In the world of constrained optimisation (and let’s be realistic, what effort isn’t constrained?), slack vectors ask: “What happens if I push back the walls?”
The canonical purpose of slack vectors is in sensitivity analysis, but I think the metaphor-that-exists-in-your-head-once-you-understand-slack-vectors applies to everyday life too.
When you build a mathematical model of something—starting with assumptions and working through to conclusions—sensitivity analysis questions the assumptions you made at the outset. “What if I was off by 1% about Factor 7? How screwed would I be? What if I was off by 1% about Factor 23?” The slack vector approach treats the optimisation problem as locally linear. So if the problem curves around 5%, 10%, 50%, you should take that into account. Also, look out for large interaction terms—say getting factors 23, 7, and 318 wrong together is much, much worse than getting any individual or pair wrong.
How about elsewhere? I see the slack vector metaphor as being appropriate to GPA. Generic human resources desks the world around want to see “a GPA of at least 3.3 / 4.0” or “a GPA of at least 3.7 / 4.0”, which is an ignorant way to go about things.
Obviously, there’s a tradeoff between how difficult a student’s classes are and how high their GPA is. Someone who challenges herself by switching out Psych 101 for Organic Chemistry is likely to experience a lower GPA — not only in O-Chem, but in other courses as well — as she converts resources that could have gone to other classes into the difficult subject. To summarise a semester (let alone 8 of them) with a single number is to ignore
- the spread and, more importantly,
- the difficulty
of the classes someone takes. Incorporating all the data would mean something like considering her place in several grade distributions per semester. (You might need to condition upon the professor or the classmates to understand the distribution across years.)
In other words, the “minimum GPA” approach ignores most of the optimisation problem. (It looks at the length of the Lagrangian without taking account of its direction or the constraints of the choice space.) Is your firm really trying to hire grade-grubbers who take the easiest classes they can? Didn’t think so.
Strategic business thinking can be thought of with slack vectors as well. (Or, if you’re labour rather than capital, substitute “strategic career management”.) Assume that you are working within certain constraints, but over a longer time horizon you have the ability to change things at least a little.
- Which walls should you push against?
- Which bonds are most worth your effort trying to loosen?
- (Conversely, over time the constraints that were working your favour might turn against you; which risks are most important to prepare for?)