pymoo

Use pymoo's minimize() function with a problem definition and an algorithm like NSGA-II. The result object contains Pareto-optimal decision variables in result.X and objective values in result.F, which you can visualize with a Scatter plot.

Use NSGA-II for problems with 2-3 objectives, where it provides well-distributed Pareto fronts with low overhead. Use NSGA-III for 4 or more objectives, since it uses reference directions to maintain diversity in high-dimensional objective spaces.

Extend the ElementwiseProblem class, set n_var, n_obj, n_ieq_constr, and bounds in __init__, then implement _evaluate to return objectives in out["F"] and constraints in out["G"]. Inequality constraints must be formulated as g(x) <= 0.

Yes, pymoo handles constraints through feasibility-first selection by default, penalty methods via ConstraintsAsPenalty, constraint-as-objective conversion, and specialized algorithms like SRES and ISRES for heavily constrained problems.

NSGA-III requires properly generated reference directions using get_reference_directions with the das-dennis method. Also check that the population size matches the reference directions, increase generations, and verify objective scaling.

Normalize the objective values to [0, 1], then apply an MCDM method such as PseudoWeights with your preference weights, or use knee point detection for balanced trade-offs. The selected index maps back to the decision variables in result.X.