SECTION 2 TANGENT CONES AND NORMAL CONES
In this section, we first try to solve the Question 1 and Question 3 of the last post, Understanding Lagrange Multipliers (1). The approach we adopt is to use a broader and more general theory to look at the nonlinear programming problems. Probably this theory is named by R.T.Rockafellar (Ralph Tyrrell Rockafellar), who is one of the founders of convex analysis. We call this theory “Variational Analysis”. The word “variational” comes from calculus of variations. Briefly, variational analysis is the study of variations of a point.
The first tool we need is tangent cones. Let us recall something learnt in differential geometry of curves and surfaces.
2.1 Tangent Space and contingent cone
Let be a regular surface in . If you don’t know what is a regular surface, then just think of as a hemisphere (say). Given a point , we can define the tangent space of at to be . We call the a tangent vector of at . Notice that by the definition of derivative, . This means, for any sequence , and let , we have . Note the above shows the following:
Proposition 2.1 If is a tangent vector of at , then there exist and such that .
It is the time to define the tangent cone. There are many types of tangent cones. The classical one is called contingent cone introduced by (Georges) Bouligand and independently (Francesco) Severi.
Without otherwise specified, we assume to be a real normed vector space under norm topology. If you do not know what is meant by normed vector space, you may let be with the standard Euclidean norm, that is, . On top of that, from here onwards, basic knowledge on point set topology is assumed.
Definition 2.2 Let be a nonempty subset of , and . is said to be a tangent vector of at if there exist and such that and (If we put , then and ). The collection of all tangent vectors of at is called the contingent cone, denoted by .
(1) Proposition 2.1 implies . From now on, tangent vector means the one in Definition 2.2.
(2) A subset of is called a cone if it is closed under nonnegative scalar multiplication, i.e. for any and , we have . In other words, . It follows that is a cone.
We are interested in whether the contingent cone is a closed convex cone, which is an object interested by “optimizationist”. We denote and the conic hull of a set to be .
(1) is a closed cone.
(2) If is starshaped at , then . Hence if is convex, then so is .
(1) Let be a sequence in convergent to some . By definition, for any tangent vector , we can find and such that . Hence for any , there exists such that
It follows that , , and , implying that . So is closed. It is a cone by the preceding remark.
(2) The inclusion “” is clear, and it does not require the starshapedness of . For the reverse inclusion, readers should be able to see that since is a closed cone, it suffices to show . Indeed, let . Owing to the starshapedness of at , for all , . Consequently, by definition. Since the conic hull of a convex set is convex, is a convex set if is convex.
One significance of a convex cone in optimization is that, it induces a reflexive, antisymmetric order on the space defined by: . Some minimality notion called “proper minimality” is defined in terms of the order induced by some contingent cone.
Some “pictures” of tangent cones will be here later.
Understanding Lagrange Multipliers (1)