I am trying to use Mosek to solve this optimization problem. I have my code which displays the optimal solution of the values of x,y,z and the radius R. I want to insert a loop in my code to test if the absolute of the optimal solutions thus absolute(x^2 + y^2 -z) <= 0.01 then it should stop and maintain the answer for x,y, z and R but if the absolute value of (X^2 + y^2-z) is not less than or equal to 0.01 it should again find R with the same formula R= max(R,(((a[i]-x)^2 + (b[i]-y)^2)^0.5 + r[i] )) using the current optimal values for x,y and calculate the new optimal value of x,y,z until the condition that the absolute of (x^2 + y^2 -z) <= 0.01 is met. It should stop and display the optimal results for x,y,z and R immediately the absolute(x^2 + y^2 -z) <= 0.01.

Any guidance to get this done will be highly appreciated please. Attached is my code.

using JuMP
using Mosek
using MosekTools
model= Model(Mosek.Optimizer)
m=15
NumVar=5
a=[0,2.5,-2.5,2,-2.5]
b=[0,0.3,0.25,-2.5,-2.25]
r=[4, 8, 9, 8, 8]
(x,y)=(0,0)
for (i,j) in enumerate(a)
R= max(R,(((a[i]-x)^2 + (b[i]-y)^2)^0.5 + r[i] ))
end
@variable(model, x>=0)
@variable(model, y>=0)
@variable(model, z>=0)
@objective(model, Min, x^2 + y^2-z )
#######Declare the constraints
for i in 1:NumVar
@constraint(model, -2*x*a[i]-2*b[i]*y + z <=(R-r[i])^2-a[i]^2-b[i]^2)
end
@show model
print(model)
optimize!(model)
@show termination_status(model)
@show primal_status(model)
@show dual_status(model)
@show objective_value(model)
@show value(x)
@show value(y)
@show value(R)
@show value(z)

Perhaps something like this? (I didnâ€™t test, so there might be typos, etc).

using JuMP
import Mosek
import MosekTools
function inner_loop(x, y)
NumVar = 5
a = [0,2.5,-2.5,2,-2.5]
b = [0,0.3,0.25,-2.5,-2.25]
r = [4, 8, 9, 8, 8]
R = 0.0
for i in 1:length(a)
R = max(R, sqrt((a[i] - x)^2 + (b[i] - y)^2) + r[i])
end
model = Model(Mosek.Optimizer)
@variable(model, x >= 0)
@variable(model, y >= 0)
@variable(model, z >= 0)
@objective(model, Min, x^2 + y^2 - z)
@constraint(
model,
[i in 1:NumVar],
-2 * x * a[i] - 2 * b[i] * y + z <= (R - r[i])^2 - a[i]^2 - b[i]^2,
)
optimize!(model)
return value(x), value(y), value(z), R
end
function main()
x, y, z, R = 0.0, 0.0, Inf, 0.0
while abs(x^2 + y^2 - z) >= 0.01
x, y, z, R = inner_loop(x, y)
end
return x, y, z, R
end

This worked after I call the function at the end of the code. Am grateful for the help. Thank you very much.
I kindly want to ask some questions.
Why did you set R=0.0 from the beginning.

and how did you get this values of x,y,z,R given below.
x, y, z, R = 0.0, 0.0, Inf, 0.0