Riemannian convexity of functionals

Abstract

Geodesic deformation, Riemannian convex functionals, Riemannian convex functions, Optimization, Variational problems, Multitime optimal control,

Efficiency for Vector Variational Quotient Problems with Curvilinear Integrals on Riemannian Manifolds via Geodesic Quasiinvexity

“…Inspired by these aspects of the scalar variational problems, Udrişte et al [16,17,[20][21][22][23], proceeded to the optimality study of variational problems in many dimensions (multitime variational problems) with constraints, using multiple integrals and curvilinear integrals. In this work, we establish the necessary efficiency conditions under new forms, and we prove the sufficient efficiency conditions for these problems.…”

“…In this paper, new classes of variational control problems of minimizing a vector of path-independent curvilinear integral (mechanical or cost) functionals ratios, were considered. Starting from scalar variational problems (SVP) elaborated by Udriste et al [16,17,20,22,23] by which optimality conditions of variational problems in the multitime approach (so called multitime variational problems) with constraints were introduced in literature, in this paper by using curvilinear integrals and generalized invex functionals, new necessary and sufficient conditions of efficiency were obtained. In particular, we have formulated and proved necessary geodesic efficiency conditions in the considered scalar, vector and vector quotient variational control problems, by using the notation of normal geodesic efficient solution and new notions of geodesic efficient solution.…”