MATLAB: Solve a System of inequalities – (fmincon or other ideas?)

system of inequations; optimization; fmincon;

Hi all,
I have a system of 3 inequalities (f1,f2,f3) and i'm searching to determine values of axi, ayi, azi(i=3:6). I've tried with the function Cylindrical Decomposition (Mathematica) but it seems that the problem is a bit complex for that function. After that I wanted to use Optimization tools like the function Fmincon but I don't know how…
If you have any suggestions, it would be wonderful. :))
Many thanks, Maria
Here's the code:
m = 0.5; g = 9.81; C1 = 1 ; C2 = 0.1; C3 = 0.1;
f1 = m*(g – az3*(120*t*(t – 1)^2 + 20*(t – 1)^3 + 30*t^2*(2*t – 2)) – az5*(60*t^2*(t – 1) + 40*t^3) + 20*az6*t^3 + az4*(60*t*(t – 1)^2 + 60*t^2*(2*t – 2) + 20*t^3))< C1
f2 = -(m^2*(2*ax3*(120*t*(t – 1)^2 + 20*(t – 1)^3 + 30*t^2*(2*t – 2)) + 2*ax5*(60*t^2*(t – 1) + 40*t^3) – 40*ax6*t^3 – 2*ax4*(60*t*(t – 1)^2 + 60*t^2*(2*t – 2) + 20*t^3))*(60*az6*t^2 – az3*(180*t*(2*t – 2) + 180*(t – 1)^2 + 60*t^2) + az4*(180*t*(2*t – 2) + 60*(t – 1)^2 + 180*t^2) – az5*(120*t*(t – 1) + 180*t^2))^2 – m^2*(120*ax6*t – ax5*(600*t – 120) + ax4*(1200*t – 480) – ax3*(1200*t – 720))*(g – az3*(120*t*(t – 1)^2 + 20*(t – 1)^3 + 30*t^2*(2*t – 2)) – az5*(60*t^2*(t – 1) + 40*t^3) + 20*az6*t^3 + az4*(60*t*(t – 1)^2 + 60*t^2*(2*t – 2) + 20*t^3))^2 + 2*m^2*(60*ax6*t^2 – ax3*(180*t*(2*t – 2) + 180*(t – 1)^2 + 60*t^2) + ax4*(180*t*(2*t – 2) + 60*(t – 1)^2 + 180*t^2) – ax5*(120*t*(t – 1) + 180*t^2))*(60*az6*t^2 – az3*(180*t*(2*t – 2) + 180*(t – 1)^2 + 60*t^2) + az4*(180*t*(2*t – 2) + 60*(t – 1)^2 + 180*t^2) – az5*(120*t*(t – 1) + 180*t^2))*(g – az3*(120*t*(t – 1)^2 + 20*(t – 1)^3 + 30*t^2*(2*t – 2)) – az5*(60*t^2*(t – 1) + 40*t^3) + 20*az6*t^3 + az4*(60*t*(t – 1)^2 + 60*t^2*(2*t – 2) + 20*t^3)) – m^2*(ax3*(120*t*(t – 1)^2 + 20*(t – 1)^3 + 30*t^2*(2*t – 2)) + ax5*(60*t^2*(t – 1) + 40*t^3) – 20*ax6*t^3 – ax4*(60*t*(t – 1)^2 + 60*t^2*(2*t – 2) + 20*t^3))*(120*az6*t – az5*(600*t – 120) + az4*(1200*t – 480) – az3*(1200*t – 720))*(g – az3*(120*t*(t – 1)^2 + 20*(t – 1)^3 + 30*t^2*(2*t – 2)) – az5*(60*t^2*(t – 1) + 40*t^3) + 20*az6*t^3 + az4*(60*t*(t – 1)^2 + 60*t^2*(2*t – 2) + 20*t^3)))/(m^3*(g – az3*(120*t*(t – 1)^2 + 20*(t – 1)^3 + 30*t^2*(2*t – 2)) – az5*(60*t^2*(t – 1) + 40*t^3) + 20*az6*t^3 + az4*(60*t*(t – 1)^2 + 60*t^2*(2*t – 2) + 20*t^3))^3) < C2
f3 = (m^2*(2*ay3*(120*t*(t – 1)^2 + 20*(t – 1)^3 + 30*t^2*(2*t – 2)) + 2*ay5*(60*t^2*(t – 1) + 40*t^3) – 40*ay6*t^3 – 2*ay4*(60*t*(t – 1)^2 + 60*t^2*(2*t – 2) + 20*t^3))*(60*az6*t^2 – az3*(180*t*(2*t – 2) + 180*(t – 1)^2 + 60*t^2) + az4*(180*t*(2*t – 2) + 60*(t – 1)^2 + 180*t^2) – az5*(120*t*(t – 1) + 180*t^2))^2 – m^2*(120*ay6*t – ay5*(600*t – 120) + ay4*(1200*t – 480) – ay3*(1200*t – 720))*(g – az3*(120*t*(t – 1)^2 + 20*(t – 1)^3 + 30*t^2*(2*t – 2)) – az5*(60*t^2*(t – 1) + 40*t^3) + 20*az6*t^3 + az4*(60*t*(t – 1)^2 + 60*t^2*(2*t – 2) + 20*t^3))^2 + 2*m^2*(60*ay6*t^2 – ay3*(180*t*(2*t – 2) + 180*(t – 1)^2 + 60*t^2) + ay4*(180*t*(2*t – 2) + 60*(t – 1)^2 + 180*t^2) – ay5*(120*t*(t – 1) + 180*t^2))*(60*az6*t^2 – az3*(180*t*(2*t – 2) + 180*(t – 1)^2 + 60*t^2) + az4*(180*t*(2*t – 2) + 60*(t – 1)^2 + 180*t^2) – az5*(120*t*(t – 1) + 180*t^2))*(g – az3*(120*t*(t – 1)^2 + 20*(t – 1)^3 + 30*t^2*(2*t – 2)) – az5*(60*t^2*(t – 1) + 40*t^3) + 20*az6*t^3 + az4*(60*t*(t – 1)^2 + 60*t^2*(2*t – 2) + 20*t^3)) – m^2*(ay3*(120*t*(t – 1)^2 + 20*(t – 1)^3 + 30*t^2*(2*t – 2)) + ay5*(60*t^2*(t – 1) + 40*t^3) – 20*ay6*t^3 – ay4*(60*t*(t – 1)^2 + 60*t^2*(2*t – 2) + 20*t^3))*(120*az6*t – az5*(600*t – 120) + az4*(1200*t – 480) – az3*(1200*t – 720))*(g – az3*(120*t*(t – 1)^2 + 20*(t – 1)^3 + 30*t^2*(2*t – 2)) – az5*(60*t^2*(t – 1) + 40*t^3) + 20*az6*t^3 + az4*(60*t*(t – 1)^2 + 60*t^2*(2*t – 2) + 20*t^3)))/(m^3*(g – az3*(120*t*(t – 1)^2 + 20*(t – 1)^3 + 30*t^2*(2*t – 2)) – az5*(60*t^2*(t – 1) + 40*t^3) + 20*az6*t^3 + az4*(60*t*(t – 1)^2 + 60*t^2*(2*t – 2) + 20*t^3))^3) < C3

Best Answer

(No, fmincon is not an option here, for SO MANY reasons.)
I think you misunderstand the problem. It is NOT an issue of the system being too complicated for a solver to deal with. What does it mean to solve such a system? I'll give an example. Suppose I were to pose the simple linear system:
x + y > 1
x > 2
There are clearly an infinite number of solutions. Even if the system was bounded, there would still be an infinite number of solutions. So it is meaningless to try to solve such a system, even such a simple one.
I suppose if there was a unique solution, it might make sense, but in your case, that cannot be true! There will NEVER be a unique solution to your system, since the inequalities posed are strict inequalities (less than) so the bounds can never be achieved. Thus any solution, if one exists, will be accompanied by infinitely many other solutions within at least an epsilon ball around that solution.
Your system is a bit more nasty looking and nonlinear, but the facts remain. It is meaningless to even try to "solve" that system. At most, you might desire to plot the solution locus of the inequalities. Of course, that is completely impossible, since it appears that you have a dozen or so unknowns, and only 3 equations. In general, such a problem will be unbounded.
Other issues ...
You do not tell us. Is t an unknown? Are you looking to solve that mess as a function of t?
I've not checked that carefully, but is the system linear in the unknowns (but not t)? If so, then a linear system of 3 inequalities with a dozen unknowns is clearly unbounded.
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