Efficiency for multitime vector variational problems on Riemannian manifolds involving geodesic quasiinvex functionals

TL;DR

This paper investigates optimality and efficiency conditions for multitime variational problems on Riemannian manifolds, introducing new notions of geodesic quasiinvexity to establish sufficient conditions for Pareto efficiency.

Contribution

It introduces the concept of $(\rho ,b)$-geodesic quasiinvexity and applies it to derive sufficient efficiency conditions for multitime vector variational problems on Riemannian manifolds.

Findings

Established necessary optimality conditions for scalar problems.

Defined Pareto and normal efficiency for vector problems.

Derived sufficient efficiency conditions using geodesic quasiinvexity.

Abstract

We study the connection between a multitime scalar variational problem (SVP), a multitime vector variational problem (VVP) and a multitime vector fractional variational problem (VFP). For (SVP), we establish necessary optimality conditions. For both vector variational problems, we define the notions of Pareto efficient solution and of normal efficient solution and we establish necessary efficiency conditions for (VVP) and (VFP) using both notions. The main purpose of the paper is to establish sufficient efficiency conditions for the vector problems (VVP) and (VFP). Moreover, we obtain sufficient optimality conditions for (SVP). The sufficient conditions are based on our original notion of $(\rho ,b)$-geodesic quasiinvexity.