Feasible solution for a probability distribution which must satisfy a system of linear equality and inequality constraints.

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Description

This function finds a feasible solution, p=(p1,...,pn), in the n-dimensional simplex of probability distributions which must satisfy A1p = b1, A2p <= b2 and A3p >= b3. All the components of the bi's must be nonnegative In addition each probability in the solution must be at least as big as eps, a small positive number.

Usage

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feasible(A1,A2,A3,b1,b2,b3,eps)

Arguments

A1

The matrix for the equality constraints.This must always contain the constraint that the sum of the pi's is one.

A2

The matrix for the <= inequality constraints. This must always contain the constraints -pi <= 0, i.e. that the pi's must be nonnegative.

A3

The matrix for the >= inequality constraints. If there are no such constraints A3 must be set equal to NULL.

b1

The rhs vector for A1, each component must be nonnegative.

b2

The rhs vector for A2, each component must be nonnegative.

b3

The rhs vector for A3, each component must be nonnegative. If A3 is NULL then b3 must be NULL.

eps

A small positive number. Each member of the solution must be at least as large as eps. Care must be taken not to choose a value of eps which is too large.

Value

The function returns a vector. If the components of the vector are positive then the feasible solution is the vector returned, otherwise there is no feasible solution.

Examples

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A1<-rbind(rep(1,7),1:7)
b1<-c(1,4)
A2<-rbind(c(1,1,1,1,0,0,0),c(.2,.4,.6,.8,1,1.2,1.4))
b2<-c(1,2)
A3<-rbind(c(1,3,5,7,9,10,11),c(1,1,1,0,0,0,1))
b3<-c(5,.5)
eps<-1/100
feasible(A1,A2,A3,b1,b2,b3,eps)