Search results
Results from the WOW.Com Content Network
Every real-valued affine function, that is, each function of the form () = +, is simultaneously convex and concave. Every norm is a convex function, by the triangle inequality and positive homogeneity .
If : is a continuous function and is open, then is closed if and only if it converges to along every sequence converging to a boundary point of . [ 2 ] A closed proper convex function f is the pointwise supremum of the collection of all affine functions h such that h ≤ f (called the affine minorants of f ).
This set is convex because is convex, the sublevel sets of convex functions are convex, affine sets are convex, and the intersection of convex sets is convex. [7]: chpt.2 Many optimization problems can be equivalently formulated in this standard form.
In convex geometry and vector algebra, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points in an affine space) where all coefficients are non-negative and sum to 1. [1]
Let X be an affine space over a field k, and V be its associated vector space. An affine transformation is a bijection f from X onto itself that is an affine map; this means that a linear map g from V to V is well defined by the equation () = (); here, as usual, the subtraction of two points denotes the free vector from the second point to the first one, and "well-defined" means that ...
An affine convex cone is the set resulting from applying an affine transformation to a convex cone. [10] A common example is translating a convex cone by a point p: p + C. Technically, such transformations can produce non-cones. For example, unless p = 0, p + C is not a linear cone. However, it is still called an affine convex cone.
A function is convex if and only if its epigraph, the region (in green) above its graph (in blue), is a convex set.. Let S be a vector space or an affine space over the real numbers, or, more generally, over some ordered field (this includes Euclidean spaces, which are affine spaces).
Uniqueness up to affine difference: = iff is an affine function. Convexity: (,) is convex in its first argument, but not necessarily in the second argument. If F is strictly convex, then (,) is strictly convex in its first argument.