In convex analysis and optimization, the normal cone to a set at a point is a convex cone consisting of vectors that make a non-acute angle with every feasible direction from the point. For a convex set, it is the polar cone of the tangent cone and gives a geometric form of first-order optimality conditions.
Definition
Let be a convex subset of a finite-dimensional real inner product space , and let . The normal cone to at is
If , the normal cone is often defined to be empty. The elements of are called normal vectors to at . The sign convention above gives outward normals.
If is an interior point of , then . If is a smooth full-dimensional convex body and is a boundary point, then is the ray generated by the outward normal vector at .
Subgradients
Normal cones are closely related to subgradients. If is a proper convex function, then its epigraph
is a convex set. A vector is a subgradient of at if and only if
Equivalently,
Sublevel sets Still with a proper convex function, consider the sublevel set through a point :
Every subgradient of at determines a normal vector to at . Indeed, if and , then
Under standard regularity hypotheses, for example when and is not a minimizer of , the converse also holds:
or equivalently the normal cone to the sublevel set is the closed convex cone generated by the subdifferential.
Notes
References
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