Distance of a point to a subset of a metric space is the smallest distance between the point and any point in the subset: $d_C(x) = \min_{y \in C} d(x, y)$. Projection of a vector on a closed subset of a normed vector space is the set of vectors in the subset closest to the vector: $P_C(x) = \arg\min_{y \in C} \| y - x \|$.
Nonnegative combination of vectors is a linear combination where the scalars are nonnegative: $\sum_{i=1}^k a^i v_i$ where $a^i \ge 0$. Convex combination of vectors is a normalized nonnegative combination: $\sum_{i=1}^k a^i v_i$ where $a^i \ge 0$ and $\sum_{i=1}^k a^i = 1$. Line segment $\overline{pq}$ between two vectors is the set of all their convex combinations: $\overline{pq} = \{\lambda p + (1-\lambda) q: \lambda \in [0,1]\}$. Convex subset in a vector space is a subset that includes the line segment between any two of its elements [@Bertsekas2003, 1.2.1]: $A = \cup_{p, q \in A} \overline{pq}$. Convexity is closely related to expectation of random vector: a random vector supported within a convex subset has its expectation within that subset. A subset is closed and convex if and only if it is the intersection of all closed half-spaces containing the subset. Extreme point of a convex subset is a vector not between any two other vectors on the subset. Extreme direction of a convex set is the direction of one of its extreme points.
Convex hull (凸包) $\text{conv}(C)$ of a subset in a vector space is the intersection of all convex sets containing a subset. Affine hull (仿射包) $\text{aff}(X)$ is the intersection of all affine subspaces containing a set. Affine dimension of a subset is the dimension of its affine hull. A set of vectors are affinely independent if when picking one as the origin, the rest are linearly independent. The number of affinely independent vectors in a set cannot be more than one plus the affine dimension of these vectors, if finite. Relative interior $\text{ri}(C)$ of a subset is the intersection of its interior and its affine hull.
A set is a cone, if it is scale invariant. A cone is pointed if it contains no line. A cone is solid if it has a nonempty interior. A proper cone (真锥) is a pointed, closed solid, convex cone. Generated cone $\text{cone}(C)$ is the cone generated by a set is the nonnegative span of it. A cone is finitely generated, if it is generated by a finite set of vectors. A vector is a direction of recession of a convex set, if arbitrary shifts of the set along this vector are within the set. The recession cone of a set $R_C$ is the cone generated by its recession directions. The lineality space of a set is the set of directions of recession whose oppose are also direction of recession: $L_C = R_C \cap -R_C$.
Generalized inequality $\preceq_K$ determined by a proper cone $K$ in a Cartesian power of the real numbers is the partial ordering such that one vector is less than or equal to another if and only if their difference lies in the cone: $x \preceq_K y$, $y \succeq_K x$ iff $y − x \in K$; $x \prec_K y$, $y \succ_K x$ iff $y − x \in \text{int}(K)$. Vector inequality is a special case of generalized inequality where the associated cone is the nonnegative cone.
The minimum element (最小元) of a multidimensional set w.r.t. a generalized inequality $\preceq_K$ is an element no greater than any other element in the set in such partial ordering: $S \subseteq \{x\} + K$. Similarly, the maximum element of a multidimensional set w.r.t. a generalized inequality $\succeq_K$ is an element no less than any other element in the set in such partial ordering: $S \subseteq \{x\} - K$.
A minimal element (极小元) of a multidimensional set w.r.t. a generalized inequality $\preceq_K$ is an element no less than any element in the set in such partial ordering: $(\{x\} - K) \cap S = \emptyset$. Similarly, a maximal element of a multidimensional set w.r.t. a generalized inequality $\succeq_K$ is an element no greater than any element in the set in such partial ordering: $(\{x\} + K) \cap S = \emptyset$.
The polar cone of a set is its shade, which is a cone: $C^∗ = \{ x | \langle x, y \rangle \le 0, \forall y \in C \}$. Dual cone of a set is the negative polar cone: $\hat{C} = -C^∗$. A set is polyhedral (多面的) if it is the intersection of finitely many closed half-spaces: ${x | A x \le b}$. A cone is polyhedral if it is the polar cone of some finite set.
The dual generalized inequality associated with a proper cone is the generalized inequality associated with its dual cone. This concept can be used for a dual characterization of the minimum/minimal elements. The minimum element of a multidimensional set w.r.t. a generalized inequality $\preceq_K$ is an element such that for all elements $\lambda$ in the interior of the dual cone $K^∗$, it is the unique minimizer for $\langle \lambda, \cdot \rangle$ in the set. A minimal element of a multidimensional set w.r.t. a generalized inequality $\succeq_K$ is an element that minimizes $\langle \lambda, \cdot \rangle$ over the set for an element $\lambda$ in the interior of the dual cone.
An extended real-valued function is a function whose range is the extended real line. The extended-value extension $\tilde{f}: \mathbb{R}^n → \mathbb{R} ∪ \{+\infty\}$ of a convex function $f : C \to \mathbb{R}$ is the function defined in its containing (Euclidean) space that takes value $+\infty$ outside its original domain.
The graph of a function is the collection of point-value pairs: $\{ (x, f(x)) | x \in \text{dom}f \}$. The epigraph (supergraph) of a function is the collection of points on and above its graph: $\text{epi}f = \{ (x, y) | x \in \text{dom}f, y \ge f(x) \}$.
Consistent definitions of convex functions:
A function is concave if its negative is a convex function. While concave set is simply any non-convex set, concave functions exactly mirror convex functions.
Level curve is the feasible set of an equality constraint: $\{x \mid h(x) = 0\}$. Level set or sub-level set is the feasible set of an inequality constraint: $\{x \mid f(x) \le 0\}$. Quasi-convex function is one whose sub-level sets (and thus its domain) are all convex. Quasi-concave function is one whose negative is quasi-convex. Unimodal function is one which is quasi-convex or quasi-concave. Quasilinear function is one whis is both quasi-convex and quasi-concave.
Log-concave/log-convex function is a positive function whose logarithm is concave/convex.
Strongly convex function is a twice-differentiable real-valued function on a closed convex domain of a Euclidean space, whose Hessian is lower bounded by a positive scalar matrix: $\exists m > 0$, $\forall x \in \mathcal{F}$, $\nabla^2 f(x) \succeq m I$. Every m-strongly convex function dominates the quadratic approximations using the m-scalar matrix as the second differential: $\forall x, y \in \mathcal{F}$, $f(y) \ge f(x) + \nabla f(x)^T (y-x) + \frac{m}{2} |y - x|^2$. Every strongly convex function is convex. Gradient dominated function of degree $p$, $p \in [1, 2]$, is a differentiable function on a closed convex domain of a Euclidean space, such that it has a global minimum and its value relative to the minimum is dominated by the norm of its gradient to the power of $p$ [@Polyak1963; @Nesterov2005]: $\exists \tau > 0$, $\forall x \in \mathcal{F}$, $f(x) - f_0 \le \tau |\nabla f(x)|^p$, where $f_0 = \min_{x \in \mathcal{F}} f(x)$. Gradient dominated functions can be nonconvex. Every $λ$-strongly convex function is $\frac{1}{2λ}$-gradient dominated of degree 2.
Given a proper cone $K$ in its domain space, a real-valued function is K-nondecreasing if it is dominated by values in its K-cone: $\forall x, y, x \preceq_K y : f(x) \le f(y)$. The function is K-increasing if it is strictly dominated by values in its K-cone other than the vertex: $\forall x \ne y, x \preceq_K y : f(x) < f(y)$. K-nonincreasing and K-decreasing are similarly defined. As a special case, a real-valued function on the space of symmetric matrices is matrix increasing (decreasing) if it is increasing (decreasing) with respect to the positive semidefinite cone.
Given a proper cone $K$ in its range space, a vector-valued function is K-convex if every line segment of the function is K-dominated by its linear interpolation: $\mathbf{f}(\boldsymbol\lambda \cdot \mathbf{x}) \preceq_K \boldsymbol\lambda \cdot \mathbf{f}(\mathbf{x})$. The function is strictly K-convex if the inequality is strict except on the endpoints. As a special case, a symmetric matrix valued function is matrix convex if it is convex with respect to the positive semidefinite cone.
Given the dual pair of vector spaces $X, Y$ w.r.t a bilinear form $\langle \cdot, \cdot \rangle$, functions $f: X \to \mathbb{R}$ and $f^\star: Y \to \mathbb{R}$ are conjugate if their sum is supported from below by the bilinear form pointwise in the dual space: $f(x) + f^\star(y) \ge \langle y,x \rangle$ (Young's inequality), and $\forall y \in Y, \exists x \in X: f(x) + f^\star(y) = \langle y,x \rangle$. Equivalently, $f^\star(y) = \sup_{x \in X} \{ \langle y,x \rangle - f(x) \}$, also known as Fenchel transformation [@Fenchel1949]. If a function is smooth, strictly convex, and increases at infinity faster than a linear function, its conjugate is equivalent to its Legendre transformation [@Legendre1789]: $f^\star(y) = \langle x^\star, \nabla f(x^\star) \rangle − f(x^\star)$, where $y = \nabla f(x^\star)$.
Common convex sets:
Common convex functions:
Common non-trivial log-concave and log-convex functions:
Common matrix monotone functions: (Real-valued functions over the symmetric matrix space.)
Common matrix convex/concave functions:
Set relations: $C \subseteq \text{cl}(C) \subseteq \text{conv}(C) \subseteq \text{cone}(C)$
Operations that preserves set convexity: [@Bertsekas2003, 1.2.1]
Properties of convex set: [@Bertsekas2003, 1.4.1]
If a set is closed with nonempty interior and has a supporting hyperplane at every point in its boundary, it is convex.
The dual cone of a proper cone is also proper.
Generated cone is convex.
Recession cone theorem [@Bertsekas2003, 1.5.1]: For a (nonempty) closed convex set,
Caratheodory's theorem [@Bertsekas2003, 1.3.1]: (minimal representation of generated cone and convex hull)
Projection theorem [@Bertsekas2003, 2.2.1]: For a (nonempty) closed convex set $C$,
Supporting hyperplane theorem: Any non-interior point of a (nonempty) convex set has some hyperplane that contains the set in one of its closed half-spaces. [@Bertsekas2003, 2.4.1]
Separating hyperplane theorem: Any two disjoint (nonempty) convex sets can be separated by some hyperplane. [@Bertsekas2003, 2.4.2]
Strict separation theorem [@Bertsekas2003, 2.4.3]: Two disjoint (nonempty) convex sets $C_1, C_2$ can be strictly separated by some hyperplane if any of the following holds:
Polar cone is closed and convex.
Polar cone of a set is also polar cone of the cone generated by the set: $C^∗ = \text{cone}(C)^∗$. [@Bertsekas2003, 3.1.1]
Polar cone theorem: Twice polar cone of a cone is the closure of its convex hull: $C^{ ** } = \text{cl}(\text{conv}(C))$. In particular, closed convex cones are self-dual to polarity: $C^{ ** } = C$.
Cone generated by a finite set is closed. (Nontrivial) The polyhedral cone and the cone generated by the same finite set of vectors are polar to each other: for a finite set $A$, $A^∗ = \text{cone}(A)^∗$, and (Farkas' Lemma) $A^{**} = \text{cone}(A)$. [@Bertsekas2003, 3.2.1]
Minkowski-Weyl Theorem: Any polyhedral cone can be represented as a finitely generated cone, and the converse is also true: $A^∗ = \text{cone}(B)$.
Minkowski-Weyl representation: Given finite sets $A, B$, a polyhedral set can be represented as the Minkowski sum of the convex hull of a finite set and a finitely generated cone; the converse is also true: $P = \text{conv}(A) + \text{cone}(B)$. [@Bertsekas2003, 3.2.2]
[@Bertsekas2003, 3.3.1] For a (nonempty) convex set, extreme points on a supporting hyperplane are also extreme points of the set. A closed convex set has at least one extreme point iff it does not have two opposing recession direction. For a bounded convex set, all extreme points form the minimal set that reproduce the set by convex hull: $C = \text{conv}(E)$, where $E$ is the set of extreme points of $C$.
For a polyhedral set with Minkowski-Weyl representation $P = \text{conv}(A) + \text{cone}(B)$, its extreme points are in $A$. [@Bertsekas2003, 3.3.2]
[@Bertsekas2003, 3.3.3]
Polyhedral proper separation theorem [@Bertsekas2003, 3.5.1]
Properties of generalized inequalities:
For a generalized inequality, minimum and maximum elements of a set are unique if they ever exist, while a set can have multiple minimal and maximal elements.
Non-decreasing in each argument is equivalent to K-nondecreasing where $K$ is the nonnegative cone.
First-order conditions for K-monotonicity: A differentiable function on a convex domain is K-nondecreasing iff its gradient is nonnegative in K-dual inequality: $\nabla f(x) \succeq_{K^\star} 0$; It is K-increasing if its gradient is positive in K-dual inequality: $\nabla f(x) \succ_{K^\star} 0$;
First-order conditions for K-convexity: A differentiable function on a convex domain is K-convex iff the function globally K-dominates its linear expansion: $f(y) \succeq_K f(x) + Df(x)(y−x)$, here $f: \mathbb{R}^n \to \mathbb{R}^m$ and $Df(x) \in \mathbb{R}^{m \times n}$. This proposition also holds when both sides are strict.
Composition theorem: A K-nondecreasing convex function of a K-convex function is convex. Note the monotonicity restriction is on the extended-value extension of the outer function.
A convex function can only be discontinuous on its relative boundary.
A function is affine iff it is simultaneously convex and concave.
Operations that preserves (vector-valued) function convexity (or concavity): [@Bertsekas2003, 1.2.4]
Characterization of differentiable convex functions:
Operations that preserve quasi-convexity:
A quasilinear function has convex level sets; the converse does not hold.
A quasi-convex function $f(x)$ can be represented by a family of convex functions $\phi_t(x)$, where the t-sublevel set of $f$ is the 0-sublevel set of $\phi_t$. Such representation is not unique.
Relations:
Properties:
Transformation properties of conjugation:
A function is closed if its epigraph is a closed set; or equivalently, if all its sub-level sets are closed. The conjugate function of any function is closed and convex. Almost every convex function can be expressed as the conjugate of some function.
An extended real-valued function is lower semi-continuous at a point if the function values in the point's neighborhood are either close to or greater than the function value at that point: $\liminf_{x \to x_0} f(x) \ge f(x_0)$. The conjugate of a function is always lower semi-continuous.
The convex closure of a function is the function whose epigraph is the closure of the convex hull of the original function. The conjugate of the conjugate (twice conjugate) of a function is its convex closure: $\text{epi}f^{\star\star} = \text{cl}(\text{conv}(\text{epi}f))$, note that $f^{\star\star} \le f$. Fenchel–Moreau theorem: If a function is convex and closed, its twice conjugate is itself: $f^{\star\star} = f$.