Problem with boundaries

[newptcai]

In the following problem, the actual minimal points are simply (0,1) and (1,0). but the package outputs a lot of boxes. It does not seem that decreasing tolerance helps.

julia> global_min, minimisers = minimise(X->((x,y)=X; -(x/(y+1)+y/(x+1))), IntervalBox(0..1, 2), tol=1e-15);

julia> minimisers
20-element Array{IntervalBox{2,Float64},1}:
 [0.999999, 1] × [0.999999, 1]             
 [0.999999, 1] × [0.999999, 1]             
 [0, 6.00051e-16] × [0.999999, 1]          
 [6.0005e-16, 1.20956e-15] × [0.999999, 1] 
 [0.999999, 1] × [0, 6.00051e-16]          
 [0.999999, 1] × [6.0005e-16, 1.20956e-15] 
 [0.999999, 1] × [0.999999, 1]             
 [0.999999, 1] × [0.999999, 1]             
 [0.999999, 1] × [0.999999, 1]             
 [0.999999, 1] × [0.999999, 1]             
 [0.999999, 1] × [1.20955e-15, 1.81906e-15]
 [1.20955e-15, 1.81906e-15] × [0.999999, 1]
 [0.999999, 1] × [0.999999, 1]             
 [0.999999, 1] × [0.999999, 1]             
 [0.999999, 1] × [0.999999, 1]             
 [0.999999, 1] × [0.999999, 1]             
 [0.999999, 1] × [0.999999, 1]             
 [0.999999, 1] × [0.999999, 1]             
 [0.999999, 1] × [0.999999, 1]             
 [0.999999, 1] × [0.999999, 1]             

[dpsanders]

I believe this is a general issue with interval optimisation methods, known as the cluster effect.
One possible solution is to use the mean value form.
Another is to combine this with https://github.com/dpsanders/IntervalConstraintProgramming.jl

Or it might just be that it struggles with points at the boundary.

[newptcai]

How does ConstraintProgramming help in this case?

[dpsanders]

Maybe it doesn't actually! Basically you contract boxes to only the region that is <= the current upper bound on the global minimum.

[yashvardhan747]

Try using CharibdeOptim.jl package and use ibc_minimise function for the problem.

julia> using CharibdeOptim
[ Info: Recompiling stale cache file /Users/yash/.julia/compiled/v1.2/CharibdeOptim/oQR8Z.ji for CharibdeOptim [be8f459e-2180-59b5-a986-74dabacdd8e3]

julia> using IntervalArithmetic

julia> ibc_minimise(X->((x,y)=X; -(x/(y+1)+y/(x+1))), IntervalBox(0..1, 2), tol=1e-15)
([-1.00001, -1], IntervalBox{2,Float64}[[0.999999, 1] × [0.999999, 1]], CharibdeOptim.Information(0, 0, 173))