Convex FunctionsConvex Optimization > Convex sets | Convex functions | Convex optimization problems | Algorithms | DCP
DefinitionDomain of a functionThe domain of a function ![]() Here are some examples:
Definition of convexityA function
![]() Note that the convexity of the domain is required. For example, the function ![]() is not convex, although is it linear (hence, convex) on its domain We say that a function is concave if Examples:
Alternate characterizations of convexityLet EpigraphA function ![]() is convex. Example: We can us this result to prove for example, that the largest eigenvalue function First-order conditionIf ![]() The geometric interpretation is that the graph of Restriction to a lineThe function Examples: Second-order conditionIf Examples: Operations that preserve convexityComposition with an affine functionThe composition with an affine function preserves convexity: if Point-wise maximumThe pointwise maximum of a family of convex functions is convex: if ![]() is convex. This is one of the most powerful ways to prove convexity. Examples:
![]() This function is convex, as the maximum of convex (in fact, linear) functions (indexed by the vector
![]() Here, each function (indexed by Nonnegative weighted sumThe nonnegative weighted sum of convex functions is convex. Example: Negative entropy function. Partial minimumIf Example:
Composition with monotone convex functionsThe composition with another function does not always preserve convexity.
However, if Indeed, the condition ![]() The condition above defines a convex set in the space of Example: proving convexity via monotonicity. More generally, if the functions For example, if |