“Supporting functional”的意思、由来-开放百科全书

词条

Supporting functional

释义

Mathematical definition
Relation to support function
Relation to supporting hyperplane
References

In convex analysis and mathematical optimization, the supporting functional is a generalization of the supporting hyperplane of a set.

Mathematical definition

Let X be a locally convex topological space, and be a convex set, then the continuous linear functional is a supporting functional of C at the point if and for every .^[1]

Relation to support function

If (where is the dual space of ) is a support function of the set C, then if , it follows that defines a supporting functional of C at the point such that for any .

Relation to supporting hyperplane

If is a supporting functional of the convex set C at the point such that

then defines a supporting hyperplane to C at .^[2]

References

1. ^{{cite book|title=Foundations of mathematical optimization: convex analysis without linearity|page=323|first1=Diethard|last1=Pallaschke|first2=Stefan|last2=Rolewicz|publisher=Springer|year=1997|isbn=978-0-7923-4424-7}}
2. ^{{cite book |last1=Borwein |first1=Jonathan |authorlink1=Jonathan Borwein |last2=Lewis |first2=Adrian |title=Convex Analysis and Nonlinear Optimization: Theory and Examples| edition=2 |year=2006 |publisher=Springer |isbn=978-0-387-29570-1 |page = 240}}

3 : Functional analysis|Duality theories|Types of functions

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