Calculates ranges of the unknowns of a linear inverse problem

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Description

Given the linear constraints

Ex=f

Gx>=h

finds the minimum and maximum values of all elements of vector x

This is done by successively minimising and maximising each x, using linear programming.

Usage

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xranges(E = NULL, F = NULL, G = NULL, H = NULL, 
        ispos = FALSE, tol = 1e-8, central = FALSE, full=FALSE)

Arguments

E

numeric matrix containing the coefficients of the equalities Ex=F.

F

numeric vector containing the right-hand side of the equalities.

G

numeric matrix containing the coefficients of the inequalities Gx>=H.

H

numeric vector containing the right-hand side of the inequalities.

ispos

if TRUE, it is imposed that unknowns are positive quantities.

tol

tolerance for equality and inequality constraints.

central

if TRUE, the mean value of all range solutions is also outputted.

full

if TRUE, all range solutions are also outputted.

Details

The ranges are estimated by successively minimising and maximising each unknown, and using linear programming (based on function lp from R-package lpSolve.

By default linear programming assumes that all unknowns are positive. If all unknowns are indeed to be positive, then it will generally be faster to set ispos equal to TRUE If ispos is FALSE, then a system double the size of the original system must be solved.

xranges outputs only the minimum and maximum value of each flow unless:

full is TRUE. In this case, all the results of the successive minimisation and maximisation will be outputted, i.e. for each linear programming application, not just the value of the unknown being optimised but also the corresponding values of the other unknowns will be outputted.

If central is TRUE, then the mean of all the results of the linear programming will be outputted. This may be a good starting value for xsample

Note: the columns corresponding to the central value and the full results are valid solutions of the equations Ex=F and Gx>=H. This is not the case for the first two columns (with the minimal and maximal values).

Value

a matrix with at least two columns:

column 1 and 2: the minimum and maximum value of each x

if central is TRUE: column 3 = the central value

if full is TRUE: next columns contain all valid range solutions

Author(s)

Karline Soetaert <karline.soetaert@nioz.nl>

References

Michel Berkelaar and others (2010). lpSolve: Interface to Lp_solve v. 5.5 to solve linear/integer programs. R package version 5.6.5. http://CRAN.R-project.org/package=lpSolve

See Also

Minkdiet, for a description of the Mink diet example.

varranges, for range estimation of variables,

xsample, to randomly sample the lsei problem

lp: linear programming from package lpSolve

Examples

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# Estimate the ranges in the Diet Composition of Mink
E <- rbind(Minkdiet$Prey, rep(1, 7))
F <- c(Minkdiet$Mink, 1)
(DC <- xranges(E, F, ispos = TRUE))

# The same, but explicitly imposing positivity
(DC <- xranges(E, F, G = diag(7), H = rep(0, 7)))