Solves the following **underdetermined** inverse problem:

*\min(∑ {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.

1 2 |

`E ` |
numeric matrix containing the coefficients of the equality
constraints |

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

`G ` |
numeric matrix containing the coefficients of the inequality
constraints |

`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 |

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 |

`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 |

`IsError ` |
logical, |

`type ` |
the string "ldei", such that how the solution was obtained can be traced. |

`numiter ` |
the number of iterations. |

One of the steps in the ldei algorithm is the creation of an orthogonal basis, constructed by Singular Value Decomposition. As this makes use of random numbers, it may happen - for problems that are difficult to solve - that ldei sometimes finds a solution or fails to find one for the same problem, depending on the random numbers used to create the orthogonal basis. If it is suspected that this is happening, trying a few times may find a solution. (example RigaWeb is such a problem).

Karline Soetaert <karline.soetaert@nioz.nl>.

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)

`Minkdiet`

, for a description of the Mink diet example.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 | ```
#-------------------------------------------------------------------------------
# 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")
``` |

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