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Question: How to produce a very accurate (and still reliable) result which is not sensitive to the precision?

Question:How to produce a very accurate (and still reliable) result which is not sensitive to the precision?

sursumCorda 1059 Maple 2023
accuracy precision optimization
June 17 2023
1

It says that one may specify the number of significant (decimal) digits when performing calculations. However, it appears that setting a higher precision does not yield more satisfactory results. Actually, the recommended solver (in the Optimization package) can report that a maximization succeeded, and yet the reported solution can be, in effect, incorrect.

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restart;
> 

kernelopts(version);

         Maple 2023.0, X86 64 WINDOWS, Mar 06 2023, Build ID 1689885

  
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fun := ((2*(x+y+z))*(sqrt(y*z*(z+x)*(x+y))/(z+2*x+y)+sqrt(z*x*(x+y)*(y+z))/(x+2*y+z)+sqrt(x*y*(y+z)*(z+x))/(y+2*z+x))-9*x*y*z/(x+y+z)-2*(x*y+x*z+y*z))/(sqrt(x*y*z/(x+y+z))*(x+y+z-sqrt(27*x*y*z/(x+y+z))))
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Digits := 5
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Optimization['Maximize'](fun, assume = nonnegative, initialpoint = ({seq})(w = (rand())(), `in`(w, [x, y, z])))

Warning, no iterations performed as initial point satisfies first-order conditions

  

[2.56454435679811255, [x = HFloat(8.44225460865e11), y = HFloat(1.5121750086e11), z = HFloat(3.58747328735e11)]]

 (1)
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Digits := 10
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Optimization['Maximize'](fun, assume = nonnegative, initialpoint = ({seq})(w = (rand())(), `in`(w, [x, y, z])))

Warning, no iterations performed as initial point satisfies first-order conditions

  

[2.54449612902387834, [x = HFloat(9.50923834782e11), y = HFloat(8.94868300574e11), z = HFloat(2.30179425858e11)]]

 (2)
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Digits := 15
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Optimization['Maximize'](fun, assume = nonnegative, initialpoint = ({seq})(w = (rand())(), `in`(w, [x, y, z])))

Warning, no iterations performed as initial point satisfies first-order conditions

  

[2.53798179305501392, [x = HFloat(9.91986977089e11), y = HFloat(5.97663762096e11), z = HFloat(4.31192847449e11)]]

 (3)
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Digits := 20
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Optimization['Maximize'](fun, assume = nonnegative, initialpoint = ({seq})(w = (rand())(), `in`(w, [x, y, z])))

[2.5281266501331255053, [x = 583804136381.99996354, y = 332585592469.99985371, z = 642614553072.00010884]]

 (4)
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Digits := 25
> 

Optimization['Maximize'](fun, assume = nonnegative, initialpoint = ({seq})(w = (rand())(), `in`(w, [x, y, z])))

Error, (in Optimization:-NLPSolve) complex value encountered

  
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Digits := 30
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Optimization['Maximize'](fun, assume = nonnegative, initialpoint = ({seq})(w = (rand())(), `in`(w, [x, y, z])))

Warning, undefined value encountered

  

[Float(infinity), [x = 996095848033.138573379019582678, y = 0., z = 473765214257.250967995851222582]]

 (5)


 

Download NMaxValue.mw

It is said that the approximate supremum of fun is 3e0 (under the assumptions). 

fun:=`assuming`((2*(x+y+z)*(sqrt(y*z*(z+x)*(x+y))/(z+2*x+y\
)+sqrt(z*x*(x+y)*(y+z))/(x+2*y+z)+sqrt(x*y*(y+z)*(z+x))/(y\
+2*z+x))-(9*x*y*z/(x+y+z)+2*(y*z+z*x+x*y)))/(sqrt(x*y*z/(x+y+z))*(x+y+z-sqrt(27*x*y*z/(x+y+z)))),[(x,y,z)>~0]):

I am not sure if the DirectSearch toolbox performs better. But is there a way to obtain significantly accurate output natively?
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