limSolve (version 1.5.3)

ldei: Weighted Least Distance Programming with equality and inequality constraints.

Description

Solves the following underdetermined inverse problem: $$\min(\sum {x_i}^2)$$ subject to $$Ex=f$$ $$Gx>=h$$

uses least distance programming subroutine ldp (FORTRAN) from Linpack

The model has to be UNDERdetermined, i.e. the number of independent equations < number of unknowns.

Usage

ldei(E, F, G = NULL, H = NULL,
     tol = sqrt(.Machine$double.eps), verbose = TRUE)

Arguments

E

numeric matrix containing the coefficients of the equality constraints $Ex=F$; if the columns of E have a names attribute, they will be used to label the output.

F

numeric vector containing the right-hand side of the equality constraints.

G

numeric matrix containing the coefficients of the inequality constraints $Gx>=H$; if the columns of G have a names attribute and the columns of E do not, they will be used to label the output.

H

numeric vector containing the right-hand side of the inequality constraints.

tol

tolerance (for singular value decomposition, equality and inequality constraints).

verbose

logical to print ldei error messages.

Value

• a list containing:
• X
  vector containing the solution of the least distance with equalities and inequalities problem.
• unconstrained.solution
  vector containing the unconstrained solution of the least distance problem, i.e. ignoring $Gx>=h$.
• residualNorm
  scalar, the sum of absolute values of residuals of equalities and violated inequalities; should be zero or very small if the problem is feasible.
• solutionNorm
  scalar, the value of the quadratic function at the solution, i.e. the value of $\sum {w_i*x_i}^2$.
• IsError
  logical, TRUE if an error occurred.
• type
  the string "ldei", such that how the solution was obtained can be traced.

References

Lawson C.L.and Hanson R.J. 1974. Solving Least Squares Problems, Prentice-Hall

Lawson C.L.and Hanson R.J. 1995. Solving Least Squares Problems. SIAM classics in applied mathematics, Philadelphia. (reprint of book)

See Also

Minkdiet, for a description of the Mink diet example.

lsei, linp

ldp

Examples

#-------------------------------------------------------------------------------
# A simple problem
#-------------------------------------------------------------------------------
# minimise x1^2 + x2^2 + x3^2 + x4^2 + x5^2 + x6^2
# subject to:
#-x1            + x4 + x5      = 0
#    - x2       - x4      + x6 = 0
# x1 + x2 + x3                 > 1
#           x3       + x5 + x6 < 1
# xi > 0

E <- matrix(nrow = 2, byrow = TRUE, data = c(-1, 0, 0, 1, 1, 0,
                                              0,-1, 0, -1, 0, 1))
F <- c(0, 0)
G <- matrix(nrow = 2, byrow = TRUE, data = c(1, 1, 1, 0, 0, 0,
                                             0, 0, -1, 0, -1, -1))
H    <- c(1, -1)
ldei(E, F, G, H)

#-------------------------------------------------------------------------------
# parsimonious (simplest) solution of the mink diet problem
#-------------------------------------------------------------------------------
E <- rbind(Minkdiet$Prey, rep(1, 7))
F <- c(Minkdiet$Mink, 1)

parsimonious <- ldei(E, F, G = diag(7), H = rep(0, 7))
data.frame(food = colnames(Minkdiet$Prey),
           fraction = parsimonious$X)
dotchart(x = as.vector(parsimonious$X),
         labels = colnames(Minkdiet$Prey),
         main = "Diet composition of Mink extimated using ldei",
         xlab = "fraction")