I have a set of equations with some inequality constraints that I expect generally does not have a unique solution.


The equations take the form below:


$$\alpha/N+(1-\alpha)x\_1=a\_1$$
$$\alpha/N+(1-\alpha)x\_2=a\_2$$
$$\vdots$$
$$\alpha/N+(1-\alpha)x\_N=a\_N$$
$$x\_1+x\_2+\dots+x\_N=1$$
$$0<x\_i<1$$
$$1>\alpha>0$$


where $N$ is fixed and $a\_i$ are known and satisfy $a\_i>0$ and $a\_1+\dots a\_N=1$.


If I toy around in Mathematica I get that if I take $N=3$ and $(a\_1,a\_2,a\_3)=(1/2,3/10,1/5)$ the feasible solutions are given as a family of solutions for $x\_1$, $x\_2$, and $x\_3$ all depending on $\alpha<3/5$. I expect in general that one can get some estimated range for $\alpha$ that depends on the values of $a\_i$, computationally for instance with enough time.


I am wondering if there is any structure I can utilize for this set of equations to simplify things, or if there is a simple reduction of this problem to a simpler one.


It seems from the examples that $\alpha<1-2\min\{a\_i\}$ is the best bound, but I don't see why this is true immediately.


P.S. The motivation for looking at these equations is related to probability, and estimation of a distribution from its marginals.



Bounding parameter satisfying a collection of inequalities
Let $n:=N$ and $t:=\alpha$. We have
$$0<x\_i=\frac{a\_i-t/n}{1-t}<1$$
for all $i\in[n]:=\{1,\dots.n\}$ --
or, equivalently,
$$t<t\_{n,a}:=\min\_{i\in[n]}\min\Big(\frac{1-a\_i}{1-1/n},na\_i\Big)
=\min\Big(\frac{1-a\_{\max}}{1-1/n},na\_{\min}\Big),$$
where $a\_{\max}:=\max\_{i\in[n]}a\_i$ and $a\_{\min}:=\min\_{i\in[n]}a\_i$.


So, $t\_{n,a}$ is the best bound on $t$.


If, as in your example, $n=3$ and $(a\_1,a\_2,a\_3)=(1/2,3/10,1/5)$, then $t\_{n,a}=3/5$, as you found. If, as in your comment, $n=4$ and $(a\_1,a\_2,a\_3,a\_4)=(1/2,1/6,1/6,1/6)$, then $t\_{n,a}=2/3$, as you also found.


