Finding a boundary of solution for Multivariate(4 variables) inequalities

Questions
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1

Hello,
I am dealing with multivariate polynomial inequalities, 4 variables in particular. I am trying to find the boundary of the solution for these 4 inequalities using “FindInstance”. However, even after running for a day I am not able to get the solution. I have tried to make the boundary more relaxed still i am unable to find the solution. Mathematica doesn’t return anything (not even a blank matrix).
I am new to Mathematica, are there any other functions I could use to deal with multivariate inequalities?
[Edit] I have attached a file to demonstrate my problem.

Multivariateineq.nb (7.69 KB)

2

If you could edit your post to include your inequalities, or at least similar inequalities that would demonstrate an equivalent problem, and describe what form you need the solution to be in then readers might be able to experiment with this and see if they can find you a solution.

Since you describe the boundaries of inequalities, would finding a solution to equalities or subsets of those equalities be close to what you are looking for?

3

Thank you Bill for your suggestion. I have formulated a case and attached with my original post. Hope this would help.

My solution would require subsets of those equalities; specifically defined by the surface under variables x2,x3,(x4/x5).
Thanks again.

4

Please check this very carefully to see if I have made any mistakes

x1=15.75*10^-6; a=x1 x2 x3^3; b=x1 x3^3; c=(9 x1+x3)x3^2; d=36 x1 x3+3(3+x4/x5)x3^2;
e=60 x1+12(3-2 x4/x5)x3; f=60(1+x4/x5); X=b c-a d; Y=b e-a f; Z=(d X-b Y)Y-X^2 f;
Con1=d X Y-(b Y^2+f X^2); Con2=(5 x1+3 x3)/(2 x3);
NMaximize[{Con1, Con2>x4/x5 && 1*10^-6<x2<4*10^-6 && 20*10^-6<x3<50*10^-6 &&
  10<x4<100 && 10<x5<100},{x2,x3,x4,x5}]
(*{7.367860243792591*^-59, {x2 -> 3.0786924361325835*^-6,
   x3 -> 0.00004972976008268178, x4 -> 10.031234356582235,
   x5 -> 44.64769318375559}}*)

Con2>x4/x5 /. %[[2]]
(*True*)

Because that value of Con1 is so small it may be nothing more than roundoff error. It may also indicate how difficult it may be to find the boundary where Con1 > 0.

5

Thanks a lot Bill ! Much appreciated. That all makes sense now coz i have been able to somehow get solutions for a more wider constraint than this particular case. Thanks again.