limSolve (version 1.4)

lsei: 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)

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 square

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 error messages.

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.

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

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))
A <- cbind(rep(1,5),x)
B <- y
cf <- lsei(A,B)$X
abline(coef=cf)
A <- cbind(A,x^2)
cf <- lsei(A,B)$X
curve(cf[1]+cf[2]*x+cf[3]*x^2,add=TRUE,lty=2)
A <- cbind(A,x^3)
cf <- lsei(A,B)$X
curve(cf[1]+cf[2]*x+cf[3]*x^2+cf[4]*x^3,add=TRUE,lty=3)
A <- cbind(A,x^4)
cf <- lsei(A,B)$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)