lsei: Least Squares with Equalities and Inequalities

In limSolve: Solving Linear Inverse Models

Least Squares with Equalities and Inequalities

Description

Solves an lsei inverse problem (Least Squares with Equality and Inequality Constraints)

\min(||Ax-b||^2)

subject to

Ex=f

Gx>=h

Uses either subroutine lsei (FORTRAN) from the LINPACK package, or solve.QP from R-package quadprog.

In case the equality constraints Ex=f cannot be satisfied, a generalized inverse solution residual vector length is obtained for f-Ex.

This is the minimal length possible for ||f-Ex||^2.

Usage

lsei (A = NULL, B = NULL, E = NULL, F = NULL, G = NULL, H = NULL,
      Wx = NULL, Wa = NULL, type = 1, tol = sqrt(.Machine$double.eps),
      tolrank = NULL, fulloutput = FALSE, verbose = TRUE, 
      lower = NULL, upper = NULL)
      

Arguments

A	numeric matrix containing the coefficients of the quadratic function to be minimised, ||Ax-B||^2; if the columns of A have a names attribute, they will be used to label the output.
B	numeric vector containing the right-hand side of the quadratic function to be minimised.
E	numeric matrix containing the coefficients of the equality constraints, Ex=F; if the columns of E have a names attribute, and the columns of A do not, 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 A and E do not, they will be used to label the output.
H	numeric vector containing the right-hand side of the inequality constraints.
Wx	numeric vector with weighting coefficients of unknowns (length = number of unknowns).
Wa	numeric vector with weighting coefficients of the quadratic function (Ax-B) to be minimised (length = number of number of rows of A).
type	integer code determining algorithm to use 1=lsei, 2=solve.QP from R-package quadprog (see note).
tol	tolerance (for singular value decomposition, equality and inequality constraints).
tolrank	only used if type = 1; if not NULL then tolrank should be a two-valued vector containing the rank determination tolerance for the equality constraint equations (1st value) and for the reduced least squares equations (2nd value).
fulloutput	if TRUE, also returns the covariance matrix of the solution and the rank of the equality constraints - only used if type = 1.
verbose	logical to print warnings and messages.
upper, lower	vector containing upper and lower bounds on the unknowns. If one value, it is assumed to apply to all unknowns. If a vector, it should have a length equal to the number of unknowns; this vector can contain NA for unbounded variables. The upper and lower bounds are added to the inequality conditions G*x>=H.

Value

a list containing:

X	vector containing the solution of the least squares problem.
residualNorm	scalar, the sum of absolute values of residuals of equalities and violated inequalities.
solutionNorm	scalar, the value of the minimised quadratic function at the solution, i.e. the value of ||Ax-b||^2.
IsError	logical, TRUE if an error occurred.
type	the string "lsei", such that how the solution was obtained can be traced.
covar	covariance matrix of the solution; only returned if fulloutput = TRUE.
RankEq	rank of the equality constraint matrix.; only returned if fulloutput = TRUE.
RankApp	rank of the reduced least squares problem (approximate equations); only returned if fulloutput = TRUE.

Note

See comments in the original code for more details; these comments are included in the ‘docs’ subroutine of the package.

Author(s)

Karline Soetaert <[e-mail]>

References

K. H. Haskell and R. J. Hanson, An algorithm for linear least squares problems with equality and nonnegativity constraints, Report SAND77-0552, Sandia Laboratories, June 1978.

K. H. Haskell and R. J. Hanson, Selected algorithms for the linearly constrained least squares problem - a users guide, Report SAND78-1290, Sandia Laboratories,August 1979.

K. H. Haskell and R. J. Hanson, An algorithm for linear least squares problems with equality and nonnegativity constraints, Mathematical Programming 21 (1981), pp. 98-118.

R. J. Hanson and K. H. Haskell, Two algorithms for the linearly constrained least squares problem, ACM Transactions on Mathematical Software, September 1982.

Berwin A. Turlach R and Andreas Weingessel (2007). quadprog: Functions to solve Quadratic Programming Problems. R package version 1.4-11. S original by Berwin A. Turlach R port by Andreas Weingessel.

See Also

ldei, linp,

solve.QR the original function from package quadprog.

Examples

# ------------------------------------------------------------------------------
# example 1: polynomial fitting
# ------------------------------------------------------------------------------
x <- 1:5
y <- c(9, 8, 6, 7, 5)
plot(x, y, main = "Polynomial fitting, using lsei", cex = 1.5, 
     pch = 16, ylim = c(4, 10))

# 1-st order
A <- cbind(rep(1, 5), x)
B <- y
cf <- lsei(A, B)$X
abline(coef = cf)

# 2-nd order
A <- cbind(A, x^2)
cf <- lsei(A, B)$X
curve(cf[1] + cf[2]*x + cf[3]*x^2, add = TRUE, lty = 2)

# 3-rd order
A <- cbind(A, x^3)
cf <- lsei(A, B)$X
curve(cf[1] + cf[2]*x + cf[3]*x^2 + cf[4]*x^3, add = TRUE, lty = 3)

# 4-th order
A <- cbind(A, x^4)
cf <- lsei(A, B)$X
curve(cf[1] + cf[2]*x + cf[3]*x^2 + cf[4]*x^3 + cf[5]*x^4, 
      add = TRUE, lty = 4)
legend("bottomleft", c("1st-order", "2nd-order","3rd-order","4th-order"),
       lty = 1:4)

# ------------------------------------------------------------------------------
# example 2: equalities, approximate equalities and inequalities
# ------------------------------------------------------------------------------

A <- matrix(nrow = 4, ncol = 3,
            data = c(3, 1, 2, 0, 2, 0, 0, 1, 1, 0, 2, 0))
B <- c(2, 1, 8, 3)
E <- c(0, 1, 0)
F <- 3
G <- matrix(nrow = 2, ncol = 3, byrow = TRUE,
            data = c(-1, 2, 0, 1, 0, -1))
H <- c(-3, 2)

lsei(E = E, F = F, A = A, B = B, G = G, H = H)

# ------------------------------------------------------------------------------
# example 3: bounds added
# ------------------------------------------------------------------------------

lsei(E = E, F = F, A = A, B = B, G = G, H = H, 
     lower = c(2, NA, 0))
lsei(E = E, F = F, A = A, B = B, G = G, H = H, 
     upper = 2)

limSolve documentation built on Sept. 22, 2023, 1:07 a.m.