  The aim of bi-objective optimization is to obtain an approximation set of
(near) Pareto optimal solutions. A decision maker then navigates this set to
select a final desired solution, often using a visualization of the
approximation front. The front provides a navigational ordering of solutions to
traverse, but this ordering does not necessarily map to a smooth trajectory
through decision space. This forces the decision maker to inspect the decision
variables of each solution individually, potentially making navigation of the
approximation set unintuitive. In this work, we aim to improve approximation
set navigability by enforcing a form of smoothness or continuity between
solutions in terms of their decision variables. Imposing smoothness as a
restriction upon common domination-based multi-objective evolutionary
algorithms is not straightforward. Therefore, we use the recently introduced
uncrowded hypervolume (UHV) to reformulate the multi-objective optimization
problem as a single-objective problem in which parameterized approximation sets
are directly optimized. We study here the case of parameterizing approximation
sets as smooth Bezier curves in decision space. We approach the resulting
single-objective problem with the gene-pool optimal mixing evolutionary
algorithm (GOMEA), and we call the resulting algorithm BezEA. We analyze the
behavior of BezEA and compare it to optimization of the UHV with GOMEA as well
as the domination-based multi-objective GOMEA. We show that high-quality
approximation sets can be obtained with BezEA, sometimes even outperforming the
domination- and UHV-based algorithms, while smoothness of the navigation
trajectory through decision space is guaranteed.
