The closure (闭包) of a set \(\text{cl}(C)\) is the collection of its limit points. The projection of a vector on a closed set is the set of vectors in the set closest to the vector: \( P_C(x) = \arg\min_{y \in C} | y - x | \). The distance of a vector to a set is the smallest distance from it to points in the set: \( d_C(x) = \min_{y \in C} | y - x | \).
A set in a linear space is convex, if any line segment between two vectors in the set is also in the set. {Bertsekas2003, 1.2.1} Dual definition of closed convex set: A set is closed and convex, if it is the intersection of all closed half-spaces containing the set. An extreme point of a (nonempty) convex set is a vector not between any two other vectors on the set. An extreme direction of a (nonempty) convex set is the direction of one of its extreme points. Convex hull (凸包) \(\text{conv}(C)\) is the smallest (intersection of all) convex set containing a set. Convex combination is a nonnegative weighted average of vectors. In comparison, nonnegative combination is a linear combination with nonnegative coefficients. Convexity is closely related to expectation: If \( P(X \in C) = 1 \) and \( C \subseteq \mathbb{R}^n \) is convex, then \( \mathbf{E}X \in C \).
Affine set is a shifted subspace. Affine hull (仿射包) \( \text{aff}(X) \) is the smallest (intersection of all) affine set containing a set. The affine dimension of a set 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. The relative interior of a set \( \text{ri}(C) \) 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 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: \( L_C = R_C \cap -R_C \).
A generalized inequality is the partial ordering on \(\mathbb{R}^n\) associated with a proper cone \(K\). Vector inequality is a special case of generalized inequality where the associated cone is the nonnegative cone.
\[\begin{align} x \preceq_K y, y \succeq_K x &\Leftrightarrow y−x \in K \ x \prec_K y, y \succ_K x &\Leftrightarrow y−x \in \text{int}(K) \end{align}\]
An element of a multidimensional set is the minimum element (最小元) w.r.t. a generalized inequality \(\preceq_K\), if it is no greater than any other element in the set in such partial ordering: \(S \subseteq \{x\} + K\). Similarly, An element of a multidimensional set is the maximum element w.r.t. a generalized inequality \(\succeq_K\), if it is no less than any other element in the set in such partial ordering: \(S \subseteq \{x\} - K\).
An element of a multidimensional set is a minimal element (极小元) w.r.t. a generalized inequality \(\preceq_K\), if it is no less than any element in the set in such partial ordering: \((\{x\} - K) \cap S = \emptyset\). Similarly, An element of a multidimensional set is a maximal element w.r.t. a generalized inequality \(\succeq_K\), if it is 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. An element of a multidimensional set is the minimum element w.r.t. a generalized inequality \(\preceq_K\), if 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. An element of a multidimensional set is a *minimal element** w.r.t. a generalized inequality \(\succeq_K\), if it 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.
A level curve is the feasible set of an equality constraint. A level set is the feasible set of an inequality constraint. A sub-level set is the feasible set of a less-than-or-equal-to constraint. A function is quasi-convex if all its sub-level sets (and thus its domain) are convex. A function is quasi-concave if its negative is quasi-convex. A function is unimodal if it is quasi-convex or quasi-concave. A function is quasilinear if it is both quasi-convex and quasi-concave.
A positive function is log-concave (log-convex) if its logarithm is concave (convex).
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: \( \text{epi}f^{\star\star} = \text{cl}(\text{conv}(\text{epi}f)) \), noticing that \( f^{\star\star} \le f \). Fenchel–Moreau theorem: If a function is convex and closed, its twice conjugate is itself: \( f^{\star\star} = f \).