1.3.4 Global minima and convex problems

Regarding global minima of $J$ over a set $A\subset V$ , much of the discussion on global minima given at the end of Section 1.2.1 carries over to the present case. In particular, the Weierstrass Theorem is still valid, provided that compactness of $A$ is understood in the sense of the second or third definition given on page  (existence of finite subcovers or sequential compactness). These two definitions of compactness are equivalent for linear vector spaces equipped with a norm (or, more generally, a metric). On the other hand, closed and bounded subsets of an infinite-dimensional vector space are not necessarily compact--we already mentioned noncompactness of the unit sphere--and the Weierstrass Theorem does not apply to them; see the next exercise. We note that since our function space $V$ has a norm, the notions of continuity of $J$ and convergence, closedness, boundedness, and openness in $V$ with respect to this norm are defined exactly as their familiar counterparts in $\mathbb{R}^n$ . We leave it to the reader to write down precise definitions or consult the references given at the end of this chapter.

If $J$ is a convex functional and $A\subset V$ is a convex set, then the optimization problem enjoys the same properties as the ones mentioned at the end of Section 1.2.1 for finite-dimensional convex problems. Namely, a local minimum is automatically a global one, and the first-order necessary condition is also a sufficient condition for a minimum. (Convexity of a functional and convexity of a subset of an infinite-dimensional linear vector space are defined exactly as the corresponding standard notions in the finite-dimensional case.) However, imposing extra assumptions to ensure convexity of $J$ would severely restrict the classes of problems that we want to study. In this book, we focus on general theory that applies to not necessarily convex problems; we will not directly use results from convex optimization. Nevertheless, some basic concepts from (finite-dimensional) convex analysis will be important for us later, particularly when we derive the maximum principle.