Computing finite size representations of the set of approximate solutions of an MOP with stochastic search algorithms

Abstract

In this work we study the convergence of generic stochastic search algorithms toward the entire set of approximate solutions of continuous multi-objective optimization problems. Since the dimension of the set of interest is typically equal to the dimension of the parameter space, we focus on obtaining a finite and tight approximation, measured by the Hausdorff distance. Under mild assumptions about the process to generate new candidate solutions, the limit approximation set will be determined entirely by the archiving strategy. We propose and investigate a novel archiving strategy theoretically and empirically. For this, we analyze the convergence behavior of the algorithm, yielding bounds on the obtained approximation quality as well as on the cardinality of the resulting approximation, and present some numerical results.

Computing the Set of Epsilon-Efficient Solutions in Multiobjective Space Mission Design

“…Therefore, for the optimal design of a generic interplanetary trajectory we can identify two objective functions: time of flight and total ∆v (or total propellant consumption). 3,19,21,24 The quality of a solution in a MOP is generally defined by its Pareto optimality, i.e. a solution is Pareto optimal if no other solution is better over all the objective values.…”

“…2 Approximate solutions in multi-objective optimization have been studied by many researchers so far, 12,13,26 however, the only studies which deal (albeit theoretically) with the computation of the entire set of approximate solutions have been undertaken by Schütze et al 18,19 A first attempt to investigate the benefit of considering approximate solutions in space mission design has been done by Vasile and Locatelli,23 albeit for the single-objective case.…”

“…The maximum norm distance d ∞ , dist(·, ·), and d H (·, ·) are defined as follows: Schütze et al 19 showed that, under certain (mild) conditions on the set P Q,ǫ of the MOP and under the assumption ∀x ∈ Q and ∀δ > 0 :…”

Approximate Solutions in Space Mission Design

“…For instance, in space mission design, which we address here, there are two crucial aims for the realization of a transfer: minimization of flight time and fuel consumption of the spacecraft ( [1], [12], [11], [10]). The former objective is related to the cost of operations which could account for roughly 50% of the cost of an interplanetary space mission.…”

“…soluttions: [9], [10] The Design Problem In the following we will analyze few examples taken from two classes of typical problems in space trajectory design: a bi-impulsive transfer from the Earth to and the asteroid Apophis, and two low-thrust multi-gravity assist transfers from the Earth to a planet.…”