KirlikSayin with unbounded objectives

[odow]

@xgandibleux says:

I have prepared an exercise for my students which is a project planning
problem with 3 objectives, all the variables are discrete (ps: setting
the variables t_i >=0 is enough), see the attached document.

When I solve it with MOA.Lexicographic(all_permutations = true) I get a
result (ps: here I have to check the implementation because line 2 and 3
in the solution returned should be exchanged because we expect to get
the minimum on fct2 when fct2 is minimized, which is not the case - but
it is late now, maybe I don't see the obvious-).

When MOA.KirlikSayin() is called for solving this problem, a warning
appears and the resolution is stopped: "objective 1 does not have a
finite domain". If I am not wrong the initial conditions for applying KS
on this problem are fulfilled, right (objectives have bounded values).
Do you have any idea why this warning is raised and then the resolution
stopped?

Reproducible example

julia> using JuMP

julia> import MultiObjectiveAlgorithms as MOA

julia> import HiGHS

julia> model = Model(() -> MOA.Optimizer(HiGHS.Optimizer))
A JuMP Model
Feasibility problem with:
Variables: 0
Model mode: AUTOMATIC
CachingOptimizer state: EMPTY_OPTIMIZER
Solver name: MOA[algorithm=MultiObjectiveAlgorithms.Lexicographic, optimizer=HiGHS]

julia> set_silent(model)

julia> set_attribute(model, MOA.Algorithm(), MOA.KirlikSayin())

julia> @variable(model, x >= 0, Int)
x

julia> @variable(model, y >= 0, Int)
y

julia> @constraint(model, x + y >= 1)
x + y ≥ 1.0

julia> @objective(model, Min, 1.0 * [x, y])
2-element Vector{AffExpr}:
 x
 y

julia> optimize!(model)
┌ Warning: Unable to solve problem using `KirlikSayin()` because objective 1 does not have a finite domain.
└ @ MultiObjectiveAlgorithms ~/.julia/packages/MultiObjectiveAlgorithms/RdHL3/src/algorithms/KirlikSayin.jl:126

[remi-garcia]

It might be confusing since the nadir point is bounded if the problem is. However in KirlikSayin it is the worst point that is computed:

y U is an over-estimator of the nadir point y N

after Equation (2).

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In the current bi-objective case it seems that we should compute the nadir point as it is really simple from the lexicographic points. That makes this warning unexpected.

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It might be a good idea to have dedicated functions to compute the nadir point and the worst case point.

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edit: https://doi.org/10.1016/S0377-2217(02)00595-7

[odow]

Does this new warning help? https://github.com/jump-dev/MultiObjectiveAlgorithms.jl/pull/64/files

We could provide a dedicated function for the nadir point in the biobjective case, but it seems like KirlikSaying etc are most useful for N > 2.

[remi-garcia]

The new warning is more explicit and will probably avoid some confusion 👍🏼

I've just suggested changes to the warning to call it anti-ideal point instead of nadir point.