How to minimize a function in R with two constraints?

Question

I want to solve a minimization problem with two constraints in R.

Problem: min x^T H x st

(1) e^T*x = 1,

(2) 0 < x_i <= 1 , i=1,..,20.

H is a 20x20 Matrix and e is vector of ones with length 20

How can I do this? I looked at optimize , optim and optimix , but somehow I don't know how to start.

I started to create my objective function:

f<- function(x) {t(x)%*%H%*%x}

and constraint (1)

g<- function(x) {t(e)*x=1}

But I don't know how to formulate constraint (2).

I also don't know which optimization function is suitable for this problem. I am thankful for any advice.

In the end I want to receive a vector x with 20 values.

Answer 1

check the quadprog package. It has a function solve.QP to solve the following quadratic programming:

min(-d^T b + 1/2 b^TD b) with the constraints A^T b >= b_0.

In your case D=H*2, d = 0.

Here is how you can construct the matrix A and vector b0 for your problem:

Amat <- t(rbind(rep(1, n), diag(1, 20), -diag(1, 20))
b0 <- c(1, rep(0, 20), rep(-1, 20))

Then you can run

library(quadprog)
solve.QP(Dmat = 2*H, dvec = 0, Amat = Amat, bvec = b0, meq = 1)

Note that meq = 1 indicates that first inequality should be an equality.

If you only want the optimal x values then try this:

solve.QP(Dmat = 2*H, dvec = 0, Amat = Amat, bvec = b0, meq = 1)$solution

Hope this helps.

Answer 2

Here is how to do with the CVXR package.

Note that (1) is the same as sum(x_i) = 1 , and (2) can be simplified to x_i > 0 because sum(x_i) = 1 and x_i > 0 => x_i <= 1 .

H <- rWishart(1, df = 30, diag(20))[,,1] # a 20x20 symmetric positive matrix

library(CVXR)

# the variable
x <- Variable(20)

# objective 
objective <- Minimize(quad_form(x, H))

# define problem
constraint1 <- sum(x) == 1
constraint2 <- x > 0
problem <- Problem(objective, constraints = list(constraint1, constraint2))

# solve problem
result <- solve(problem)

# results
result$getValue(x) # optimal x
result$value # value of objective at optimal x