The following example solves problem 16 from the Hock-Schittkowski-Collection
.
minimize \ f(x) = 100 (x_2 - x_1^2)^2 + (1 - x_1)^2
subject \ to: \ \ x_1 + x_2^2 ≥q 0 \ \ \ x_1^2 + x_2 ≥q 0
-2 ≥q x_1 ≥q 0.5 \ \ \ x_2 ≥q 1
Solution: c(0.5, 0.25)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 | library(ROI)
f <- function(x) {
return( 100 * (x[2] - x[1]^2)^2 + (1 - x[1])^2 )
}
f.gradient <- function(x) {
return( c( -400 * x[1] * (x[2] - x[1] * x[1]) - 2 * (1 - x[1]),
200 * (x[2] - x[1] * x[1])) )
}
x <- OP( objective = F_objective(f, n=2L, G=f.gradient),
constraints = c(F_constraint(F=function(x) x[1] + x[2]^2, ">=", 0,
J=function(x) c(1, 2*x[2])),
F_constraint(F=function(x) x[1]^2 + x[2], ">=", 0,
J=function(x) c(2*x[1], x[2]))),
bounds = V_bound(li=1:2, ui=1:2, lb=c(-2, -Inf), ub=c(0.5, 1)) )
nlp <- ROI_solve(x, solver="alabama", start=c(-2, 1))
nlp
## Optimal solution found.
## The objective value is: 2.499999e-01
solution(nlp)
## [1] 0.5000001 0.2499994
|
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