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I'm doing a constrained optimization function, where I maximize a function with two arguments subject to a simple constraint:

Maximize[4*(x^(1/2) + y^(1/2))^2, 5 x + 4 y == 170, {x, y}]

This successfully returns

{306, {x -> 136/9, y -> 425/18}}

This is great. No problem here. But, when I simply replace the power (1/2) with (0.5), as follows,

Maximize[4*(x^(0.5) + y^(0.5))^2, 5 x + 4 y == 170, {x, y}]

I get the following errors:

    NMaximize::nrnum: The function value -33.8342-10.7215 I is not a real number at {x,y} = {34.6632,-0.829053}. >>
    NMaximize::nrnum: The function value -33.8342-10.7215 I is not a real number at {x,y} = {34.6632,-0.829053}. >>
    NMaximize::nrnum: The function value -33.8342-10.7215 I is not a real number at {x,y} = {34.6632,-0.829053}. >>
    Further output of NMaximize::nrnum will be suppressed during this calculation. >>

What's the matter here? I'm very new to mathematica and I don't understand if this is a syntax error or a mathematical error. Any ideas most appreciated.

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    $\begingroup$ It appears that using machine precision numbers (e.g., 0.5) causes Maximize to invoke NMaximize. Performing NMaximize[4*(x^(1/2) + y^(1/2))^2, 5 x + 4 y == 170, {x, y}] gives the same error. $\endgroup$ – bbgodfrey Oct 7 '16 at 22:05
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    $\begingroup$ This works: Maximize[4*(x^(0.5) + y^(0.5))^2, 5 x + 4 y == 170 && x > 0 && y > 0, {x, y}], although I find it peculiar. $\endgroup$ – corey979 Oct 7 '16 at 22:08
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    $\begingroup$ To elaborate on my earlier comment, Maximize definitely calls NMaximize, when finite-precision numbers are used. According to its documentation, Maximize excludes function arguments that lead to complex numbers, but the NMaximize documentation is silent on this point. Evidently, NMaximize does not exclude arguments that lead to complex numbers, and so it fails when they occur. $\endgroup$ – bbgodfrey Oct 7 '16 at 23:09
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From the docs for Maximize:

  • If Maximize is given an expression containing approximate numbers, it automatically calls NMaximize.

That the functions Maximize[] and NMaximize[] have different algorithms is well-documented.

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