A chain rule for ɛ-subdifferentials with applications to approximate solutions in convex Pareto problems

In this work we obtain a chain rule for the approximate subdifferential considering a vector-valued proper convex function and its post-composition with a proper convex function of several variables nondecreasing in the sense of the Pareto order. We derive an interesting formula for the conjugate of a composition in the same framework and we prove the chain rule using this formula. To get the results, we require qualification conditions since, in the composition, the initial function is extended vector-valued. This chain rule extends analogous well-known calculus rules obtained when the functions involved are finite and it gives a complementary simple expression for other chain rules proved without assuming any qualification condition. As application we deduce the well-known calculus rule for the addition and we extend the formula for the maximum of functions. Finally, we use them and a scalarization process to obtain Kuhn-Tucker type necessary and sufficient conditions for approximate solutions in convex Pareto problems. These conditions extend other obtained in scalar optimization problems.