What if we Increase the Number of Objectives? Theoretical and Empirical Implications for Many-objective Optimization

In practical applications, there can be significantly more than three objectives accounting for a mix of performance, resource, environmental and other goals; for example, Kollat et al. (2011) optimize the position and frequency of tracer sampling of groundwater using six design objectives, Eikelboom et al. (2015) address land use planning problems using cost, benefit and spatial objectives for the individual parcels of land, and Fleming et al. (2005) design optimal control systems for aircraft engines using eight objectives, each corresponding to a different design specifications. Multi-objective optimization (MO) (Deb, 2001; Miettinen, 2012) is the area looking at the development and application of algorithms to problems with multiple conflicting objectives; problems with more than three objectives have also been termed as many-objective problems (Kollat et al., 2011) and are less studied. In the absence of any user preferences about desired ideal solutions, the goal of MO algorithms (MOAs) is not to identify a single optimal solution but to approximate the set of best trade-off solutions to a problem, also known as the Pareto (optimal) set. We say that a solution Pareto dominates, or simply dominates, another solution if it is not worse in any objective and if it is strictly better in at least one objective. Solutions in the Pareto set are non-dominated by any other solution from the feasible solution space. The traditional approach to tackle a MO problem is to convert it into a single-objective problem using a scalarizing function (Eichfelder, 2008), and then solve the problem repeatedly using different'configurations' of the scalarizing function (e.g.