Box-constrained multi-objective optimization: A gradient-like method without "a priori" scalarization

Let Rn be the n-dimensional real Euclidean space, x = (x1, x2, . . . , xn) ∈ Rn be a generic vector, where the superscript T means transpose. Let x, y ∈ Rn, we denote by yTx the Euclidean scalar product, by ‖x‖ = (xTx)1/2 the Euclidean norm and by the symbols x < y and x ≤ y, x, y ∈ Rn the componentwise inequalities, that is: xi < yi, xi ≤ yi, i = 1, 2, . . . , n. Let us denote by Uδ(x̂) the open ball with center at x̂ and radius δ, i.e. Uδ(x̂) = {x ∈ Rn | ‖x− x̂‖ < δ } and by B ⊂ Rn the following box: B = { x ∈ R | l ≤ x ≤ u } (1)