# testing if system of inequalities has solution

Hi there,

I have a big system of inequalities (~1500 inequalities, 45 variables) and want to check, if there exists a real solution to it. Trying out the 'solve' and 'solve_ineq' takes a huge amount of time or it breaks during calculation and after checking some of the questions here I even assume the solve-function is broken and sometimes gives wrong results. Does anybody know about a function/system which returns in a responsible time and reliable if there exists a solution or not (existence is enough) (I want to use this in an actual proof, so it would be useless, if I can't trust the result).

In my use case I have the variables $a1, ..., a15, b1,...,b15,c1,...,c15 \in \mathbb R^+$ and my inequalities are all of the form $$\frac{f(a1,...,a15)}{g(c1,\dots, c15)} \geq \frac{f'(a1,\dots,a15)}{g'(c1,\dots, c15)}$$ (and same for combinations of (a,b) and (b,c)) for given linear functions $f,f',g,g'$ (i.e. multivariate polynomials with degree at most 1), so restricting to the variables $ai$ we get a system of linear inequalities (but even trying to solve these takes long/doesn't work with the solve function).

Actually more accuratly I have indexed sets $$F_{a,b} ={{(f_i,g_i) | i \in I }, F_{c,b} ={(p_i,q_i) | i \in I } ,F_{a,c} ={(r_i,s_i) | i \in I }$$ and want to show, that if there exists a solution $(a,b)$ of $$\frac{f_k(a)}{g_k(b)} = max_i \frac{f_i(a)}{g_i(b)}$$ then there exists a solution $(c)$ to $$\frac{r_k(a)}{s_k(c)} = max_i \frac{r_i(a)}{s_i(c)}$$ $$\frac{p_k(c)}{q_k(b)} = max_i \frac{p_i(c)}{q_i(b)}$$

So far I have a the follwing snippet:

#fractionAB are the saved fractions from above, a,b,c are arrays with e.g. a=[a1,a2,...,a15]
stopIt=False
for maxStretch in cands:    #cands is the index set I
ineq=[fractionAB.get(maxStretch) >= fractionAB.get(cand)  for cand in cands]
if solve(ineq,a+b):
#try here if there exists a middle point on the geodesic, i.e. geodesic exists
ineq.extend([fractionAC.get(maxStretch) >= fractionAC.get(cand)  for cand in cands])
ineq.extend([fractionCB.get(maxStretch) >= fractionCB.get(cand)  for cand in cands])
if not solve(ineq, a+b+c):
stopIt=True
print 'Tested for candidate ' , maxStretch
if stopIt:
break


I know, there is room for improvement e.g. at reusing to first solution from (a+b), the problem is, that even that first system pretty much kills the calculation. Also multiplying the denominators on each side doesn't seem to help.

PS: The mathjax seems to be broken on this site, since the code for leftbraces seems to vanish (hence the ugly "fix" above).

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Could you please provide the code for the inequalities ? Without a concrete example, it is hard to say.

( 2018-01-22 14:18:55 -0600 )edit

The problem is, there are too many inequalities to post them here and to get there I already used a few hundreds of lines. I could give you access on CoCalc instead, there I have the fractions saved as an object. The only thing there is, that the calculation stops because of ram restrictions. (It even kills my computer when I try to run it one mine.) Since solve and solve_ineq returns the actual solutions, I hope there was a lighter method which returns only the existence of a solution.

( 2018-01-22 17:14:34 -0600 )edit

To summarize, we do not have the code, we do not know how the functions are implemented, we do not have a similar minimal situation with only 2 or 3 (instead of 15) variables, $f'$ is possibly not the derivative of $f$, we can not specialize (e.g. $a=b=c$) to have a start, then start a random walk to find the better direction to move... In such a case i can only suggest to pick random variables / values for $a,b,c$, then use a "win function" telling how far is the random point under examination from a valid solution, e.g. taken from the negative differences of fractions of functions, that should become positive, by summing them... Then take some $3^{15}$ points in the neighborhood by adding $\{-\epsilon,0,\epsilon\}^{\times15}$ and recalculate. Braces were typed as slash-slash-brace.

( 2018-03-02 05:55:10 -0600 )edit

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Using qepcad might give better results than using solve.

Install by doing

sage -i qepcad


and then follow

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