quadprog fails to find a solution

Question

I am trying to optimize layout of a set of boxes w.r.t. their hanger locations s.t. the boxes are most aligned with their hangers and do not crowd out each other. Using quadprog.

Givens:

1.  box hanger x-locations (P). =710  850  990 1130
2.  box-sizes (W). =690 550 690 130 
3.  usable x-spread tuple (S). =-150 2090
4.  number of boxes (K). =4
5.  minimum interbox spread (G). =50
6.  box x-locations (X). =objective

We can see that the total required x-spread is sum(W) + 3G = 2060 + 150 = 2210 whereas the available x-spread is S[2] - S1 = 2240. So, a solution should exist.

Min:

sumof (P[i] – X[i])^2

s.t.: 

(1) X[i+i] – X[i] >= G + ½ ( W[i+1] + W[i] ); i = 1..(K-1), i.e. the boxes do not crowd out each other

        -X[i] + X[i+1] >= -( -G – ½ (W[i+1] + W[i]) )

(2) X1 >= S[left] + ½ W1, and (3) X[K] <= S[right] – ½ W[K], i.e. the boxes are within the given x-spread

        X[1] >= - ( S[left] + ½ W[1] )
        -X[K] >= - ( S[right] – ½ W[K] )

for a total of 5 constraints - 3 for the inter-box spread, and 2 for extremities.

in R:

> Dmat = matrix(0,4,4)
> diag(Dmat) = 1
> dvec = P, the hanger locations
[1]  710  850  990 1130
> bvec 
[1] -670 -670 -460 -195 2025
> t(Amat)
     [,1] [,2] [,3] [,4]
[1,]   -1    1    0    0
[2,]    0   -1    1    0
[3,]    0    0   -1    1
[4,]    1    0    0    0
[5,]    0    0    0   -1
> solve.QP(Dmat, dvec, Amat, bvec)
Error in solve.QP(Dmat, dvec, Amat, bvec) : 
  constraints are inconsistent, no solution!

Quite obviously I have missed or mis-specified the problem (Package 'quadprog')! I am using quadprog as I found a JavaScript port of it.

Thanks a lot.

Answer 1

I'm not sure that this solves your physical problem but the code below seems to solve the optimization problem as you stated it. I've generalized it to a variable number of boxes and included a plot to check the solution.

  library(quadprog)
  p  <- c(710,  850,  990, 1130)   # hanger positions
  w  <- c(690, 550, 690, 130)      # box widths
  g <- 50                          # min box separation
  s <- c(-150, 2390)               # min and max postions of box edges

  k <- length(w)                   # number of boxes
  Dmat <- 2*diag(nrow=k)
  dvec <- p
# separation constraints
  Amat <- -diag(nrow=k,ncol=(k-1))
  Amat[lower.tri(Amat)] <- unlist(lapply((k-1):1, function(n) c(1,numeric(n-1))))
  bvec <- sapply(1:(k-1), function(n) g + (w[n+1]+w[n])/2)
# x-spread constraints
  Amat <- cbind(Amat, c(1,numeric(k-1)), c(numeric(k-1),-1))
  bvec <- c(bvec, s[1] + w[1]/2, -(s[2] - w[k]/2))

  sol <- solve.QP(Dmat, dvec, Amat, bvec)
  plot(x=s, y=c(0,0), type="l", ylim=c(-2.5,0))
  points(x=p, y=numeric(k), pch=19)
  segments(x0=sol$solution, y0=-1, x1=p, y1=0)
  rect(xleft=sol$solution-w/2, xright=sol$solution+w/2, ytop=-1.0, ybottom=-2, density=8)