A question on maximization

NaturePaper

Hi everyone,

Suppose f =f(x_1, x_2,...,x_n) be a real-valued, any-time differentiable function. Let each x_i=x_i(u_1, u_2,......,u_{2^n-1}) be a linear function of reall u_i's. Let f=g(u_1, u_2,...,u_{2^n-1}). Then is it right that Max f w.r.t. x_i=Max of g w.r.t. u_i?

Sorry for the inconvenience of typo. I don't know how to use LateX fonts here.

Regards,
NaturePaper

HallsofIvy

If a f has a maximum, then that maximum is a specific number. That number is larger than all other values of f no matter what variables you are using. Thus, the answer is "yes". The maximum value of f is independent of the variables.

NaturePaper

@HallsofIvy,

Does the result will change if all the primary variables x_i's are restricted to have values from an interval [a,b] subset of R?

Thanks & regards,
NaturePaper

HallsofIvy

If you also restrict the "u" variables so that the range is the set of permissible x variables, no the result does not change. Of course, if you allow values of the u variables that would give unallowed x values, then maxima might well be different.

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