial members of the BOD, saved FedEx twice, but one manager tried to fire me,
another said there was "no money in the budget" for me, the stock promised in 3 weeks was WAY late, and I'd been commuting between Memphis and Maryland.
Could have done better with a lawn mowing company, LITERALLY.
Actually, big picture: Heavily long, likely still, the main US customer, apparently 99+%, for optimization of any kind was US national
security.
I was manipulated, fooled, and very seriously hurt by the claim "applied".
A fact of life: Nearly always people with money, power, and optimization problems don't understand optimization, fear and resent those who do, and choose just to avoid the subject.
So, no wife, no cats, starting a business, no boss, sole-solo founder, do have some original applied math deep in the technology, and face just US Internet startup reality.
Optimization? Get a 2 hour overview and otherwise f'get about it.
The point of the post is that in my long experience there was not much about optimization that in any significant practical career sense was "applied", i.e., no jobs even to keep one from living on the streets, far from a career to buy a house and support a family.
The point about my Ph.D. with a lot in optimization is that I was quite well qualified in the field, but even with all those qualifications "applied" was not real, i.e., there just was nothing like a career in optimization applications.
Can suspect that, yes, "Ph.D." did seriously damage all career prospects, optimization, math, even computing.
The successes I did have were in computing. At the time the math was a small aid.
Did get paid for some work in applied optimization on some military problems. Looking back, there was some stranger in the office eager to discuss the weather, etc. with me. Maybe I said the wrong things about some of the US foreign wars, and then the stranger was gone. Might have been some high end military job interview that needed only gung ho attitudes toward foreign policy.
Delicate political situation, and I was oblivious about politics. Like, "stick to two subjects, the weather and everyone's health" and avoid "sex, politics, and religion".
Early on while I was in a grad program teaching math, some recruiters came from DC desperate for anyone with some math/physics education. I interviewed: Got a offer and took a job right away. Soon bought a new car and got married.
In those days, around DC, to get a job, just look in the WaPo, apply, go on the interview, show some knowledge of some of computing, get an offer, compare a few offers, with, say, a 15% raise, and accept one -- worked great. For a while, at GE time sharing national HQ, was the main guy for the applied math library, e.g., the FFT, regression analysis. Later a good background in "applied" optimization, worthless.
You should have tried wall street. At this point they are the real supporters of mathematicians.
We had optimization problems everywhere and had physics PhDs reinventing mathematical algorithms and keeping things “proprietary”.
Right now i work in a startup that essentially writes optimization routines for portfolio problems.
I was in NY and close enough to NYC. I'd just published a paper in anomaly detection in complex systems, gave a talk at the main NASDAQ server farm, and later at Morgan Stanley. No real interest.
Sent a copy of my anomaly paper to a hedge fund, got an interview, was asked by one of their junior people "If know the correlation between A and B and that between B and C, what about A and C"? Okay, maybe: Start with the cosine of the sum of two angles???
Asked them for a reference on investing math -- did already know about the old Markowitz work, the efficient frontier, the role of quadratic optimization, about to do more with stochastic differential equations for the Black-Scholes work -- got the book they mentioned, saw that its math was all junk, and didn't follow up with the A and C contact, a mistake.
Did send a resume to Simons.
Looked into deterministic optimal control, Athens and Falb, talked with Athens, later talked with an Athens student who knew something about Simons and claimed that he hired mostly Russian mathematicians. So gave up. A mistake. I was naive.
Later, of course, Simons explained that he liked people who, say, via math but any math, had shown some ability, and I had some evidence I could have shown. My Math SAT was high enough that maybe I even beat Simons?
I was naive: Assumed that a carefully written resume was necessary and sufficient and that anything else was superfluous and unwelcome.
Nope: In practice in the real world, keep trying different things. Do send reprints of published papers. E.g., when I was at Georgetown, computer center staff and teaching computer science, a prof had some teaching software as a front end to the IBM SSP (scientific subroutine package) and in testing found that two of the IBM routines were too slow and the third had poor numerical accuracy. So, I wrote plug compatible versions -- used some (n)ln(n) software and some tricky double use of memory to replace the n^2 software and used some Forsythe and Moler work to fix the accuracy problem -- seemed too simple to me, but COULD have sent Simons that work. Once did get a lecture on differential geometry from a student of A. Gleason and had a copy of some S. Chern notes -- could have studied those and sent something to Simons. How'd I know Simons knew Chern??
There is a recent remark: "Don't give up. Keep plugging".
I was naive. Knew much more about math and computing than people and personality.
Since WWII, the US military has pushed hard to have more -- students, professors, and research -- in math and science. In high school, taught myself the math, learned the physics at a glance, otherwise goofed off (had a girlfriend drop dead gorgeous), but did well on the state standardized tests, so got sent to summer math/physics enrichment programs. I swallowed the bait hook, line, and sinker. I'd recommend:
"Always look for the hidden agenda."
"Believe none of what you hear, half of what you see, still that will be twice too much."
"Who you know can be more important than what you know."
There were some opportunities to "know" some powerful people, but I was naive.
My Ph.D. advisor was a nice guy, but I got to him after the fallout of a bad civil war in the faculty and never much talked with him. For my dissertation, some applied math, and had the main idea on an airplane flight before the Ph.D. program, in the first year wrote a 50 page first draft, later cleaned up the math, used Fubini's theorem in a short proof that my math was optimal, wrote some illustrative software, typed in the paper, showed it to my advisor and the rest of the department, had a famous guy a Chair of an orals committee to review the dissertation, and graduated. My advisor and one of the faculty (connected in DC and later President at an Ivy) knew a LOT about politics, but I was naive.
For a while, my career, in computing but with some math on the side, e.g., the FFT and digital filtering of Navy sonar signals, was going well, so I got the Ph.D. in applied math just to do better in THAT career and with ZERO intentions to be a professor or do academic research. That career direction was MY idea, mine alone, ..., a BAD situation!!! I was naive.
> I was naive. Knew much more about math and computing than people and personality.
Do you think not learning math would have helped you understand people at a younger age? It sounds like you just needed time to grow socially and in practicality. For most people on this forum, that’s a challenge regardless.
About people in math and the more technical parts of computing, I've guessed that poor socialization has played a role.
But when my career was okay, it was in computing, and I did well enough in the socialization.
Can consider these and those issues, but my experience was that "applied" optimization, as in the book title in the OP here, was too near the empty set.
It isn't just me: My professors in applied math and the ones in optimization were not getting much if anything in consulting. I've been recruited and hired, but never for optimization.
Here I'm trying to do a service to the readers: Be very careful about the idea that there is significant career help via "applied" optimization.
Sounds like you opened up the newspaper and scanned for “mathematician”. Leveraging phd research into a great job is a tough. Re-skilling into a normie engineer/technician/analyst, is not.
My point is not to criticize your job hunting skills, it’s to suggest that this an undue psychological burden in your life and is perhaps masking other causes and personal challenges.
Naw: The WaPo period was before my Ph.D. The ads were for computing -- math not mentioned. For some years, the career was computing but with some math, e.g., the FFT (fast Fourier transform), ....
I never wanted the Ph.D., what I learned there, the research I did there, to be the basis of a career. Instead, before the Ph.D. I had a good career going with computing and, at times a crucial help, some math, and went for the Ph.D. ONLY to do better at THAT career. For my career, the day I entered the Ph.D. program was a BIG step down, and what I'd learned about optimization was, in a word, WORTHLESS.
My main point here is on the word "applied" for optimization: I was well qualified, and happened to publish some research in optimization, but discovered that "applied" optimization was not the basis of a good career. Here I'm just reporting that fact. I doubt that there is still any real career opportunity in "applied" optimization.
So, a book title with "Applied" Optimization is to me a outrage.
I wasn't stuck on "optimization". For a while worked in the first wave of AI (artificial intelligence via the Rete algorithm). Then published in mathematical statistics. I was perfectly willing to mow grass, shine shoes, ..., do anything that would support me financially, be reasonably safe, and not seriously illegal but discovered that "Ph.D."
on the resume blocked any such. Thought about taking "Ph.D." off the resume but was afraid that I'd get into trouble due to the gap in time.
Here my point, complaint, warning, contribution to others, is: My long experience was that there is nearly no career in "applied" optimization. A second point could be, outside of academics, a Ph.D. can hurt your career. Try leaving it off your resume. A Ph.D. might be worse for your career than a felony conviction; no joke (my legal history is totally clean).
In life, we are forced to make important decisions without good information. In my career, at times I did well, and at times I didn't.
E.g., by middle school it seemed accepted and true that education helps, more education helps more, education in the STEM fields is the best, a Ph.D. is the best education, and, thus, a Ph.D. in a STEM field should be really good, e.g., easily enough to buy a house and support a family.
Truth: Nope, too simple. I couldn't take care of my wife, kitty cats, get a job, any job, at all, ANY job, got run out of the house by the Sheriff with guns.
With a BS "With Honors" in math, I got strongly recruited. With a Ph.D. in applied math, including optimization, I got strongly rejected.
Yup, it hurt. I was manipulated, lied to, and hurt.
"psychological burden": Maybe those are the right words. But millions of people have suffered worse, e.g., The Great Depression, wars, Covid in the family, and much more, and still did well.
Don't know the solution in general.
For me, now, still good in math and computing, with .NET, etc. got a Web site, with some math at the core, running easily enough, and intending to go live, get some viewers, run simple ads (standard sized rectangles), and make some money. In this, want to remain anonymous and not be a public person.
And want to OWN the business. Have someone list what papers I need to file for a business, an LLC, etc. Get an accountant. Get and receive revenue. In simple terms, add up the expenses and keep the rest. Eventually sell the business and pursue, say, mathematical physics.
> I doubt that there is still any real career opportunity in "applied" optimization.
I agree there are no ready made jobs for that.
But you yourself know there are optimization problems all over real life. It’s a sales problem. Companies don’t know what they need or who has it.
> there is nearly no career in "applied" optimization
Agreed. But that’s true of all PhDs. The only difference is business guys see “computer science” and have an idea of where it fits in their org. It’s easier to sell. But in reality there is no business for experts in complexity theory or category theory type systems.
Making money involves solving practical problems. Even professors take a two job approach, mixing official research to get tenure with stuff they are actually interested in.
> With a BS "With Honors" in math, I got strongly recruited
This is very unfortunate. Because professors grew up competing in an academic tournament for their jobs they think that’s how the whole world works.
> A fact of life: Nearly always people with money, power, and optimization problems don't understand optimization, fear and resent those who do, and choose just to avoid the subject.
Food for thought: the solution to real-world optimization problems is often dictated by constraints instead of optimal values.
This means that if you fail to understand the constraints, or even fail to identify them, then whatever your solution to the problem is, it will be wrong. And it will be obvious to those who are aware of the constraints.
Now, you're complaining that those presenting you with problems "don't understand optimization".
From your anecdotes you were the one tasked with clarifying things to them. From the sound of it, you didn't accomplished that, and it was unclear to stakeholders whether your output even provided any value worth keeping.
Have you ever considered the possibility that you failed to understand the actual problems presented to you and even failed to clarify why your output was aligned with anyone's best interests?
Naw, you list some mistakes, but I didn't make any of those.
> From your anecdotes you were the one tasked with clarifying things to them. From the sound of it, you didn't accomplished that, and it was unclear to stakeholders whether your output even provided any value worth keeping.
First, for any application, there has to be some practical interest. My view, there isn't much. The schools of math, engineering, and business have given optimization a big push, back at least to Dantzig, but from my long experience the interest was and is still just too low for "applied" optimization to have much in applications.
Cases: Sure, there has been a professor at Princeton who applied optimization to oil refining: What mixture of crude oil to mix into the refinery and what mixture of refined products to take out. Maybe a few, large livestock operations actually do run some diet problem solutions. And can use 0-1 optimization for Sudoku problems? What path for picking orders in a big warehouse, for Amazon or Walmart? A simple traveling salesman problem, and for a good enough solution build a minimum spanning tree and walk around that -- maybe they are doing that already. Assembly line balancing: Assign workers to positions to maximize the speed of the slowest worker assignment. Is anyone actually doing that? Even if they are, the solution is quite simple. Yes, a start on P vs NP was at Bell Labs designing networks. So, maybe with the Internet there are still valuable applications? Considered that. Got an interview at a company trying that. They were impressed by what I'd done at FedEx, but they were nearly dead and, I suspect, soon died. Maybe with big logistics, ocean, rail, trucks, warehouses, there are some big logistics problems where optimization could save a lot -- applications enough for careers? Better than grass mowing? When I got my Ph.D., the Chair of my dissertation orals committee was a big name in logistics -- saw no evidence
of significant interest in applications. No ones in the halls. Phone not ringing. No suggestions of contacts for me.
Look, when there is a big need, ESPECIALLY when there is big money involved, it soon gets obvious, and the US economy gets to it right away. In that, "applied" optimization is not hot, warm, or much above freezing.
Right, you are mentioning formulation:
(1) They had already formulated a 0-1 optimization problem. It had 40,000 constraints and 600,000 variables. They had tried the then popular simulated annealing, ran for days, and quit. So, the formulation was done and not mine.
I worked hard, with the IBM OSL (optimization subroutine library), did 900 primal-dual iterations, Lagrangian relaxation, got a feasible solution within 0.025% of optimality, within two weeks, for free, a free sample, and never heard from them again. They resented and were afraid of my success.
(2) Another company was working a little more generally in optimization. Had a crude heuristic running. On some of their problems, 0-1, linear, again was successful with the OSL, and got only insults and resentment. Continued on, gave them a nice formulation, better than their heuristic, and path through optimization, and got fired. They'd hired me and wanted to fire me before 6 months was up. They were not very good with linear programming at all, and I was a LOT better at what they were doing in the formulation, math, and computing, and their reaction was they didn't want me for competition.
(3) In a military group, did well with some non-linear optimization (their formulation). Then they had a challenging strategic problem. I did a formulation of a Monte-Carlo solution and wrote and ran the code (used an Oak Ridge random number generator I'd programmed in assembler). They called in a famous probability professor for a review. His remark was that there was no way the Monte-Carlo could "fathom" the tree. He was right; the tree was huge. But each trial of my Monte-Carlo yielded at each point in time a random variable on 0-15, and the law of large numbers applied right away. It wasn't D-day, but suppose it was: The tree of possibilities was enormous, but the, say, number of Allied soldiers killed was, what, 0-200,000. So, each trial give a random variable value at, say, each second, for, say, 48 hours -- the law of large numbers applies and could tell Ike the distribution of number of deaths, the expected values, the median, the variance for each second of the 48 hours. Passed the review. One guy there used my random number generator on one of his old problems, got significantly different results, was afraid, said "I don't want you in the center of all my projects", and I got ignored on the way to being fired.
(4) At FedEx, had written a program that showed the BOD that the program made the fleet scheduling easy enough and saved the company. So, to do better, formulated a set covering direction. Savings? 1% would have been $millions a year. The founder, COB, CEO wrote a memo making that my project, but my boss, a Senior VP, said that there was no money in the budget for me; I'd been commuting between Memphis and Maryland where my wife was in her Ph.D. program; the stock promised in three weeks was very late; and I went for a Ph.D.
Actually another student at another school ran with my set covering formulation for his dissertation.
The high level, overview, simple fact of life, is as I described: There just is no real career in "applied" optimization. That horse is nearly dead and should not be further flogged. Millions of US families have a house, stable marriage, and healthy children, and I'd believe that fewer than 20 of those families are supported by careers in "applied" optimization -- maybe 0 families.
I don't think I got my point across. The constraints I've referred to aren't a reference to how problems are formulated, but what leads decision-makers to make decisions. The goal of any number-cruncher in a corporate environment, whether they are data scientists, machine learning engineers, operations research specialist, etc., is to advise decision-makers on what are their options. If they stop collaborating and start to lift barriers and create problems, instead of adding value they turn themselves into a bigger problem. And I'm not even touching on the problem of gains.
Adding to that, decision-making is all about tradeoffs. All problems have sensitivity to input parameters. This means that there are always choices that can be made to have different solutions if decision-makers are willing to accept the tradeoffs. They always do, because not all constraints and requirements are expressed or expressable in a problem statement. More to the point, the output of an optimization problem is not reaching the optimal point, but to improve on the current performance.
Not everything in life can be limited or summarized in crisp values. Moreso in the business world. Do you understand what I'm saying?
> Savings? 1% would have been $millions a year.
That's your projection. And 1% of anything is completely irrevelant, I might add. No wonder the project was killed.
I worked in projects that we could advise cost improvements of around 4% and the project was slashed as well. What's the year on year variance though? 1% is a fraction of inflation. How many meetings would they need to meet a ceiling of 1%? Is 1% the value-added of a PhD? Do you get what I'm saying?
At one point, we need a correction: The optimization was to schedule the fleet for FedEx. They were spending big time money on the airplane operations, and a better schedule, saving even 1% of that, would amount to maybe $millions a year, at any rate, money worth saving and where, in comparison, my work cost peanuts. At FedEx at the time, the 1%, the automation of the fleet scheduling, the business planning potential, made the project worthwhile.
The "decision maker" was F. Smith, founder, COB, CEO. My office was next to his. On paper I reported to a Senior VP, but in reality reported directly to Smith. Smith wrote a memo giving the optimization problem to me -- I still have a copy. The project was approved, and by passing all the executive considerations you mentioned. The project was not "killed". Instead, due to (a) commuting between Memphis and Maryland where my wife was in her Ph.D. program, (b) wanting to stay in Maryland, where I'd done the best work for FedEx and saved it the first time (access to consultants and time sharing computing), (c) the Senior VP I was reporting to telling me there "was no money in the budget for me", and (d) the promised stock very late, I left for the Ph.D.
I got Smith to approve the project by explaining "Integer linear programming set covering" to him for this application: (i) Take all or a reasonably large subset of all the reasonable tours from Memphis and back. All the planes were the same, French Falcons, .... (ii) For each tour, program standard means of evaluating flight times and costs and find the cost of the tour. Throw out some tours for however goofy reasons, e.g., can't fly over this city at that time of day. Note, some of the costs and constraints were really goofy, way beyond what could be converted to software even for non-linear integer programming. Such is the magic of set covering enumeration.
(iii) Intuitively regard each tour as a piece in a jigsaw puzzle where have lots of extra pieces, each piece has a cost, and want to cover. "set covering", the board exactly at minimum total cost. That's when Smith understood the work well enough to approve it as my project. Considering what airplanes cost, my project was not very expensive.
(iv) Now have a 0-1 integer linear program with one row for each city and one column for each tour. In a column, a row i = 1 to 90 is 1 if the tour serves city i and a 0 otherwise. The cost of that column is just the cost of the tour. The right side is all 1s. The constraints are all >=. With 90 US cities, there were only 90 rows in the linear program. So, use some LP (linear programming) software to get solutions feasible and optimal or nearly so. At some point if necessary just use old branch and bound. Worth a try. My computing was VM/CMS time sharing on relatively large IBM mainframes.
"1% is a fraction of inflation." Maybe you are saying that CEOs should have optimization involved only for super big aspects of the company, like Ike's work on D-day, and ignoring a 1% reduction in the cost of M1 rifle ammunition. Well, not then at FedEx. Fleet operating costs were by far the biggest expense of the company and .... Even for D-day, such a 1% reduction might have been worthwhile but to be handled by some Colonel and not Ike!!
Your other points seem to be correct. One response is when working for US national security, some of the problems were
"strategic" for the US and where ... would be as you describe "advising" "options" at no higher than some mid-level uniform in the Pentagon and not for the POTUS. But for my work in the commercial economy, I never was trying to advize the CEO of some $800 billion company on some crucial decision. And I didn't work on enough problems to encounter many of the considerations you listed.
Again, my main point here was just the "applied" optimization in the book title in the OP. My experience, academic, military, commercial, student, professor, research and applications, indicates that outside of the US military there is nearly nothing real about "applied" optimization -- optimization applications are nearly like hen's teeth.
Maybe Amazon, Walmart, Google, Microsoft, and a few of the largest companies have an office of planning and analysis on the organization chart reporting to some VP for something or other and there occasionally develop/run some optimization models, but I never saw any evidence of such. Sure, such an "office" would have to do the TLC of the CEO you mentioned. I'd guess that such an "office" would get their optimization done with a lot of contact with some professors, but the only professors I ever saw doing any such work were the few I contacted. I just didn't see any credible evidence of "applied" optimization in the US commercial economy.
Now, in the last 10 years or so, everything about such optimization -- data collection, data manipulation, word processing for the math, the basic desktop computing, statistical tools, and, sure, Gurobi, etc. are just MUCH better. Soooo, maybe now the US commercial world is ready to exploit optimization like Dantzig, Kuhn, Tucker, Arrow, Hurwicz, Nemhauser, etc. intended. Maybe.
If I got an offer, now, for such work I'd turn it down and stay with my startup. (a) You're talking too much in office politics; that's not one of my specialties; and I have none of that in my startup. (b) I know the math for my startup, all nicely written up with TeX, but I'm rusty on a lot of the rest of math. E.g., for Lagrangian relaxation I'd look at my notes for the last time I did that. For some of the classic integer programming on graphs, I'd go for my grad school notes. For anything in probability, I'd want to review the Radon-Nikodym theorem and conditional expectation, all the standard theorems on convergence of random variables, etc. (c) From the time I got the offer, I'd have to start spending my money, and I might get fired before I even got that back -- such a job would be a gamble where I could lose money significant for me.
Again, my post here was just to object to "applied" for optimization. No way am I looking for a job; not now; not any more; not again. Instead I'm staying with my startup.
> A fact of life: Nearly always people with money, power, and optimization problems don't understand optimization, fear and resent those who do, and choose just to avoid the subject.
Worth considering this hard question: who optimized their life better? Them or us?
Yup. Applied Optimization 101: For decisions in life and career, avoid the applied math approach to optimization.
In particular, except maybe for some work in US national security, don't try to have the applied math of applied optimization for a career. Don't spend a lot of time studying optimization.
Later might be able to be the founder, COB, CEO of a startup where some math is a big advantage.
Fine. But I'm still not finding the private planes, yachts, mansions, or even houses with wife and children of people with careers in "applied" optimization. Computing, startups, venture capital, lots of careers. "Applied" optimization? In the years after my Ph.D., I didn't hear of them. As I was teaching optimization in a business school, the phone didn't ring. The phone did ring for some statistics, one from a company with sales districts and one from a law firm.
Hey I'm just replying because I currently have a BSc with Honors in math and a double major with econ but I'm having trouble in the current market. I also am interested in pursuing a master's in combinatorics or optimization. Is there anything you recommend besides not getting the PhD?
I know it goes against the thread but I am considering a career in academia. I'd like to become a Prof if possible. If that doesn't work out then a job that allows me to solve interesting math problems everyday. I was thinking optimization research but maybe not from how this thread is going. Besides that working for Renaissance tech or a similar company also interests me.
I'd strongly encourage you to leave academia for at least 2 years, preferably a little more -- and go intern, or work at, at least 2 or 3 different companies. This will benefit your mental health, working practices, and sense of perspective.
Academia is full of superstitions and misunderstandings about 'industry', and is an ideologically closed system designed to present itself and its practices as of the highest value. Including, as mentioned here, hijacking 'industrial values' (such as utility, applicability, etc.) and presenting itself with them.
This is in part a straight deception: academics need to lie about this to governments and other funding agencies in order to get funding. And in part naivety, their sense of 'application' is so far removed from any, that they are poor judges of it.
Enabling these deceptions is, of course, that academic output does occasionally have an impact over longer time horizons. But it is more the case that something written ~30 years ago will today find some use by a small team somewhere -- which might have a lot of impact, but isnt on the scale or time lines you'd need for your research today to give you an ROI.
Indeed, as you'll see if you're in industry long enough, there is often something very suspicious and deceptive about the values of various research communities. Even when researchers are told: no one cares about this, this does not have the value you say it does, this isn't a real problem etc.-- they continue. Why? Because they aren't interested in solving problems people have, they are interested in whatever puzzles are current within their community.
If you want to solve real problems using research, it is better to go out and undertand what problems people actually have, then go back into research aiming to help them.
Bro, you spend a really long time getting great at this and from the bottom of my heart I feel your pain and believe we need to improve a lot as a society to take advantage of talents like you better.
Thanks for investing so much time to it, I share your sadness for your wife and hope you get ao much good as you tried to bring to the world, man
Just yesterday I sketch a solver for a board game: "Search for planet X".
The objective of the game is to figure out what kind of planets are hidden in sector of the boards using clues like: "There are 2 comets between secotr 3 and 7" or "No comet is next to an asteroid ".
Sketching the solver was incredibly fun and rewarding!
This was posted as a comment on a thread about a new Google tool for LP [0]. It was in response to someone asking for resources on learning linear programming for business applications. It looks like the examples have been solved using Excel, and it's for business students at MIT. Definitely not cutting edge.
The original posting is about new tools and algorithms, with some more analysis. Well beyond my background from undergrad courses in LP and OR, but probably more relevant and insightful to you.
The tools probably have changed but the fundamental language is the same. The same way that you need to wire your brain to see how a problem can be casted as a dynamic programming one, you also need to learn how to formulate problems as integer/linear programming ones.
For example all of the "hard" leetcode problems can be casted as math programming ones. But the interviewers will not appreciate this solution approach lol.
Once you conquer the logic/language then learning the tools is the easy part.
> But the interviewers will not appreciate this solution approach lol.
I once witnessed a programmer with a PhD in Maths find closed form formulas for a lot of questions where it was expected to write some code with loops building/accumulating a result. As a simple example, to explain what was going on, if the question would be "calculate the 100th fibonacci number", she would just use Binet's formula to do so (as opposed to using a loop). I was rather impressed how often that happened.
Is Binet's formula really that practical a way to calculate the Fibonacci numbers (except asymptotically)? The problem is, you have this nice clean expression, but you'd still have to implement a bunch of fancy arbitrary-precision arithmetic to approximate the golden ratio through Newton's method. In other words, the formula gives much more information about the structure of the Fibonacci numbers than their actual values.
For evaluating the Fibonacci numbers (as with any other integer linear recurrence), I'd generally prefer the matrix-exponentiation-by-squaring approach, or one of the simplified formulas based on it. Those don't need anything more complicated than bigint multiplication. [And from there, taking the ratio between two values gives you a quick way to approximate the golden ratio!]
Once you have postulated BigInt as available, the mathematician is going to make a rational approximation for phi using the continued fraction expansion that they know by heart (because of its “simplicity”).
Calculating φ from its continued-fraction expansion is equivalent to just iterating the Fibonacci sequence normally, since its convergents are precisely the ratios between the Fibonacci numbers. At that point, it's totally redundant to use Binet's formula on the approximation, since you have the values already!
If you want to beat the O(n^2) runtime of the trivial iteration, you pretty much have to use Newton's method for φ, exponentiation by squaring on the matrix form, or another method with faster-than-linear convergence.
If you think it is you should read up on linear algebra, specifically its use in finite difference equations and how that relates to linear differential equations.
The astonishment doesn't get less, but it shifts from Binet's single formula to the exponential map, and maybe the fundamental theorem of algebra (or generalisations).
Depending on the job, interviewer gripes may be legitimate, or at least they should give you the opportunity to write a different type of solution.
If they’re writing a compartmentalized library specific to their domain, it’s fine. I’ve worked with a Stats PhD doing that.
If you’re dropping them into a shared codebase, the comprehensibility of their code to the other people on their team is essential. Great code that depends on knowledge no one else on the team has is not great. You end up with “That’s going to take forever to change, Steve wrote it”.
Agree, I‘d say also the term „mathematical programming“ sounds really old school and never was that fitting to begin with.
„Learning how to formulate problems as integer/linear programming ones“, as another commenter put it, works great if it‘s a natural fit and sure is fun for idk 7th grade math text problems I guess but OTOH squeezing realistic problems into systems of hundreds of equations (or more if dealing with linearizations of inherently non-linear/concave/multi-step problems) to satisfy tool idiosyncracies calls for additional tools in your arsenal.
Based on the code output in the book, it is old. But the book seems pretty easy to follow even if you are not strong in math. Hopefully, in the near future I will be able to pass a book like this to an LLM and have it enrich it with code examples in a programming language I am familiar with.
