Module multiobjective 

Multi-objective optimization

Multi-objective Pareto Optimization

This module implements advanced multi-objective optimization algorithms, focusing on Pareto-optimal solutions and evolutionary approaches for handling conflicting objectives.

Mathematical Background

Multi-objective optimization deals with problems of the form:

minimize f(x) = [f₁(x), f₂(x), ..., fₘ(x)]ᵀ
subject to g(x) ≤ 0, h(x) = 0

where multiple objectives f₁, f₂, …, fₘ are optimized simultaneously.

Key Concepts

• Pareto Dominance: Solution x dominates y if x is no worse in all objectives and strictly better in at least one objective
• Pareto Front: Set of all non-dominated solutions
• Hypervolume: Volume of objective space dominated by a solution set
• Crowding Distance: Measure of solution density for diversity preservation

Algorithms

• NSGA-II: Non-dominated Sorting Genetic Algorithm II
• NSGA-III: Many-objective optimization with reference points
• MOEA/D: Multi-objective Evolutionary Algorithm based on Decomposition
• Hypervolume-based Selection: HypE, SMS-EMOA variants

Structs

Individual

Individual solution in multi-objective optimization

MultiObjectiveConfig

Configuration for multi-objective optimization algorithms

MultiObjectiveResult

Results from multi-objective optimization

NsgaII

NSGA-II optimizer for multi-objective problems

ParetoFront

Pareto front representation

Traits

MultiObjectiveFunction

Trait for multi-objective function evaluation