# 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 ).