Search results
Results from the WOW.Com Content Network
In particular, the practitioner can specify a convex criterion using the maxima of convex optimality-criteria and nonnegative combinations of optimality criteria (since these operations preserve convex functions).
A convex optimization problem is defined by two ingredients: [5] [6] The objective function, which is a real-valued convex function of n variables, :;; The feasible set, which is a convex subset.
The concept of strong convexity extends and parametrizes the notion of strict convexity. Intuitively, a strongly-convex function is a function that grows as fast as a quadratic function. [11] A strongly convex function is also strictly convex, but not vice versa.
then is called strictly convex. [1]Convex functions are related to convex sets. Specifically, the function is convex if and only if its epigraph. A function (in black) is convex if and only if its epigraph, which is the region above its graph (in green), is a convex set.
In mathematical optimization, the ellipsoid method is an iterative method for minimizing convex functions over convex sets.The ellipsoid method generates a sequence of ellipsoids whose volume uniformly decreases at every step, thus enclosing a minimizer of a convex function.
Convex series allow for the definition of special types of subsets that are well-behaved and useful with very good stability properties. If S {\displaystyle S} is a subset of a topological vector space X {\displaystyle X} then S {\displaystyle S} is said to be a:
Set-theoretic, algebraic and topological operations on set-valued functions (like union, composition, sum, convex hull, closure) usually preserve the type of continuity. But this should be taken with appropriate care since, for example, there exists a pair of lower hemicontinuous set-valued functions whose intersection is not lower hemicontinuous.
In convex analysis, a non-negative function f : R n → R + is logarithmically concave (or log-concave for short) if its domain is a convex set, and if it satisfies the inequality