I have a system of 5 non-linear equations and 5 unknowns (p1,p2,p3,p4,p5 bellow), and a few inequalities. I am trying to understand whether the solution is unique. For this purpose, I tried to run FindInstance asking for 2 solutions, but the system could not provide an answer even after 2 days. I also tried NSolve and had the same problem. (I also tried FindInstance asking for 1 solution and that didn’t work either, even though, analytically I know that a solution must exist). I would really appreciate your help. Here is the code:
SetOptions[EvaluationNotebook[], CellContext -> Notebook]
ClearAll["Global`*"];
k1 = RandomInteger[{1, 100}];
k2 = RandomInteger[{1, 100}];
k3 = RandomInteger[{1, 100}];
k4 = RandomInteger[{1, 100}];
k5 = RandomInteger[{1, 100}];
k12 = RandomInteger[{1, 100}];
k13 = RandomInteger[{1, 100}];
k14 = RandomInteger[{1, 100}];
k15 = RandomInteger[{1, 100}];
k23 = RandomInteger[{1, 100}];
k24 = RandomInteger[{1, 100}];
k25 = RandomInteger[{1, 100}];
k34 = RandomInteger[{1, 100}];
k35 = RandomInteger[{1, 100}];
k45 = RandomInteger[{1, 100}];
k123 = RandomInteger[{1, 100}];
k124 = RandomInteger[{1, 100}];
k125 = RandomInteger[{1, 100}];
k134 = RandomInteger[{1, 100}];
k135 = RandomInteger[{1, 100}];
k145 = RandomInteger[{1, 100}];
k234 = RandomInteger[{1, 100}];
k235 = RandomInteger[{1, 100}];
k245 = RandomInteger[{1, 100}];
k345 = RandomInteger[{1, 100}];
k1234 = RandomInteger[{1, 100}];
k1235 = RandomInteger[{1, 100}];
k1245 = RandomInteger[{1, 100}];
k1345 = RandomInteger[{1, 100}];
k2345 = RandomInteger[{1, 100}];
k = {k1, k2, k3, k4, k5, k12, k13, k14, k15, k23, k24, k25, k34, k35,
k45, k123, k124, k125, k134, k135, k145, k234, k235, k245, k345,
k1234, k1235, k1245, k1345, k2345};
n = k1 + k2 + k3 + k4 + k5 + k12 + k13 + k14 + k15 + k23 + k24 + k25 +
k34 + k35 + k45 + k123 + k124 + k125 + k134 + k135 + k145 +
k234 + k235 + k245 + k345 + k1234 + k1235 + k1245 + k1345 + k2345;
kk = N[{k1/n, k2/n, k3/n, k4/n, k5/n, k12/n, k13/n, k14/n, k15/n,
k23/n, k24/n, k25/n, k34/n, k35/n, k45/n, k123/n, k124/n, k125/n,
k134/n, k135/n, k145/n, k234/n, k235/n, k245/n, k345/n, k1234/n,
k1235/n, k1245/n, k1345/n, k2345/n}];
A = { k1/p1 + k12/(p1 + p2) + k13 /(p1 + p3) + k14/(p1 + p4) +
k15/(p1 + p5) + k123/(p1 + p2 + p3) + k124/(p1 + p2 + p4) +
k125/(p1 + p2 + p5) + k134 /(p1 + p3 + p4) + k135/(p1 + p3 + p5) +
k145/(p1 + p4 + p5) + k1234/(p1 + p2 + p3 + p4) +
k1235/(p1 + p2 + p3 + p5) + k1245/(p1 + p2 + p4 + p5) +
k1345 /(p1 + p3 + p4 + p5) == n,
k2/p2 + k12/(p1 + p2) + k23 /(p2 + p3) + k24 /(p2 + p4) +
k25/(p2 + p5) + k123 /(p1 + p2 + p3) + k124 /(p1 + p2 + p4) +
k125/(p1 + p2 + p5) + k234/(p2 + p3 + p4) + k235 /(p2 + p3 + p5) +
k245/(p2 + p4 + p5) + k1234/(p1 + p2 + p3 + p4) +
k1235/(p1 + p2 + p3 + p5) + k1245/(p1 + p2 + p4 + p5) +
k2345/(p2 + p3 + p4 + p5) == n,
k3/p3 + k13 /(p1 + p3) + k23/(p2 + p3) + k34/(p3 + p4) +
k35 /(p3 + p5) + k123/(p1 + p2 + p3) + k134/(p1 + p3 + p4) +
k135/(p1 + p3 + p5) + k234/(p2 + p3 + p4) +
k235/(p2 + p3 + p5) + k345/(p3 + p4 + p5) +
k1234/(p1 + p2 + p3 + p4) + k1235/(p1 + p2 + p3 + p5) +
k1345/(p1 + p3 + p4 + p5) + k2345/(p2 + p3 + p4 + p5) == n,
k4/p4 + k14/(p1 + p4) + k24/(p2 + p4) + k34/(p3 + p4) +
k45/(p4 + p5) + k124 /(p1 + p2 + p4) + k134/(p1 + p3 + p4) +
k145/(p1 + p4 + p5) + k234 /(p2 + p3 + p4) + k245/(p2 + p4 + p5) +
k345 /(p3 + p4 + p5) + k1234/(p1 + p2 + p3 + p4) +
k1245/(p1 + p2 + p4 + p5) + k1345/(p1 + p3 + p4 + p5) +
k2345/(p2 + p3 + p4 + p5) == n ,
k5/p5 + k15/(p1 + p5) + k25/(p2 + p5) + k35/(p3 + p5) +
k45/(p4 + p5) + k125 /(p1 + p2 + p5) + k135/(p1 + p3 + p5) +
k145/(p1 + p4 + p5) + k235/(p2 + p3 + p5) + k245/(p2 + p4 + p5) +
k345 /(p3 + p4 + p5) + k1235/(p1 + p2 + p3 + p5) +
k1245/(p1 + p2 + p4 + p5) + k1345/(p1 + p3 + p4 + p5) +
k2345/(p2 + p3 + p4 + p5) == n , p1 >= 0, p1 <= 1, p2 >= 0,
p2 <= 1, p3 >= 0, p3 <= 1, p4 >= 0, p4 <= 1, p5 >= 0, p5 <= 1,
p1 + p2 + p3 + p4 + p5 == 1}
Print["===================="];
f = FindInstance[A, {p1, p2, p3, p4, p5}, Reals, 2]
p = NSolve[A, {p1, p2, p3, p4, p5}, Reals]
dimp = Dimensions[p];
Together[A[[1]] /. Equal -> Subtract] // Numerator
it's equivalent to finding the roots of a massive multivariate polynomial. Add in the other equations and inequalities that must be satisfied and this problem is infinitely more difficult. It might be possible to show that two 'close' numerical results exist, though numerical results tell us nothing about uniqueness. Again, I think you should show how you constructed these equations because they are too complex in this form. $\endgroup$