MUTIPLE-GRADIENT DESCENT ALGORITHM FOR MULTIOBJECTIVE OPTIMIZATION
Jean‐Antoine Désidéri · 2012
Abstract. The steepest-descent method is a well-known and effective single-objective descent algorithm when the gradient of the objective function is known. Here, we propose a particular generalization of this method to multi-objective optimization by considering the concurrent minimization of n smooth criteria {Ji} (i = 1,...,n). The novel algorithm is based on the following observation: consider a finite set of vectors {ui} (ui ∈ R N, n ≤ N); in the convex hull of this family, there exists a unique element of minimal norm, say ω ∈ R N; then, the scalar product of ω with any vector in the convex hull, and in particular, with any ui, is at least equal to ‖ω ‖ 2 ≥ 0. Applying this to the objective-function gradients (ui = ∇Ji), we conclude that either ω = 0, and the current design point belongs to the Pareto set, or −ω is a descent direction common to all objective functions. We propose to construct a fixed-point iteration in which updates of the element ω are used as successive directions of search. This method converges to a point on the Pareto set. This result applies to both finite-dimensional and functional design spaces. Numerical illustrations have been provided in both cases using either analytical objective functions, or (discretized) functionals in [9] [5]. Here, following [6], a domain-decomposition framework is used to illustrate the necessity, in a (discretized) functional setting, to scale the gradients appropriately. Jean-Antoine Désidéri hal-00764318, version 1- 12 Dec 2012 1