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Continuing the question: Formalization of one optimization problem or solution of inequalities

Let's consider a more complex problem. We have two polynomial:

$p_1=A_2t^2+A_1t+A_0$

$p_2=B_2t^2+B_1t+B_0$

$p=p_1p_2$

$A_0=(x^2-y^2-z^2-w^2-v^2-u^2)+xzy+wvu$

$A_1=x^2+y^2+z^2+w^2+\sin(x)\cos(w)\cos(u)$

$A_2=x^4+y^3+z^2-w^4-v^4-u^4$

$B_0=(x^2+y^2+z^2+w^2)+(x+y+z+w)^2+u^2$

$B_1=x^2+y^2+z^2+w^4+v^2+u^2$

$B_2=x^4+y^3+z^2+w^3+v^2+u^3$

$x,y,z,w,v,u$ - parameters, moreover $w,v,u$ - the value of which varies in the range $[-1,1]$.

Polynomial $p$ has all real solutions if and only if:

$d_1=A_1^2-4A_0A_2>0$

$d_2=B_1^2-4B_0B_2>0$

There are known bounds:

$-8.5<A_0<1,-0.2<A_1<6.7,-3<A_2<5,0<B_0<27.5,0<B_1<8.25,-2<B_2<8$

$-1<x<1,-\frac{1}{2}<y<\frac{1}{2},0<z<1$

Task: find the intervals for the parameters $x,y,z$ at which the polynomial $p_1$ and $p_2$ has only real solutions, i.e. the inequality $d_1>0$ and $d_2>0$ (taking into account the existing bounds) for any $w,v,u$ from the specified range $[-1,1]$.

Problem: In this problem, the number of parameters complicates visualization and analytical research, as does the number of free parameters $w,v,u$.

How to solve the problem and get the required intervals for $x,y,z$ out of the box? I.e. the answer should be something like:

x=[?;?] and y=[?;?] and z=[?;?]

Assumptions: maybe we need to choose a random combination of free parameters $w,v,u$ and find a solution for this random combination? And do it repeatedly?

My code:

Clear["Derivative"]; ClearAll["Global` "];

Remove[A, B, d, x, y, z, w, v, u]

Subscript[A, 0] = (x^2 - y^2 - z^2 - w^2 - v^2 - u^2) + x y z + w v u;

Subscript[A, 1] = x^2 + y^2 + z^2 + w^2 + Sin[x] Cos[w] Cos[u];

Subscript[A, 2] = x^4 + y^3 + z^2 - w^4 - v^4 - u^4;

Subscript[B, 0] = (x^2 + y^2 + z^2 + w^2) + (x + y + z + w)^2 + u^2;

Subscript[B, 1] = x^2 + y^2 + z^2 + w^4 + v^2 + u^2;

Subscript[B, 2] = x^4 + y^3 + z^2 + w^3 + v^2 + u^3;

Subscript[d, 1] = 
  Subscript[A, 1]^2 - 4 Subscript[A, 0] Subscript[A, 2];

Subscript[d, 2] = 
  Subscript[B, 1]^2 - 4 Subscript[B, 0] Subscript[B, 2];

NMinimize[{x, Subscript[d, 1] > 0, 
   Subscript[d, 2] > 0, -8.5 < Subscript[A, 0] < 1, -0.2 < 
    Subscript[A, 1] < 6.7, -3 < Subscript[A, 2] < 5, 
   0 < Subscript[B, 0] < 27.5, 
   0 < Subscript[B, 1] < 8.25, -2 < Subscript[B, 2] < 8, -1 < x < 
    1, -1 < y < 1, 0 < z < 1, -1 < w < 1, -1 < v < 1, -1 < u < 1}, {x,
    y, z, w, v, u}, Method -> {"RandomSearch", "SearchPoints" -> 3}];
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1 Answer 1

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NMinimize[{x, Subscript[d, 1] > 0, Subscript[d, 2] > 0, -8.5 < Subscript[A, 0] < 1, -0.2 < 
Subscript[A, 1] < 6.7, -3 < Subscript[A, 2] < 5, 0 < Subscript[B, 0] < 27.5, 
0 < Subscript[B, 1] < 8.25, -2 < Subscript[B, 2] < 8, -1 < x < 1, -1/2 < y < 1/2, 
0 < z < 1, -1 < w < 1, -1 < v < 1, -1 < u < 1}, {x, y, z, w, v, u}, 
Method -> "DifferentialEvolution"]

{-1., {x -> -1., y -> 0.5, z -> 0., w -> -0.969478, v -> 0.666549, u -> -0.892578}}

and with -1<y<1

{-1., {x -> -1., y -> -1., z -> 1., w -> -0.94647, v -> 0.597063, u -> -0.970962}}

Addition. The following works.

Table[{ Subscript[d, 1] > 0, Subscript[d, 2] > 0, -8.5 < Subscript[A, 0] < 1, 
-0.2 <  Subscript[A, 1] < 6.7, -3 < Subscript[A, 2] < 5, 
0 < Subscript[B, 0] < 27.5,  0 < Subscript[B, 1] < 8.25, -2 < Subscript[B, 2] < 8}, 
{x, 0, 0.3, 0.05}, {y, 0,0.3, 0.05}, {z, 0, 0.3, 0.05}]

The output of Trues may be visualized as PointPlot3D.

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  • $\begingroup$ This is the same line that is in my code. It minimizes $x$. The difference is in the optimization method. This is the lower bound for $x$ only. I don't quite understand how to get the intervals for all three parameters $x,y,z$ so that an arbitrary triple would satisfy the inequalities? $\endgroup$
    – dtn
    Jan 10, 2022 at 8:29
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    $\begingroup$ @dtn: Even u = 1/2; v = -1/3; w = 1/10; FindInstance[{ Subscript[d, 1] > 0, Subscript[d, 2] > 0, -8.5 < Subscript[A, 0] < 1, -0.2 < Subscript[A, 1] < 6.7, -3 < Subscript[A, 2] < 5, 0 < Subscript[B, 0] < 27.5, 0 < Subscript[B, 1] < 8.25, -2 < Subscript[B, 2] < 8}, {x, y, z}, Reals] fails. $\endgroup$
    – user64494
    Jan 10, 2022 at 11:03
  • $\begingroup$ I'm not surprised. Because this is a command from a group of analytical tools. Maybe try to run the optimization many times and build a picture that will show the area of feasible solutions? Ideally, the program should provide such information in some way. $\endgroup$
    – dtn
    Jan 10, 2022 at 11:22
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    $\begingroup$ @dtn: Disagree concerning FindInstance. This command sometimes works when Solve/Reduce fails. $\endgroup$
    – user64494
    Jan 10, 2022 at 11:41
  • $\begingroup$ Have you tried this command for one triple of free parameters $w,v,u$. How to make a calculation for any combination of triplets? $\endgroup$
    – dtn
    Jan 10, 2022 at 11:46

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