R: How to add additional constraints to DEoptim - r

Question

I am trying to minimize an objective function using DEoptim, subject to a simple constraint. I am not clear as to how to add the simple constraint to the call to DEoptim. Here is the objective function:
obj_min <- function(n,in_data) {
gamma <- in_data$Gamma
delta <- in_data$Delta
theta <- in_data$Theta
gammaSum <- sum(n * gamma)
deltaSum <- sum(n * delta)
thetaSum <- sum(n * theta)
abs((EPC * gammaSum - 2 * abs(deltaSum)) / thetaSum )
}
My mapping function (to impose integer constraints) is as follows:
mappingFun <- function(x) {
x[1:length(x)] <- round(x[1:length(x)], 0)
}
My call to DEoptim is:
out <- DEoptim(DTRRR_min, lower = c(rep(-5, length(in_data[, 1]))),
upper = c(rep(5, length(in_data[, 1]))),
fnMap = mappingFun, DEoptim.control(trace = F),in_data)
My in_data object (data frame) is:
Underlying.Price Delta Gamma Theta Vega Rho Implied.Volatility
1 40.69 0.9237 3.2188 -0.7111 2.0493 0.0033 0.3119
2 40.69 0.7713 6.2267 -1.6352 4.3240 0.0032 0.3402
3 40.69 0.5822 8.4631 -2.0019 5.5782 0.0338 0.3229
4 40.69 0.3642 8.5186 -1.8403 5.3661 0.0210 0.3086
5 40.69 0.1802 6.1968 -1.2366 3.7517 0.0093 0.2966
I would like to add a simple constraint that:
sum(n * delta) = target
In other words, the summation of the optimized parameters, n, multiplied by the deltas in my in_data data frame sum to a target of some sort. For simplicity, lets just say 0.5. How do I impose
sum(n * delta) = 0.5
as a constraint? Thank you for your help!

Answer 1

OK, thank you for all of your suggestions. I have researched and worked through my problem from many angles, and I wanted to share my thoughts with everyone, in case they can be helpful to some of you.
Most obvious, in my particular objective function, deltaSum is a variable, and I am attempting to constrain it to a particular value. Simple substitution of this constrained value into the objective function is the solution to this (trivial). However, assuming I was to introduce a constraint on a variable which is not already a variable in the objective function, I can simply run a for loop which returns Inf for any constraint I wish to impose, ie:
obj_func_sum_RRRs <- function(n, in_data) {
#Declare deltaSum, gammaSum, thetaSum, vegaSum, and rhoSum from in_data
#Impose constraints
#No dividing by 0:
if (thetaSum == 0) {
return(Inf)
}
#Specify that regardless of the length of vector of variables to
#be optimized, we only want our final results to include either 4 or 6
#nonzero n's in our final optimized solution
if (!sum(n[1:length(n)] != 0) == 4 &
!sum(n[1:length(n)] != 0) == 6) {
return(Inf)
}
(deltaSum + gammaSum)/thetaSum
}
The first for loop, (thetaSum == 0, return Inf) works because while Inf is a solution which the optimizer understands (and will never select as optimal), division by 0 in R returns NaN, which "breaks" the optimization process. This is a bit "hacky", in that it is likely NOT the most computationally efficient way to approach the problem, but to be honest, with the infrastructure that I am developing with a close friend and software architect guru (which utilizes microservices deployed through the Microsoft Service Fabric), our long-range backtesting is still lightening quick. This methodology actually allows you to impose any number of constraints on your problem, although further testing would need to be done to see how burdensome the computational complexity could become using this technique...
The Lagrange technique above can be viable, but only if you derive an analytical form of lambda on paper, then implement in code. It is not always practical in application, and while you may be able to code up an algorithm to optimize the parameter, it sounds like a bad idea to paint yourself into a corner where you have to optimize a parameter which is, in turn, necessary to the optimizing of the original objective function. Just setting a for loop as advised above seems the better way to go.
Food for thought....

Answer 2

DEOptim package description says
Implements the differential evolution algorithm for global
optimization of a realvalued function of a real-valued parameter
vector.
The concept of global optimization doesn't have place for constraints and it is also known as unconstrained optimization. So sorry but its not possible directly. Having said that you can always use "Lagrange's multiplier" hack if you must do it. To do it you need to do something like:
abs((EPC * gammaSum - 2 * abs(deltaSum))/thetaSum) - lambda* (sum(n * delta) - 0.5)
where you penalizing slack of your constraint.

Answer 3

I am using a wrapper which customises the call of DEoptim based on external constraints. Not very elegant I admit it but it works to some extent.
My objective function - a Monte Carlo simulation - is quite time consuming
so constraints are really helpful...
Chris
Due to the very specific character of what I am doing (Monte Carlo raytracing for the optimisation of neutron beam optics) I did not see any reason to add code. I think it is really the concept what matters here. I'll gladly share what I have with anybody interested. Just let me know.... Chris