Functions and MapsOptimization Models > Gauss’ Bet | Functions and Maps | Standard Forms | Nomenclature | Problem Classes | Complexity | History
DefinitionsFunctionsIn this course we define functions as objects which take an argument in ![]() to refer to a function with ‘‘input’’ space Example: The function ![]() gives the distance from the point DomainWe allow functions to take infinity values. The domain of a function Example: Define the logarithm function as the function Two functions can differ not by their formal expression, but but because they have different domains. Example: The functions ![]() have the same formal expression inside their respective domains. However, they are not the same functions, since their domain is different. MapsWe reserve the term map to refer to vector-valued functions. That is, maps are functions which return more than a single value. We use the notation ![]() to refer to a map with input space Example: a map. Graph and EpigraphConsider a function GraphThe graph of ![]() EpigraphThe epigraph, denoted ![]() Level and Sub-level SetsLevel and sub-level sets correspond to the notion of contour of a function.
Both are indexed on some scalar value Level setsA level set is simply the set of points that achieve exactly some value for the function ![]() Sub-level setsA related notion is that of sub-level set. This is now the set of points that achieve at most a certain value for ![]()
|