Pareto–Fenchel ϵ ‐subdifferential sum rule and ϵ ‐efficiency

The paper is centered around a sum rule for the efficient (Pareto) ϵ ‐subdifferential of two convex vector mappings, having the property to be exact under a qualification condition. Such a formula has not been explored previously. Our formula which holds under the Attouch–Brézis as well as Moreau–Rockafellar conditions, reveals strangely a primordial presence of the convex (Fenchel) ϵ ‐subdifferential. This appearance turns out to be rather favorable. This effectively permits to derive approximate efficiency conditions in terms of Pareto subgradient and vectorial normal cone, which completely characterizes an ϵ ‐efficient solution in constrained convex vector optimization in (partially) ordered spaces. Our sum rule also allows a fundamental deduction of relation between Pareto and Fenchel ϵ‐subdifferentials, which, in reality, brings out a certain gap linking ϵ ‐efficiency with ϵ ‐optimality. Scalarization approaches in connection with ϵ ‐subdifferentials are first established by simple proofs. This principle has contributed for a large part, not only for discovering the sum formula, but also for establishing some punctual necessary and/or sufficient conditions for Pareto ϵ ‐subdifferentiability.