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rmathematical-optimizationsolver

Optimisation problem with constraint in R: how to solve problem with "complex" logarithmic benefit functions?

I have the following optimisation problem:

Maximisise combined benefit functions

F(x) = 2*10^9 + 160*x - 7*x*ln(x) 
F(y) = 3*10^9 + 170*y - 7.5*y*ln(y)

subject to 

x + y <= 3*10^9
x,y >= 0

I set up a Lagrangian

L(x,y,λ) = F(x) + F(y) + λ(3*10^9-(x+y))

I then reduced it algebraically before solving it with a graphing calculator

y = 3*10^9 - x  
e^(19/15)*x^(14/15) + x - 3*10^9 = 0
y^(15/14) + e^(19/14)*y - e^(19/14)*3*10^9 = 0

x = 1.61 * 10^9
y = 1.39 * 10^9

Now, I would like to be able to achieve this solution in R because the actual function parameters may change. I have tried adapting a few different solutions I found online, but none seem to work.

If I understand the problem correctly, I need a non-linear solver. I therefore set up the problem in Rsolnp as such (inspired from this answer):

library(Rsolnp)

opt_func_log <- function(x) {
  a <- x[1] 
  b <- x[2] 
  
  ben_func <-  2e9 + 160*a - 7*a*log(a) + 3e9 + 170*b - 7.5*b*log(b)
  
  -ben_func #invert to find minimum
}

equal_const <- function(x) {
  a <- x[1] 
  b <- x[2] 
  
  a + b # budget constraint formula
  
}

eps <- .Machine$double.eps*10^2 # low number, but not zero due to logs
x0 <- c(0.1, 0.1) # starting values
budget <- 3e9 # overall budget constraint value

opt_solution_log <- solnp(pars = x0,
                          fun = opt_func_log,
                          eqfun = equal_const,
                          eqB = budget,
                          LB = c(eps,eps))

Unfortunately I don't get a viable solution

Iter: 1 fn: -5032442923.2173     Pars:  213333.43335 213333.43335
solnp-->Redundant constraints were found. Poor
solnp-->intermediate results may result.Suggest that you
solnp-->remove redundant constraints and re-OPTIMIZE

Iter: 2 fn: -5032442923.2173     Pars:  213333.43335 213333.43335
solnp--> Solution not reliable....Problem Inverting Hessian.

What am I doing wrong? What constraint is redundant in this problem? Have I defined the problem wrongly or is it just not solvable in this way?

Solution

  • Below are two options to solve the constrained optimization problem.

    If you work with base R

    You can use constrOptim if you need to set up linear constraints, e.g.,

    ui <- rbind(-c(1, 1), c(1, 0), c(0, 1))
ci <- c(-3e9, 0, 0)
theta <- runif(2)
constrOptim(theta,
    f = opt_func_log,
    grad = NULL,
    ui = ui,
    ci = ci,
    control = list(reltol = .Machine$double.eps)
)

    and you will obtain

    $par
[1] 1609771281 1390228719

$value
[1] -40508597161

$counts
function gradient
     754       NA 

$convergence
[1] 0

$message
NULL

$outer.iterations
[1] 3

$barrier.value
[1] -6339423

    If you would like to use fmincon from package pracma

    library(pracma)
fmincon(runif(2),
    opt_func_log,
    A = t(c(1, 1)),
    b = 3e9,
    lb = c(0, 0),
    tol = .Machine$double.eps
)

    which gives

    $par
[1] 1609771176 1390228824

$value
[1] -40508597161

$convergence
[1] 0

$info
$info$lambda
$info$lambda$lower
     [,1]
[1,]    0
[2,]    0

$info$lambda$upper
     [,1]
[1,]    0
[2,]    0

$info$lambda$ineqlin
[1] 4.604494 0.000000 0.000000


$info$grad
          [,1]
[1,] -4.604494
[2,] -4.604494

$info$hessian
             [,1]         [,2]
[1,] 1.778707e+00 5.176693e-09
[2,] 5.176693e-09 2.693166e-09