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The aim of bi-objective optimization is to obtain an approximation set of(near) Pareto optimal solutions. A decision maker then navigates this set toselect a final desired solution, often using a visualization of theapproximation front. The front provides a navigational ordering of solutions totraverse, but this ordering does not necessarily map to a smooth trajectorythrough decision space. This forces the decision maker to inspect the decisionvariables of each solution individually, potentially making navigation of theapproximation set unintuitive. In this work, we aim to improve approximationset navigability by enforcing a form of smoothness or continuity betweensolutions in terms of their decision variables. Imposing smoothness as arestriction upon common domination-based multi-objective evolutionaryalgorithms is not straightforward. Therefore, we use the recently introduceduncrowded hypervolume (UHV) to reformulate the multi-objective optimizationproblem as a single-objective problem in which parameterized approximation setsare directly optimized. We study here the case of parameterizing approximationsets as smooth Bezier curves in decision space. We approach the resultingsingle-objective problem with the gene-pool optimal mixing evolutionaryalgorithm (GOMEA), and we call the resulting algorithm BezEA. We analyze thebehavior of BezEA and compare it to optimization of the UHV with GOMEA as wellas the domination-based multi-objective GOMEA. We show that high-qualityapproximation sets can be obtained with BezEA, sometimes even outperforming thedomination- and UHV-based algorithms, while smoothness of the navigationtrajectory through decision space is guaranteed.

Ensuring smoothly navigable approximation sets by Bezier curve  parameterizations in evolutionary bi-objective optimization -- applied to  brachytherapy treatment planning for prostate cancer