Often, there is no good answer. There are only tradeoffs.
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Something similar to what you said: if you stray from the optimum a bit, some factors will get better, and some will get worse. These will almost completely cancel, so if you’re near the optimum, you needn’t worry too much about getting it exactly right.
For example, if there’s one parameter x, and two costs, one x^2, and the other (x-2)^2, the derivatives of the costs are 2x and 2(x-2), so the minimum is at x=1. At this point, the derivatives are 2 and -2. If you move a little from the optimum, to first order your objective function will not change. For example, At x=1.1, cost one is 1.21, cost 2 is 0.81, but the total cost is now 2.02 – barely any different from the value at x=1.
Corollary: in debates such as abortion, there are strong reasons both ways, *but*, since they nearly cancel out near the optimum, exactly where you draw the line is much less important than the canceling factors that go into it. So concentrate on optimizing problems you didn’t know existed and are therefore likely to be far from optimal.
You’re usually a fountain of geeky charm and wisdom, but I really have to take exception to the use of a cost function (implying a utilitarian view) in making decisions about abortion. :)
For once, , I completely agree with you. :)
“Cost function” is perhaps a little misleading; one shouldn’t decide to ban or allow abortions based on short-term factors such as whether diapers or abortions are cheaper. But, there’s really no way to make an informed decision without some sort of evaluation function telling you how much you like different possibilities, that includes terms for things such as the mother and child’s freedom.
Assuming that any kind of evaluation function is appropriate in this case, the notion of such a function is not at all instructive here. How one chooses to define the inputs (and there is no objective way to do so, no matter how much an analytical mind might will it) is far more pertinent than the behavior of such a function.
Something similar to what you said: if you stray from the optimum a bit, some factors will get better, and some will get worse. These will almost completely cancel, so if you’re near the optimum, you needn’t worry too much about getting it exactly right.
For example, if there’s one parameter x, and two costs, one x^2, and the other (x-2)^2, the derivatives of the costs are 2x and 2(x-2), so the minimum is at x=1. At this point, the derivatives are 2 and -2. If you move a little from the optimum, to first order your objective function will not change. For example, At x=1.1, cost one is 1.21, cost 2 is 0.81, but the total cost is now 2.02 – barely any different from the value at x=1.
Corollary: in debates such as abortion, there are strong reasons both ways, *but*, since they nearly cancel out near the optimum, exactly where you draw the line is much less important than the canceling factors that go into it. So concentrate on optimizing problems you didn’t know existed and are therefore likely to be far from optimal.
You’re usually a fountain of geeky charm and wisdom, but I really have to take exception to the use of a cost function (implying a utilitarian view) in making decisions about abortion. :)
For once,, I completely agree with you. :)
“Cost function” is perhaps a little misleading; one shouldn’t decide to ban or allow abortions based on short-term factors such as whether diapers or abortions are cheaper. But, there’s really no way to make an informed decision without some sort of evaluation function telling you how much you like different possibilities, that includes terms for things such as the mother and child’s freedom.
Assuming that any kind of evaluation function is appropriate in this case, the notion of such a function is not at all instructive here. How one chooses to define the inputs (and there is no objective way to do so, no matter how much an analytical mind might will it) is far more pertinent than the behavior of such a function.