How to express x1*x2 in the quadprog of R - r

Question

I have a question about how to express x1x2 in objective function.
Here is an example in the Internet.
##
## min x1^2 +2x2^2 + 4x3^2 - x1 - x2 + 5x3
## x1 + x3 <= 1
## x1 >= 5
## x2 <= 0
##
P = 2*diag (c (1, 2, 4));
d = c (-1, -1, 5);
A = matrix (0, nrow=3, ncol=3);
A[1,] = c(-1, 0, -1);
A[2,] = c( 1, 0, 0);
A[3,] = c( 0, -1, 0);
b = c(-1, 5, 0);
From the example, the objective finction is x1^2 +2x2^2 + 4x3^2 - x1 - x2 + 5x3
In R, it is P = 2*diag (c (1, 2, 4)); d = c (-1, -1, 5);
However, if i have an objective function like x1x2 or x1^2
How to key the commands in R ?
Thanks a lot in advance.

Answer 1

solve.QP function minimizes the following expression:
min 1/2 * (t(x) %*% D %*% x) - t(d) %*% x
where x is the vector of n variables (e.g. x1, x2 ...), D a symmetric matrix of n x n coefficients and d a vector of n coefficients.
In order to understand how those matrix products translate to an objective function expression, I've created the following simple function to expand the quadprog objective given the inputs :
# Given matrix D and vector d that will be passed to solve.QP,
# it returns the objective function expression as string
expandQPObj <- function(D, d, var.name = 'x') {
# helper function that expands a matrix product. NB it does not compute the result,
# it just returns the expressions of each element of the resulting matrix
expandMatrixProd <- function(m1, m2) {
m1 <- as.matrix(m1)
m2 <- as.matrix(m2)
if (ncol(m1) != nrow(m2)) { stop("incompatible dimensions") }
n <- nrow(m1)
m <- ncol(m2)
res <- matrix('', nrow = n, ncol = m)
for (i in 1:n) {
for (j in 1:m) {
a <- m1[i, ]
b <- m2[, j]
a <- ifelse(grepl('[+*]', a), paste0('(', a, ')'), a)
b <- ifelse(grepl('[+*]', b), paste0('(', b, ')'), b)
res[i, j] <- gsub('+-','-',paste(a, b, sep = '*', collapse = '+'),fixed = TRUE)
}
}
return(res)
}
D <- as.matrix(D)
d <- as.vector(d)
n <- length(d)
if (!all(dim(D) == n) || n == 0) {
stop('Dimensions problem: D should be an nxn matrix and d a vector of length n (n>0)')
}
xvec <- paste(var.name, 1:n, sep = '')
quadComp <- as.vector(Reduce(list(t(xvec), D, xvec),f=expandMatrixProd))
linearComp <- as.vector(strMatrixMult(t(d), xvec))
return(paste0('1/2*(', quadComp, ') - (', linearComp, ')'))
}
By calling it with literal coefficients we can easily understand how to set these coefficients to get our desired result :
Dliteral <- matrix(paste0('D',1:9),nrow=3,byrow = T)
# [,1] [,2] [,3]
#[1,] "D1" "D2" "D3"
#[2,] "D4" "D5" "D6"
#[3,] "D7" "D8" "D9"
dliteral <- paste0('d',1:3)
#[1] "d1" "d2" "d3"
expandQPObj(Dliteral,dliteral)
#"1/2*((x1*D1+x2*D4+x3*D7)*x1+(x1*D2+x2*D5+x3*D8)*x2+(x1*D3+x2*D6+x3*D9)*x3) - d1*x1+d2*x2+d3*x3"
So, in order to get :
min x1*x2
we need to set D2=1 and D4=1 (not just D2=2 since matrix D must be symmetric!) and all the others coefficients = 0, hence :
D = rbind(c(0,1,0),c(1,0,0),c(0,0,0))
# [,1] [,2] [,3]
#[1,] 0 1 0
#[2,] 1 0 0
#[3,] 0 0 0
d <- rep(0,3)
#[1] 0 0 0
expandQPObj(D,d)
[1] "1/2*((x1*0+x2*1+x3*0)*x1+(x1*1+x2*0+x3*0)*x2+(x1*0+x2*0+x3*0)*x3) - 0*x1+0*x2+0*x3"
Similarly, to obtain :
min x1^2 (== min x1*x1)
we need to set D1=2 and all the others coefficients = 0 :
D = rbind(c(2,0,0),c(0,0,0),c(0,0,0))
# [,1] [,2] [,3]
#[1,] 2 0 0
#[2,] 0 0 0
#[3,] 0 0 0
d <- rep(0,3)
#[1] 0 0 0
expandQPObj(D,d)
# "1/2*((x1*2+x2*0+x3*0)*x1+(x1*0+x2*0+x3*0)*x2+(x1*0+x2*0+x3*0)*x3) - 0*x1+0*x2+0*x3"