Abstract
In the paper, we analyze the necessary efficiency conditions for scalar, vectorial and vector fractional variational problems using curvilinear integrals as objectives and we establish sufficient conditions of efficiency to the above variational problems. The efficiency sufficient conditions use of notions of the geodesic invex set and of (strictly, monotonic) ( ρ , b)-geodesic quasiinvex functions.
Highlights
We establish the necessary efficiency conditions under new forms, and we prove the sufficient efficiency conditions for Mathematics 2020, 8, 1054; doi:10.3390/math8071054
Suppose that x0 (·) ∈ D is an optimal solution to the variational problem (SVP) and that we choose the functions (Lagrange multipliers): λ β (t) ≥ 0, t ∈ Ω, β = 1, . . . , m, such that λ β (t) g β = 0 and μκ (t) ∈ R, κ = 1, . . . , q, t ∈ Ω
Theorem 2. (Fritz–John conditions) If x0 (·) ∈ D is an optimal solution of the variational problem (SVP) there exist the real scalar τ ∈ R and the piecewise smooth functions λ = (λ β (t)) ∈ Rm and μ = (μθ (t)) ∈ Rq defined on Ω, satisfying the following conditions:
Summary
Valentine [1] began the study of scalar variational problems with constraints by establishing the necessary conditions of optimality in 1937. The sufficient conditions of efficiency are based on the notions of the geodesic invex set and (strictly) (ρ, b)-geodesic quasiinvex functions on Riemannian manifolds. Let S ⊂ M be an open η-geodesic invex set and f : S → R be a C1 function. A set G ⊂ F (Ω, M) is called η-geodesic invex if, for every x0 (·), x (·) ∈ G, there exists exactly one geodesic deformation φ(t, θ ), t ∈ Ω, θ ∈ [0, 1] such that the vector function η (t) = η ( x0 (t), x (t)) = (η 1 (t), .
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