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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.

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

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.

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 take the form $x_3=1/3$ with a family of solutions for $x_1$ and $x_2$ 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.

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

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.

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