Advanced Physics for Brachytherapy 2018

Inverse Optimisation and Planning: A Multi-Objective (MO) Problem

Dominance & Pareto Front

A plan/solution x 1 conditions are true:

dominates a plan/solution x 2

if and only if the two following

▪

2 ) ∀ j=1,..., M.

x 1 is no worse than x 2

in all objectives, i.e. f j (x 1

) ≤ f

(x

j

▪

x 1 is strictly better than x 2

in at least one objective ,

2 ) for at least one j ∈ {1,...,M}.

i.e. f j

( x

) < f

(x

1

j

Among a set of solutions P , the non-dominated set of solutions P' are those that are not dominated by any other member of the set P : The Pareto Optimal Set P’ . For the case that the set P is the entire feasible search space then the set P' is called the global Pareto Optimal Set . The image f(x) of the Pareto Optimal Set is called the Pareto Front (PF) . The Pareto Optimal Set is defined in the Parameter Space , while the Pareto Front is defined in the Objective Space .