Minimization of Convex Functions

Abstract

Utilizing the Gâteaux variations from the preceding chapter, it is straightforward to characterize convexity for a function J on a subset $\mathcal{D}$ of a linear space $\mathcal{Y}$, such that a convex function is automatically minimized by y ∈ $\mathcal{D}$ at which its Gâteaux variations vanish.¹ Moreover, in the presence of strict convexity, there can be at most one such y. A large and useful class of functions is shown to be convex. In particular, in §3.2, the role of [strongly] convex integrands f in producing [strictly] convex integral functions F is examined, and a supply of such f is made accessible through the techniques and examples of §3.3. Moreover, the Gâteaux variations of integral functions will, in general, vanish at each solution y of an associated differential equation (of Euler-Lagrange).