set.seed(1)
n=500
p=1
a=rnorm(n*p, mean = 10, sd = 1)
x=matrix(a,n,p)
beta=rnorm(p,1,1)
beta=matrix(beta,p,1)
y=x%*%beta-matrix(rnorm(n,0.1,1),n,1)
tau=0.01 # example tau, here to get underapproximation

library(microbenchmark)

# quantreg solution
library(quantreg)
microbenchmark(rq(y~0+x, tau)) # quantreg, 1.0 ms, beta=-0.023

# using Brent line search, using [0, least square estimate] search interval
quantloss = function(beta, x, y, tau) { r=y-beta*x   # quantile loss function
                                        sum(r*(r>0) - (1-tau)*r) }
microbenchmark(optimize(quantloss,
                       interval=c(0, mean(x*y)/mean(x*x)), # mean(x*y)/mean(x*x) = least square beta
                       x=x, y=y, tau=tau)$minimum) # 227 µs, beta=0.84