Solve Quadratically Constrained Linear Objective Function in R

Question

Context

A bit of context is in order. I am currently migrating an Excel tool (sigh!) to n Shiny app. Excel is using the solver Addin to find an "optimal" solution. Now I am looking for a method to do the same in R.

Analyzing the Objective Function

I did my homework and analyzed the function currently "optimized" in Excel and saw that it can be formalized as a linear objective function with a quadratic constraint, that is, I am effectively trying to optimize this program:

     min P'x
s.t. x'Qx + q'x >= a given const
     0 <= x <= 1 (piecewise)

Currently my problem involves just 2 variables and I tried to solve it in R using library(optiSolve):

library(optiSolve)

Q <- matrix(c(0, 3897.6, 3897.6, 0), 2)
q <- matrix(c(2, 0), ncol = 1)
P <- matrix(c(1.24, 10), ncol = 1)
x_cur <- matrix(c(0.1, 0.01), ncol = 1)

ref_val <- t(x_cur) %*% Q %*% x_cur + t(q) %*% x_cur 

of <- linfun(P)
qc <- quadcon(-Q, -q, val = -ref_val)
lb <- lbcon(rep(0, ncol(Q)))
ub <- ubcon(rep(1, ncol(Q)))

mod <- cop(of, qc = qc, lb = lb, ub = ub)
solvecop(mod, quiet = TRUE)
# No solution exists.
# $x
#             1             2 
# -0.0000219045 -0.0005667524 
# 
# $solver
# [1] "cccp2"
#
# $status
# [1] "No solution exists."

However, the solver is able find a slightly better solution (0.0909556361478811, 0.0110198817266055) with the following settings:

(K3:K4 hold the variables to change (x), C7 calculates the quadratic form, B7 holds my reference value and C8 is the linear objective function).

x_excel <- matrix(c(0.0909556361478811, 0.0110198817266055), ncol = 1)
all.equal(t(x_excel) %*% Q %*% x_excel + t(q) %*% x_excel, ref_val,
          tolerance = 0.000001) # excel solver tolerance
# [1] TRUE
c(t(P) %*% x_cur > t(P) %*% x_excel)
# [1] TRUE
all(x_excel >= 0 & x_excel <= 1)
# [1] TRUE

Goal

I am fully aware that Excel simply uses another solver ( GRG2 if I am not mistaken) and also that my constraint matrix is not positive (semi-) definite.

Thus, it seems that despite my problem is a QC, I cannot rely on dedicated solvers. Hence, I need to solve it differently. I do not need a global optimum, a local one would be fine. Ideally, I could use the same solver to the twin problem, where we want to maximize the Quadratic form, while the linear element becomes a constraint.

My naive approach would be to use optim and add the quadratic constraint as a penalty to the objective function, but this feels very hackish and i fear scaling effects need to be considered carefully (the linear funciton and the quadratic form are not at all on the same scale).

How could I get a (local) optimum with R with a quadratic constraint?