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In variational analysis, set-valued analysis and optimization, the concept of a normal cone to a subset of a space generalizes that of the orthogonal complement / annihilator of a vector space, (outward) normal vector fields to surfaces — or more generally of the normal bundle of an embedded submanifold — to possibly non-smooth subsets of vector spaces.

Normal cones provide, among other things, the geometrical foundation for generalizing the convex subdifferential to non-convex functions. Of particular note is also their role in generalizing Fermat's rule to give necessary (and sometimes also sufficient) first order optimality conditions for constrained and non-smooth optimization problems. Moreover they play a role in defining coderivatives of set-valued maps. ^[1]

In the non-convex case there are several inequivalent definitions for a normal cone that all turn out to be useful and interesting for different problems — whereas in the convex case these all coincide which greatly simplifies things. For clarity and approachability this article first discusses the convex case over Hilbert spaces before going into the non-convex case over more general spaces.

All vector spaces in this article are assumed to be real. The set ${\displaystyle {\bar {\mathbb {R} }}}$ denotes the extended real numbers and a function ${\displaystyle f:X\to {\bar {\mathbb {R} }}}$ is called proper if it is nowhere equal to ${\displaystyle -\infty }$ and also somewhere finite. We also assume that all cones contain zero. Sums of sets throughout the article are to be interpreted as Minkowski sums.

The convex normal cone ${\displaystyle N_{C}^{\operatorname {conv} }}$ to a non-empty convex subset ${\displaystyle C\subseteq H}$ of a (pre-)Hilbert space is defined by$${\displaystyle {\begin{alignedat}{2}N_{C}^{\operatorname {conv} }({\bar {x}}):=&\{v\in H:\langle v,x-{\bar {x}}\rangle \leq 0~{\text{for all}}~x\in C\}\\=&\{v\in H:\sup _{x\in C}\langle v,x-{\bar {x}}\rangle \leq 0\}\end{alignedat}}}$$for all ${\displaystyle {\bar {x}}\in C}$, and ${\displaystyle N_{C}^{\operatorname {conv} }({\bar {x}})=\emptyset }$ for all ${\displaystyle {\bar {x}}\not \in C}$. ^[2][3] Here ${\displaystyle \langle \cdot ,\cdot \rangle :H\times H\to \mathbb {R} }$ denotes the inner product of ${\displaystyle H}$.

This definition can intuitively be understood as follows: suppose ${\displaystyle {\bar {x}},x\in C}$ and ${\displaystyle v\in H}$, then ${\displaystyle x-{\bar {x}}}$ is the vector from ${\displaystyle {\bar {x}}}$ to ${\displaystyle x}$. If ${\displaystyle \langle x-{\bar {x}},v\rangle =0}$ then ${\displaystyle x-{\bar {x}}}$ would be orthogonal to ${\displaystyle v}$, while the condition ${\displaystyle \langle x-{\bar {x}},v\rangle \leq 0}$ additionally allows ${\displaystyle v}$ to "point away" from ${\displaystyle x-{\bar {x}}}$ more than 90°. So the (convex) normal cone at ${\displaystyle {\bar {x}}}$ is the set of all vectors ${\displaystyle v}$ that point at least 90° away from all the vectors from ${\displaystyle {\bar {x}}}$ to ${\displaystyle C}$; but translated to the origin.

• When ${\displaystyle C}$ is a subspace of ${\displaystyle H}$, then ${\displaystyle N_{C}^{\operatorname {conv} }({\bar {x}})=C^{\bot }}$ is the linear-algebraic normal space / orthogonal complement to ${\displaystyle C}$ for ${\displaystyle {\bar {x}}\in C}$. If ${\displaystyle C}$ is instead an affine subspace then ${\displaystyle N_{C}^{\operatorname {conv} }({\bar {x}})}$ is instead the orthogonal complement to the underlying parallel vector space of ${\displaystyle C}$.
• Let ${\displaystyle C\subseteq H}$ be a non-empty, closed, convex set and consider its convex indicator function ${\displaystyle \delta _{C}(x)={\begin{cases}0&x\in C\\+\infty &x\in H\setminus C\end{cases}}.}$ Then ${\displaystyle \partial \delta _{C}=N_{C}^{\operatorname {conv} }.}$
• For a proper, lower-semicontinuous, convex function ${\displaystyle f:H\to {\bar {\mathbb {R} }}}$ we have ${\displaystyle s\in \partial f({\bar {x}})\iff (s,-1)\in N_{\operatorname {epi} (f)}^{\mathrm {conv} }({\bar {x}},f({\bar {x}}))}$, where ${\displaystyle \operatorname {epi} (f)\subseteq H\times \mathbb {R} }$ denotes the epigraph of ${\displaystyle f}$ and ${\displaystyle \partial f}$ the convex subdifferential of ${\displaystyle f}$. This relationship serves to define further subdifferentials based on normal cones in the non-convex case.

• ${\displaystyle N_{C}^{\operatorname {conv} }({\bar {x}})}$ is a non-empty, convex cone for all ${\displaystyle {\bar {x}}\in C}$.
• If ${\displaystyle \dim(H)<\infty ,{\bar {x}}\in C}$, then ${\displaystyle {\bar {x}}\in \operatorname {int} (C)\iff N_{C}^{\operatorname {conv} }({\bar {x}})=\{0\}}$.^[3] The forward direction ${\displaystyle (\Rightarrow )}$ holds even if ${\displaystyle H}$ is infinite dimensional, while the backward direction ${\displaystyle (\Leftarrow )}$ may fail unless ${\displaystyle C}$ has nonempty interior.^[4]
• Let ${\displaystyle C\subseteq H}$ be non-empty, closed, convex. Then ${\displaystyle c=\operatorname {proj} _{C}(x^{*})\iff x^{*}-c\in N_{C}^{\operatorname {conv} }(c)}$, where ${\displaystyle \operatorname {proj} _{C}(x^{*})}$ is the metric projection of ${\displaystyle x^{*}}$ onto ${\displaystyle C}$. ^[3]
• Intersection Rule: ${\displaystyle N_{C_{1}\cap C_{2}}^{\operatorname {conv} }({\bar {x}})\supseteq N_{C_{1}}^{\operatorname {conv} }({\bar {x}})+N_{C_{2}}^{\operatorname {conv} }({\bar {x}})}$ for any two nonempty, convex subsets ${\displaystyle C_{1},C_{2}\subseteq X}$ of a tvs ${\displaystyle X}$ and ${\displaystyle {\bar {x}}\in C_{1}\cap C_{2}}$. Provided that the qualification condition ${\displaystyle \operatorname {int} (C_{1})\cap C_{2}\neq \emptyset }$ holds, the reverse inclusion is also true such that $${\displaystyle N_{C_{1}\cap C_{2}}^{\operatorname {conv} }({\bar {x}})=N_{C_{1}}^{\operatorname {conv} }({\bar {x}})+N_{C_{2}}^{\operatorname {conv} }({\bar {x}}).}$$The qualification condition may be further weakened, especially if ${\displaystyle X}$ is a nicer space. ^[4] For example if ${\displaystyle X}$ is Banach and the two convex sets are closed, it is sufficient that ${\displaystyle 0\in \operatorname {int} (C_{1}-C_{2})}$, while in finite dimensions one can show that $${\displaystyle N_{\bigcap _{i}C_{i}}^{\operatorname {conv} }({\bar {x}})=\sum _{i}N_{C_{i}}^{\operatorname {conv} }({\bar {x}})}$$for any finite family of convex sets ${\displaystyle (C_{i})_{i}}$ whose relative interiors intersect.
• For further calculus rules cf. the properties of the non-convex generalizations below.

Another very important property is what is sometimes called Fermat's rule: let ${\displaystyle f:H\to \mathbb {R} }$ be convex, ${\displaystyle C\subseteq H}$ nonempty, closed and convex. Then, assuming some very mild constraint qualifications (c.f. for example Proposition 27.8 in ^[3]), ${\displaystyle {\bar {x}}\in H}$ solves the constrained minimization problem ${\displaystyle \min _{x\in C}f(x)}$ if and only if ${\displaystyle 0\in \partial f({\bar {x}})+N_{C}^{\operatorname {conv} }({\bar {x}})}$. If ${\displaystyle f}$ is differentiable this reduces to ${\displaystyle -\nabla f({\bar {x}})\in N_{C}^{\operatorname {conv} }({\bar {x}})}$ (here ${\displaystyle \nabla f}$ is the Fréchet gradient of ${\displaystyle f}$; i.e. the pointwise Riesz representative of the Fréchet derivative of ${\displaystyle f}$).

More generally, one may define $${\displaystyle N_{C}^{\operatorname {conv} }({\bar {x}}):=\{\varphi \in X^{*}:\langle \varphi ,x-{\bar {x}}\rangle \leq 0~{\text{for all}}~x\in C\}=\{\varphi \in X^{*}:\sup _{x\in C}\langle \varphi ,x-{\bar {x}}\rangle \leq 0\}}$$for any topological vector space ${\displaystyle X}$, with topological dual ${\displaystyle X^{*}}$. In this case ${\displaystyle \langle \cdot ,\cdot \rangle :X^{*}\times X\to \mathbb {R} ,(\varphi ,x)\mapsto \varphi (x)}$ is the duality pairing of ${\displaystyle X}$ and ${\displaystyle X^{*}.}$ In this more general setting the normal cone can be recognized to be the set of all ${\displaystyle \varphi \in X^{*}}$ that attain their maximum on ${\displaystyle C}$ at ${\displaystyle {\bar {x}}}$. ^[1] This connects normal cones to another important class of objects, the so-called support function ${\displaystyle \sigma _{C}}$ of a set. One has ${\displaystyle \varphi \in N_{C}^{\operatorname {conv} }({\bar {x}})\iff \varphi ({\bar {x}})=\sigma _{C}(\varphi )}$.

In the remainder of this article we will assume the space we're working on to be Asplund for ease of exposition. While some of the following definitions and properties are fine / remain true for more general spaces (e.g. general normed spaces), this is not true for all of them. The class of Asplund spaces contains many important Banach spaces, for example all ${\displaystyle L^{p}}$-spaces for ${\displaystyle p\in (1,\infty )}$, but not the cases ${\displaystyle p\in \{1,\infty \}.}$

Let ${\displaystyle X}$ be an Asplund space with topological dual ${\displaystyle X^{*}}$ and ${\displaystyle A\subseteq X}$ a (potentially non-convex) nonempty subset of ${\displaystyle X}$. There are three main normal cones defined for this case:

1. The Fréchet normal cone ${\displaystyle N_{A}^{F}}$ is defined by $${\displaystyle N_{A}^{F}({\bar {x}}):=\{\varphi \in X^{*}:\limsup _{x\to {\bar {x}},x\neq {\bar {x}}}{\frac {\langle \varphi ,x-{\bar {x}}\rangle }{\lVert x-{\bar {x}}\rVert }}\leq 0\}}$$for any ${\displaystyle {\bar {x}}\in \operatorname {cl} A}$, and empty outside of that. Alternatively it may be defined as the polar of the Bouligand tangent cone ${\displaystyle T_{A}({\bar {x}})=\{x\in X:\exists (x^{k})\subseteq X,\exists (t_{k})\searrow 0:x^{k}\to x{\text{ and }}{\bar {x}}+t_{k}x^{k}\in A\}}$^[1]. Note that these definitions generally only coincide if ${\displaystyle A}$ is closed. Alternatively one may define ${\textstyle {\begin{alignedat}{2}N_{A}^{F}({\bar {x}})&:=\{\varphi \in X^{*}:\langle \varphi ,x-{\bar {x}}\rangle \leq o(\lVert x-{\bar {x}}\rVert )~{\text{as}}~x\to _{A}{\bar {x}}\}\end{alignedat}}}$where ${\displaystyle o(\lVert x-{\bar {x}}\rVert )~{\text{as}}~x\to _{A}{\bar {x}}}$ denotes a "small-o" Landau symbol; this in turn is equivalent to the first definition^[5].
2. The Mordukhovich (or limiting) normal cone ${\displaystyle N_{A}^{M}}$ is defined by $${\displaystyle N_{A}^{M}({\bar {x}}):=\operatorname {Lim\,sup} _{x\to _{A}{\bar {x}}}N_{A}^{F}(x)=\{\varphi \in X^{*}:\exists (x^{k})\subseteq X:\exists (\varphi ^{k})\subseteq X^{*}:x^{k}\to _{X}{\bar {x}},\varphi ^{k}\to \varphi ,\varphi ^{k}\in N_{A}^{F}(x^{k})\}.}$$ Here ${\displaystyle \operatorname {Lim\,sup} }$ is an outer limit of sets where X carries its usual norm topology while ${\displaystyle X^{*}}$ is endowed with its weak* topology and the notation ${\displaystyle x\to _{A}{\bar {x}}}$ means that "${\displaystyle x}$ converges to ${\displaystyle {\bar {x}}}$ along ${\displaystyle A}$", i.e. the sequence of points in the definition of the outer limit is a sequence of points in ${\displaystyle A}$.
3. The Clarke normal cone ${\displaystyle N_{A}^{C}}$ is defined by $${\displaystyle N_{A}^{C}({\bar {x}}):=\operatorname {cl} (\operatorname {conv} (N_{A}^{M}({\bar {x}}))),}$$where ${\displaystyle \operatorname {cl} }$ is the topological closure and ${\displaystyle \operatorname {conv} }$ the convex hull of a set.

These normal cones form a nested hierarchy ${\displaystyle N_{A}^{F}({\bar {x}})\subseteq N_{A}^{M}({\bar {x}})\subseteq N_{A}^{C}({\bar {x}})}$.

The Fréchet normal cone is also called the firm, regular and prenormal Normal cone in the literature and often times denoted by ${\displaystyle {\hat {N}}_{A}({\bar {x}}).}$ The Mordukhovich normal cone is also called the limiting or basic normal cone and sometimes denoted by ${\displaystyle N_{A}({\bar {x}})}$ or ${\displaystyle N_{A}^{L}({\bar {x}})}$. The Clarke normal cone is also called the convexified normal cone and sometimes denoted by ${\displaystyle {\bar {N}}_{A}({\bar {x}}).}$ ^[5][6][7]

Suppose, for simplicity, that we have some Fréchet differentiable function ${\displaystyle f:X\to \mathbb {R} }$ and consider the stationarity condition ${\displaystyle -\nabla f({\bar {x}})\in N_{A}({\bar {x}})}$ corresponding to each of the three normal cones. If ${\displaystyle N_{A}({\bar {x}})}$ is "too large" — in the extreme case we may even find that ${\displaystyle N_{A}({\bar {x}})=X}$ — we can't expect these conditions to actually tell us anything about the (non-)optimality of ${\displaystyle {\bar {x}}}$. Conversely, if ${\displaystyle N_{A}({\bar {x}})}$ is "too small" — in the extreme case we might find it to be empty or to only contain zero — we will find it difficult or even impossible to verify that ${\displaystyle -\nabla f({\bar {x}})\in N_{A}({\bar {x}})}$, or to computationally determine ${\displaystyle {\bar {x}}}$ such that this holds. Because of this (and differences in their associated calculi) all three cones (as well as further ones) have their place in the theory and applications.

• Let ${\displaystyle A=\{(x,y)\in \mathbb {R} ^{2}:xy=0,y\geq 0\}}$. Then, identifying ${\displaystyle (\mathbb {R} ^{2})^{*}\simeq \mathbb {R} ^{2}}$, we have ${\displaystyle N_{A}^{F}(0)=\{0\}\times (-\infty ,0]}$, ${\displaystyle N_{A}^{M}(0)=\{xy=0\}}$ and ${\displaystyle N_{A}^{C}(0)=\mathbb {R} ^{2}.}$
• Suppose ${\displaystyle X\subseteq \mathbb {R} ^{n}}$ is a smooth embedded submanifold. Let ${\displaystyle p\in X}$ and denote by ${\displaystyle N_{p}X\subseteq T_{p}\mathbb {R} ^{n}}$ the normal space of ${\displaystyle X}$ at ${\displaystyle p}$. Then ${\displaystyle N_{p}X\simeq _{\mathbf {Vect} }N_{X}^{F}(p)=N_{X}^{C}(p)=N_{X}^{M}(p)}$.^[5]

• The inclusions ${\displaystyle N_{A}^{F}({\bar {x}})\subseteq N_{A}^{M}({\bar {x}})\subseteq N_{A}^{C}({\bar {x}})}$ always hold, in general they are strict.
• ${\displaystyle N_{A}^{F}({\bar {x}})}$ may be trivial, while ${\displaystyle N_{A}^{M}({\bar {x}})}$ is always nontrivial provided that ${\displaystyle \dim(X)<\infty }$, ${\displaystyle A}$ is closed and ${\displaystyle {\bar {x}}\in \operatorname {bd} (A)}$.
• Since the Clarke normal cone is always convex it can often end up being "too large" in practice and may degenerate to ${\displaystyle N_{A}^{C}({\bar {x}})=X^{*}}$. In contrast to this, the Mordukhovich normal cone can be nonconvex which allows it to remain "small enough to be useful" in some cases where Clarke is too large.
• It holds that ${\displaystyle N_{A}^{C}({\bar {x}})=((N_{A}^{M}({\bar {x}}))_{\circ })^{\circ }}$ where ${\displaystyle S_{\circ }\subseteq X}$ is the prepolar of a set ${\displaystyle S\subseteq X^{*}}$ and ${\displaystyle T^{\circ }\subseteq X^{*}}$ is the polar of a set ${\displaystyle T\subseteq X}$ so that ${\displaystyle N_{A}^{C}({\bar {x}})}$ is the bipolar of ${\displaystyle N_{A}^{M}({\bar {x}})}$.

• Let ${\displaystyle C\subseteq X}$ be a convex set, then ${\displaystyle N_{C}^{\operatorname {conv} }=N_{C}^{F}=N_{C}^{M}=N_{C}^{C}}$.
• ${\displaystyle N_{A}^{M}}$ is stable, that is ${\displaystyle N_{A}^{M}({\bar {x}})=\operatorname {Lim\,sup} _{x\to _{A}{\bar {x}}}N_{A}^{M}(x).}$
• Let ${\displaystyle X}$ be Hilbert, ${\displaystyle C\subseteq X}$ non-empty and closed, and suppose ${\displaystyle c\in \operatorname {proj} _{C}(x^{*})}$. Then ${\displaystyle x^{*}-c\in N_{C}^{F}(c)}$. (Note that in contrast to the convex case this fails to be an equivalence in general. Moreover, if ${\displaystyle X}$ is infinite dimensional there may be no such ${\displaystyle c}$)
• Product Rule: ${\displaystyle N_{A_{1}\times A_{2}}^{F}({\bar {x}}_{1},{\bar {x}}_{2})=N_{A_{1}}^{F}({\bar {x}}_{1})\times N_{A_{2}}^{F}({\bar {x}}_{2})}$ for any two nonempty, closed subsets ${\displaystyle A_{1},A_{2}\subseteq X}$, ${\displaystyle {\bar {x}}_{1}\in A_{1},{\bar {x}}_{2}\in A_{2}}$ of a normed space ${\displaystyle X}$. The analogous statement holds for ${\displaystyle N_{A_{1}\times A_{2}}^{M}}$. ^[1]
• Intersection rule: Suppose ${\displaystyle (A_{i})_{i}}$ is a finite family of closed subsets in ${\displaystyle X}$ that is allied at ${\displaystyle {\bar {x}}\in \bigcap _{i}A_{i}}$. Then$${\displaystyle N_{\bigcap _{i}C_{i}}^{\operatorname {conv} }({\bar {x}})\subseteq \sum _{i}N_{C_{i}}^{\operatorname {conv} }({\bar {x}})}$$holds. Alliedness is a somewhat lengthy, technical condition that may be found in Penot's book. Alternatively one may assume the so-called Fuzzy qualification condition. ^[1]
• Chain Rule: Let ${\displaystyle F:\mathbb {R} ^{n}\to \mathbb {R} ^{m}}$ be a ${\displaystyle C^{1}}$-function, and ${\displaystyle C=\{x\in A:F(x)\in D\}}$ where ${\displaystyle A,D}$ are closed in ${\displaystyle \mathbb {R} ^{n},\mathbb {R} ^{m}}$ respectively. Then $${\displaystyle N_{C}^{M}({\bar {x}})=N_{A}^{M}({\bar {x}})+F'({\bar {x}})^{*}N_{D}^{M}(F({\bar {x}})),}$$ subject to some constraint qualifications and regularity conditions. ^[5] Here ${\displaystyle F'}$ is the Jacobian of ${\displaystyle F}$.