Multi-Objective Optimisation

Multi-Objective Optimisation

Many real-world engineering problems involve optimizing multiple competing objectives simultaneously.

Problem Definition

A general multi-objective optimization problem is defined as:

$\arg {\min }_{\theta }L(\theta )=[L_1(\theta ),L_2(\theta ),\dots ,L_p(\theta )]$

Where:

• $L(\theta )$ is a vector of $p$ objective functions to be optimized: ${\mathbb{R}}^D\to {\mathbb{R}}^p$
• The goal is to find a solution that balances the trade-offs between objectives

Key Concepts

Pareto Dominance

A solution ${\theta }_a$ dominates another solution ${\theta }_b$ if:

• $L_i({\theta }_a)\le L_i({\theta }_b)$ for all $i$, and
• $L_j({\theta }_a)<L_j({\theta }_b)$ for at least one $j$

Pareto Optimality

A solution ${\theta }^{\ast }$ is Pareto optimal if no other solution dominates it.

Pareto Front

The set of all Pareto optimal solutions forms the Pareto Front.

Key Challenge

Due to trade-offs, there are typically multiple (potentially infinite) Pareto optimal solutions – uniqueness is lost (even in the local sense).

Solution Methods

Weighted Sum Method

${\min }_{\theta }{\sum }_{i=1}^p{w}_i{L}_i(\theta ),w_i\ge 0,{\sum }_{i=1}^p{w}_i=1$

This transforms multi-objective optimization into a single-objective problem by assigning weights to each objective.

ε-Constraint Method

${\min }_{\theta }{L}_1(\theta ),\text{ subject to }{L}_i(\theta )\le {\epsilon }_i\text{ for }i=2,\dots ,p$

This optimizes one objective while constraining the others.

Other Approaches

Various other scalarization methods exist to convert the vector of objective functions into a scalar quantity.