Approximate first-order and second-order directional derivatives of a marginal function in convex optimization

Given a convex functionf:^p×^q (–, +], the marginal function is defined on ^p by (x)=inf{f(x, y)|y^q}. Our purpose in this paper is to express the approximate first-order and second-order directional derivatives of atx₀ in terms of those off at (x₀,y₀), wherey₀ is any element for which (x₀)=f(x₀,y₀).The author is indebted to one referee for pointing out an inaccuracy in an earlier version of Theorem 4.1.