[R] Problem with minimization that I failed to understand

[Daniel Lobo]

Hi,

I have below minimization problem


MyDat = structure(list(c(50L, 0L, 0L, 50L, 75L, 100L, 50L, 0L, 50L, 0L,
25L, 50L, 50L, 75L, 75L, 75L, 0L, 75L, 75L, 75L, 0L, 25L, 75L,
75L, 0L, 75L, 100L, 0L, 25L, 100L), c(75L, 0L, 0L, 50L, 100L,
50L, 75L, 75L, 100L, 25L, 0L, 25L, 100L, 0L, 50L, 0L, 25L, 25L,
100L, 75L, 0L, 0L, 0L, 50L, 0L, 75L, 75L, 0L, 50L, 25L), c(50L,
0L, 0L, 0L, 100L, 25L, 0L, 0L, 25L, 50L, 0L, 25L, 75L, 50L, 100L,
50L, 0L, 75L, 25L, 50L, 0L, 0L, 25L, 0L, 50L, 100L, 100L, 0L,
75L, 50L), c(25L, 0L, 0L, 75L, 75L, 25L, 50L, 50L, 100L, 25L,
0L, 100L, 50L, 25L, 100L, 25L, 25L, 100L, 50L, 100L, 0L, 0L,
100L, 50L, 0L, 50L, 75L, 0L, 50L, 25L), c(50L, 0L, 0L, 75L, 75L,
75L, 25L, 25L, 0L, 100L, 0L, 25L, 25L, 75L, 100L, 0L, 25L, 0L,
75L, 25L, 25L, 25L, 75L, 25L, 0L, 75L, 100L, 0L, 100L, 100L),
    c(50L, 0L, 0L, 50L, 100L, 25L, 25L, 25L, 50L, 50L, 0L, 50L,
    75L, 0L, 100L, 50L, 25L, 100L, 50L, 75L, 0L, 0L, 50L, 25L,
    0L, 100L, 100L, 0L, 75L, 50L), c(50L, 0L, 0L, 50L, 75L, 25L,
    75L, 50L, 100L, 25L, 0L, 75L, 25L, 0L, 50L, 0L, 50L, 75L,
    100L, 75L, 0L, 0L, 100L, 0L, 0L, 50L, 75L, 0L, 100L, 100L
    ), c(25L, 75L, 50L, 25L, 75L, 50L, 100L, 75L, 100L, 25L,
    0L, 75L, 25L, 50L, 25L, 25L, 75L, 75L, 100L, 75L, 75L, 100L,
    75L, 25L, 0L, 75L, 75L, 0L, 75L, 100L), c(55L, 30L, 20L,
    30L, 45L, 30L, 30L, 30L, 70L, 30L, 10L, 45L, 45L, 45L, 45L,
    30L, 30L, 55L, 45L, 45L, 30L, 30L, 30L, NA, 30L, 55L, 45L,
    20L, 45L, 70L), c(85L, 40L, 40L, 40L, 55L, 40L, 20L, 30L,
    30L, 30L, 20L, 30L, 70L, 40L, 85L, 55L, 30L, 40L, 30L, 55L,
    20L, 30L, 55L, 0L, 40L, 55L, 70L, 40L, 85L, 70L), c(45L,
    45L, 0L, 45L, 45L, 45L, 0L, 0L, 100L, 45L, 0L, 100L, 45L,
    45L, 100L, 45L, 45L, 100L, 45L, 45L, 45L, 45L, 25L, 45L,
    0L, 100L, 45L, 0L, 45L, 45L), c(55L, 45L, 45L, 45L, 55L,
    45L, 45L, 45L, 45L, 45L, 45L, 45L, 45L, 45L, 55L, 55L, 45L,
    55L, 45L, 45L, 45L, 45L, 45L, 45L, 45L, 55L, 45L, 45L, 45L,
    45L), c(100L, 100L, 50L, 100L, 100L, 100L, 100L, 100L, 100L,
    100L, 50L, 100L, 100L, 100L, 100L, 100L, 100L, 100L, 100L,
    100L, 100L, 100L, 100L, 50L, 100L, 100L, 100L, 100L, 100L,
    100L), c(100L, 25L, 25L, 0L, 100L, 60L, 0L, 0L, 25L, 60L,
    0L, 60L, 100L, 60L, 100L, 100L, 25L, 100L, 60L, 100L, 100L,
    60L, 100L, 60L, 100L, 100L, 100L, 100L, 60L, 60L), c(0L,
    0L, 50L, 50L, 100L, 100L, 0L, 0L, 100L, 100L, 0L, 100L, 100L,
    0L, 100L, 100L, 0L, 100L, 100L, 100L, 100L, 100L, 100L, 0L,
    100L, 100L, 100L, 100L, 100L, 100L), c(40L, 100L, 40L, 100L,
    100L, 40L, 100L, 100L, 100L, 40L, 100L, 100L, 100L, 100L,
    100L, 100L, 100L, 100L, 100L, 100L, 100L, 100L, 100L, 100L,
    100L, 100L, 100L, 0L, 100L, 100L), c(100L, 100L, 100L, 100L,
    100L, 100L, 100L, 100L, 100L, 100L, 100L, 100L, 100L, 100L,
    100L, 100L, 100L, 100L, 100L, 100L, 100L, 100L, 100L, -10L,
    100L, 100L, 100L, -10L, 100L, 100L), c(70L, 0L, 25L, 0L,
    100L, 25L, 0L, 0L, 0L, 45L, 0L, 25L, 100L, 100L, 100L, 100L,
    0L, 70L, 0L, 100L, 45L, 45L, 0L, 0L, 100L, 100L, 100L, 0L,
    100L, 100L), c(55L, 55L, 55L, 55L, 55L, 55L, 55L, 55L, 55L,
    55L, 55L, 55L, 55L, 55L, 55L, 55L, 20L, 55L, 20L, 55L, 20L,
    20L, 100L, 55L, 55L, 55L, 55L, 0L, 55L, 55L), c(65L, 65L,
    100L, 65L, 100L, 100L, 100L, 100L, 100L, 100L, 100L, 100L,
    100L, 100L, 100L, 100L, 65L, 100L, 100L, 100L, 65L, 100L,
    0L, 65L, 100L, 100L, 100L, 100L, 100L, 100L), c(85L, 85L,
    85L, 85L, 85L, 85L, 85L, 85L, 85L, 85L, 85L, 85L, 56L, 85L,
    100L, 85L, 85L, 85L, 0L, 85L, 85L, 85L, 85L, 85L, 85L, 85L,
    85L, 28L, 56L, 56L)), row.names = c(NA, -30L), class = "data.frame")

Fn = function(Wts) return(-Kendall::Kendall(1:Nobs,
rank(-as.vector(as.matrix(MyDat) %*% matrix(Wts, nc = 1)[, 1, drop =
T])))$tau[1])
q1 = pracma::fmincon(c(0.12, 0.04, 0.07, 0.03, 0.06, 0.07, 0.07, 0.04,
0.09, 0.08, 0.02, 0.02, 0.03, 0.06, 0.02, 0, 0.07, 0.05, 0.02, 0.02, 0.02),
fn = Fn,
          A = matrix(c(rep(0, 20), -1), nrow = 1), b = -2.05/100, Aeq =
matrix(c(rep(1, 20), 1), nrow = 1), beq = 1,
          lb = rep(0.01, 21),
          tol = 1e-16, maxfeval = 10000000, maxiter = 5000000)


However with above code, I got sub-optimal value in terms of minimization
of the objective function:

q1$value
#0.1632184
Fn(c(0.12, 0.04, 0.07, 0.03, 0.06, 0.07, 0.07, 0.04, 0.09, 0.08, 0.02,
0.02, 0.03, 0.06, 0.02, 0, 0.07, 0.05, 0.02, 0.02, 0.02))
#0.1586207

Could you please help me to understand what went wrong with my code and how
to correct that?

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