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4

Let's assume that (a) the full polyhedron is not empty (a solution to the inequalities exists) and (b) you have identified the extreme points of the unit simplex that remain extreme points after the new constraint has been added. Any other extreme points will necessarily satisfy the new constraint as an equality ($\mathbf{b}^T\mathbf{x}=c$). Since they are ...

4

Have a look at PANDA. It's a new alternative to porta/polymake with some improvements. One of the features it has is the possibility for specifying symmetry properties to accelerate the computation. Section 2.3 (Exploitation of symmetry) of the paper provides relevant details (your example is also mentioned). Edit: Here is the website where PANDA can be ...

3

I think it should be possible. Firstly, let us see if we can establish that $A$ is convex. Take \begin{align}X &= (x_1,\ldots,x_M)\in A\\Y&=(y_1,\ldots,y_M)\in A.\end{align} Let $0\leq\lambda\leq 1$. Then $$\lambda X + (1-\lambda) Y = (\lambda x_1 + (1-\lambda)y_1,\ldots,\lambda x_M + (1-\lambda)y_M).$$ Since \begin{align}N_1x_1 + \ldots + N_Mx_M &...

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