# Weighted Least Distance Programming with equality and inequality constraints.

### Description

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.

### Usage

1 2 |

### Arguments

`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 |

### 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 |

`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. |

### Note

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).

### Author(s)

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

### 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

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")
``` |