# K-convex function

A K-convex function is a generalization of the notion of convexity of a function to real-vector-valued functions . For this, the strict order is weakened and it is worked with partial orders , the so-called generalized inequalities . ${\ displaystyle \ mathbb {R}}$ ${\ displaystyle \ mathbb {R} ^ {m}}$ ## definition

Given is a closed, pointed and convex cone with a non-empty interior and / or the partial order or strict partial order induced by this cone . Furthermore, let us be a convex subset of the . The function ${\ displaystyle K \ subset \ mathbb {R} ^ {m}}$ ${\ displaystyle \ preccurlyeq _ {K}}$ ${\ displaystyle \ prec _ {K}}$ ${\ displaystyle D}$ ${\ displaystyle \ mathbb {R} ^ {n}}$ ${\ displaystyle f \ colon D \ to \ mathbb {R} ^ {m}}$ is called K-convex on the set if and only if ${\ displaystyle D}$ ${\ displaystyle f (\ lambda x + (1- \ lambda) y) \ preccurlyeq _ {K} \ lambda f (x) + (1- \ lambda) f (y)}$ applies to everyone and everyone . The function is called strictly K-convex on the set if ${\ displaystyle \ lambda \ in [0,1]}$ ${\ displaystyle x, y \ in D}$ ${\ displaystyle f}$ ${\ displaystyle D}$ ${\ displaystyle f (\ lambda x + (1- \ lambda) y) \ prec _ {K} \ lambda f (x) + (1- \ lambda) f (y)}$ applies to everyone and everyone in . ${\ displaystyle \ lambda \ in (0,1)}$ ${\ displaystyle x \ neq y}$ ${\ displaystyle D}$ ## Examples and characteristics

• If one sets that the function is real-valued and chooses the set as the cone , then the K-convex functions are exactly the convex functions. This is because the order induced by the cone is the ordinary order on the real numbers.${\ displaystyle m = 1}$ ${\ displaystyle K = \ {x \ geq 0 \}}$ • On the other hand, if you choose the set as the cone , then the K-convex functions are exactly the concave functions, since the cone reverses the order on the real numbers.${\ displaystyle K = \ {x \ leq 0 \}}$ • The cone is the crowd
${\ displaystyle K = \ {x \ in \ mathbb {R} ^ {m} \, | \, x_ {i} \ geq 0 \, {\ text {for all}} \, i = 1, \ dots, n \}}$ , the induced general inequality is less than or equal to the component wise . The K-convex functions are then the functions whose components are all convex.
• Affine functions are always K-convex, regardless of the cone used. This follows directly from the linearity of the function and the reflexivity of the generalized inequality.
• The sub-level set of a K-convex function is a convex set.
• A function is K-convex if and only if its epigraph is a convex set. The epigraph is defined in this case by means of the generalized inequality and not with the conventional less or equal .

## Alternative characterizations

The K-convexity of a function can also be well described by means of the partial order induced by the dual cone . A function is then exactly (strictly) K-convex if for each different from the zero vector Vector with valid that is (strictly) is convex in the conventional sense. ${\ displaystyle K}$ ${\ displaystyle K ^ {*}}$ ${\ displaystyle v}$ ${\ displaystyle 0 \ preccurlyeq _ {K ^ {*}} v}$ ${\ displaystyle v ^ {T} f}$ ### For differentiable functions

If there is a differentiable function , then it is K-convex if and only if ${\ displaystyle f}$ ${\ displaystyle f (x) + Df (x) (yx) \ preccurlyeq _ {K} f (y)}$ for everyone . Here is the Jacobian matrix .${\ displaystyle x, y}$ ${\ displaystyle D}$ ## Concatenations of K-convex functions

The compositions of K-convex functions are again convex under certain circumstances.

• If K-convex and convex and if the extended function K-monotonically increasing , then is convex. In particular, the two cones that define the K-convexity and the K-monotony must match.${\ displaystyle g \ colon \ mathbb {R} ^ {n} \ to \ mathbb {R} ^ {m}}$ ${\ displaystyle h \ colon \ mathbb {R} ^ {m} \ to \ mathbb {R}}$ ${\ displaystyle {\ tilde {h}}}$ ${\ displaystyle h (g (x))}$ ## Matrix convex functions

If one considers mappings from into the space of the symmetrical real matrices , provided with the Loewner half-order , then the corresponding K-convex functions are also called matrix-convex functions. An equivalent characterization of matrix convexity is that the function is convex for all if and only if matrix is ​​convex. ${\ displaystyle \ mathbb {R} ^ {n}}$ ${\ displaystyle S ^ {n}}$ ${\ displaystyle \ succcurlyeq _ {S _ {+} ^ {n}}}$ ${\ displaystyle g (x) = z ^ {T} f (x) z}$ ${\ displaystyle z \ in \ mathbb {R} ^ {n}}$ ${\ displaystyle f (x)}$ For example, the function defined by is matrix-convex because convex is the norm because of convexity. ${\ displaystyle f \ colon \ mathbb {R} ^ {n \ times m} \ to S ^ {n}}$ ${\ displaystyle f (X) = X \ cdot X ^ {T}}$ ${\ displaystyle z ^ {T} XXz = \ Vert X ^ {T} z \ Vert _ {2} ^ {2}}$ ## use

K-convex functions are used, for example, in the formulation of conical programs or generalizations of the Lagrange duality .

## Generalizations

In some cases, mappings between two real vector spaces are also considered and only provided with an order cone , not with a generalized inequality. The requirement is attached to the figure ${\ displaystyle f \ colon V_ {1} \ mapsto V_ {2}}$ ${\ displaystyle V_ {2}}$ ${\ displaystyle K}$ ${\ displaystyle \ lambda f (x) + (1- \ lambda) f (y) -f (\ lambda x + (1- \ lambda) y) \ in K}$ for all and put out of the convex set . Then the map is called a convex map again . ${\ displaystyle \ lambda \ in [0,1]}$ ${\ displaystyle x, y}$ ${\ displaystyle M}$ ${\ displaystyle f}$ ## literature

• Stephen Boyd, Lieven Vandenberghe: Convex Optimization . Cambridge University Press, Cambridge, New York, Melbourne 2004, ISBN 978-0-521-83378-3 ( online ).