Abstract
Multi-objective optimization problems (MOPs) naturally arise in many applications. Since for such problems one can expect an entire set of optimal solutions, a common task in set based multi-objective optimization is to compute N solutions along the Pareto set/front of a given MOP. In this work, we propose and discuss the set based Newton methods for the performance indicators Generational Distance (GD), Inverted Generational Distance (IGD), and the averaged Hausdorff distance Δp for reference set problems for unconstrained MOPs. The methods hence directly utilize the set based scalarization problems that are induced by these indicators and manipulate all N candidate solutions in each iteration. We demonstrate the applicability of the methods on several benchmark problems, and also show how the reference set approach can be used in a bootstrap manner to compute Pareto front approximations in certain cases.
Highlights
Multi-objective optimization problems (MOPs), i.e., problems where multiple incommensurable and conflicting objectives have to be optimized concurrently, arise in many fields such as engineering and finance (e.g., [1,2,3,4,5])
Continuous unconstrained multi-objective optimization problems are expressed as min F ( x ), x where F : Rn → Rk, F ( x ) = ( f 1 ( x ), . . . , f k ( x )) T denotes the map that is composed of the individual objectives f i : Rn → R, i = 1, . . . , k, which are to be minimized simultaneously
There exist set oriented methods that are capable of obtaining the entire solution set in a global manner. Examples for the latter are subdivision [34,35,36] and cell mapping techniques [37,38,39]. Another class of set based methods is given by multi-objective evolutionary algorithms (MOEAs) that have proven to be very effective for the treatment of MOPs [14,16,40,41,42,43]
Summary
Multi-objective optimization problems (MOPs), i.e., problems where multiple incommensurable and conflicting objectives have to be optimized concurrently, arise in many fields such as engineering and finance (e.g., [1,2,3,4,5]). Many numerical methods take this fact into account and generate an entire (finite) set of candidate solutions so that the decision maker (DM) obtains an overview of the possible realizations of his/her project For such set based multi-objective optimization algorithms a natural question that arises is the goodness of the obtained solution set A (i.e., the relation of A to the Pareto set/front of the underlying MOP). MOEAs evolve entire sets of candidate solutions (called populations or archives) and are capable of computing finite size approximations of the entire Pareto set/front in one single run of the algorithm They are of global nature, very robust, and require only minimal assumptions on the model (e.g., no differentiability on the objective or constraint functions).
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