Unconstrained Steepest Descent Method for Multicriteria Optimization on Riemannian Manifolds

Abstract

In this paper, we present a steepest descent method with Armijo's rule for multicriteria optimization in the Riemannian context. The sequence generated by the method is guaranteed to be well defined. Under mild assumptions on the multicriteria function, we prove that each accumulation point (if any) satisfies first-order necessary conditions for Pareto optimality. Moreover, assuming quasiconvexity of the multicriteria function and nonnegative curvature of the Riemannian manifold, we prove full convergence of the sequence to a critical Pareto point.

Iteration-Complexity and Asymptotic Analysis of Steepest Descent Method for Multiobjective Optimization on Riemannian Manifolds

“…Hereafter, we assume that M is a complete Riemannian manifolds with sectional curvature K ≥ κ, where κ < 0. We point out that for Riemannian manifold with nonnegative sectional curvature, the convergence analysis of the steepest descent method for convex and quasi-convex vector functions is well understood; see for example [35,36].…”

“…First, asymptotic analysis will be done for quasi-convex and convex vectorial functions. In fact, in [35] asymptotic analysis of this method has already been done in Riemannian context; see also [36]. However, the analysis asymptotic presented in these previous works is just to stepsize given by Armijo rule and it demand that the Riemannian manifolds have nonnegative sectional curvature.…”

Riemannian optimization and multidisciplinary design optimization

“…In Riemannian geometry, however, convexity is defined with geodesics instead of straight lines (Bento et al 2012); assuming that there is a unique geodesic c t ð Þ connecting x and y such that c 0 ð Þ ¼ x and c 1 ð Þ ¼ y, a function f is convex if…”

“…Convexity is not, however, determined by the coordinate system used (Udrişte 1996a). This means that it is sometimes possible to transform nonconvex problems into convex problems through the choice of a favourable metric tensor (Bento et al 2012).…”

“…Firstly, Bento et al (2012) touched on Multi-Objective Optimization (MOO) on Riemannian manifolds in their paper. We have not seen any other work in this area, but given that single-objective RO is as established (though small) as it is, it makes sense to expand into multiple objectives.…”