quasi
Posts:
12,023
Registered:
7/15/05


Re: An optimization problem
Posted:
Sep 8, 2013 6:40 AM


Peter Percival wrote: >quasi wrote: >> Peter Percival wrote: >>> quasi wrote: >>>> William Elliot wrote: >>>>> analyst41@hotmail.com wrote: >>>>> >>>>>> consider f(x1,x2,...xn) = {1 + sum(xi)^2)} / {1 +sum{x(i)} >>>>>> 0 <=xi <=1 for all i. >>>>>> >>>>>> The maximum function value of 1 occurs at either >>>>>> all x's = 0 or all x's = 1. >>>>>> Can an explicit formula be given for the minimum? >>>>> >>>>> Let g(x) = (1 + x^2)/(1 + x) and look for extreme values of g. >>>> >>>> The function g is not applicable. >>>> >>>> From the context, the OP intended >>>> >>>> f(x_1, ..., x_n) = (1 + sum((x_i)^2))/(1 + sum(x_i)) >>>> >>>> whereas you interpreted it (incorrectly) as >>>> >>>> f(x_1, ..., x_n) = (1 + (sum(x_i))^2))/(1 + sum(x_i)) >>>> >>>> Of the two possible interpretations above, only the first is >>>> consistent with the OP's claim that for all n, the maximum >>>> value of f is 1. >>> >>> I know nothing about such problems, but is it not clear that >>> _if_ there is a minimum, >> >> Let X denote the standard unit cube in R^n. >> >> Since f is a continuous realvalued function with domain X, and >> since X is compact, it follows that f achieves a minimum value >> on X. >> >>> it occurs at some point where all the x_i's are equal? >> >> No, that's not automatic  it requires proof. > >Indeed so. What I meant when I said it was clear is that the >proof is very simple and can be be done in ones head. > >For simplicity's sake let n = 2, x_1 = y and x_2 = z, then my >claim is that if > > (1 + y^2 + z^2)/(1 + y + z) > >has a minimum, then y = z there. Why is that clear? Because >if it weren't so then > > (1 + y^2 + z^2)/(1 + y + z) =/= (1 + z^2 + y^2)/(1 + z + y).
No, your appeal to symmetry doesn't support your claim unless you know that there is a _unique_ point in R^2 for which the function achieves a minimum.
If all you know is that the function f is symmetric in the variables y and z and that f achieves a global minimum value somewhere in the domain, that's not enough to conclude that the global minimum value can be achieved at some point for which y = z.
quasi

