Quadratic Optimization in R with Equality and Inequality Constraints
I'm trying to find how to solve a quadratic problem in R with equality and inequality constraints and upper and lower bounds:
min 0.5*x'*H*x + f'*x
subject to: A*x <= b
Aeq*x = beq
LB <= x <= UB
I checked the "quadprog" and "kernlab" packages, but ... I must be missing something as I have no idea how to specify both "A" and "Aeq" for solve.QP () or ipop ()
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1 answer
Here's a working example:
library('quadprog')
# min
# -8 x1 -16 x2 + x1^2 + 4 x2^2
#
# s.t.
#
# x1 + 2 x2 == 12 # equalities
# x1 + x2 <= 10 # inequalities (N.B. you need to turn it into "greater-equal" form )
# 1 <= x1 <= 3 # bounds
# 1 <= x2 <= 6 # bounds
H <- rbind(c(2, 0),
c(0, 8))
f <- c(8,16)
# equalities
A.eq <- rbind(c(1,2))
b.eq <- c(12)
# inequalities
A.ge <- rbind(c(-1,-1))
b.ge <- c(-10)
# lower-bounds
A.lbs <- rbind(c( 1, 0),
c( 0, 1))
b.lbs <- c(1, 1)
# upper-bounds on variables
A.ubs <- rbind(c(-1, 0),
c( 0,-1))
b.ubs <- c(-3, -6)
# solve
sol <- solve.QP(Dmat = H,
dvec = f,
Amat = t(rbind(A.eq, A.ge, A.lbs, A.ubs)),
bvec = c(b.eq, b.ge, b.lbs, b.ubs),
meq = 1) # this argument says the first "meq" rows of Amat are equalities
sol
> sol
$solution
[1] 3.0 4.5
$value
[1] -6
$unconstrained.solution
[1] 4 2
$iterations
[1] 3 0
$Lagrangian
[1] 10 0 0 0 12 0
$iact
[1] 1 5
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