We compute two-sided second-order epi-derivatives for certain composite functionals f = g o F where F is a 𝒞1 mapping between two Banach spaces X and Y, and g is a convex extended real-valued function on Y. These functionals include most essential objectives associated with smooth constrained minimization problems on Banach spaces. Our proof relies on our development of a formula for the second-order upper epi-derivative that mirrors a formula for a second-order lower epi-derivative obtained elsewhere, and the two-sided results we obtain promise to support a more precise sensitivity analysis of parameterized optimization problems than has been previously possible.
