Convex polyhedron P is a subset of R n that satisfies system of linear inequalities...

Cristopher Knox

Cristopher Knox

Answered

2022-07-15

Convex polyhedron P is a subset of R n that satisfies system of linear inequalities
a 11 x 1 + + a 1 n x n 1 c 1 a p 1 x 1 + + a p n x n p c p ,
where i { , }. It can be alternatively represented by two finite sets of generators V , W R n :
P = conv ( V ) + cone ( W ) ,
where conv(V) denotes all convex combinations of points in V and cone(W) all nonnegative linear combinations of points in W.
Now, what if we allow i to be from { , > , , < }. Is there some similar representation in terms of generating points for such sets?

Answer & Explanation

Sophia Mcdowell

Sophia Mcdowell

Expert

2022-07-16Added 14 answers

No. Unfortunately, it can't even be done in the 1D case. For instance, consider the inequalities x > 0 and x < 1. Then P is the open interval ( 0 , 1 ). You can't take V = { 0 , 1 } because then conv ( V ) is the closed interval [ 0 , 1 ]. Taking V to be any other two points between 0 and 1 would generate a conv ( V ) that is a strict subset of P.
vortoca

vortoca

Expert

2022-07-17Added 2 answers

Seems like such polyhedra are called not necessarily closed (NNC) and are usually represented as closed polyhedra with additional dimension ε: every strict inequality a i 1 x 1 + + a i n x n > c i is replaced by a i 1 x 1 + + a i n x n ε c i and two additional inequalities 0 ε 1 are added to the system. If we call this polyhedron P , the desired polyhedron is the set { ( x 1 , , x n ) ( x 1 , , x n , ε ) P , ε > 0 }. Such a system of constraints can be converted to representation by generators.
Alternatively, they can be characterized directly by three sets of generators R , P , C R n i.e. every point can be obtained as
α 1 r 1 + + α k r k + β 1 p 1 + + β l p l + γ 1 c 1 + + γ m p m ,
where r i R , p i P , c i C and α i , β i , γ i R + and i = 1 l β i + i = 1 m γ i = 1 and there is 1 i l such that β i 0. The trick here is that points in C don't have to lie within the NNC polyhedron, but its closure, and whenever they appear in the sum, there must also be point from P with nonzero coefficient. This representation can easily be converted to the one mentioned above.

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