In the following figure, the minimization problem on the left side of the equation is illustrated. One seeks to vary x such that the vertical distance between the convex and concave curves at x is as small as possible. The position of the vertical line in the figure is the (approximate) optimum.
The next figure illustrates the maximization problem on the right hand side of the above equation. Tangents are drawn to each of the two curves such that both tangents have the same slope p. The problem is to adjust p in such a way that the two tangents are as far away from each other as possible (more precisely, such that the points where they intersect the y-axis are as far from each other as possible). Imagine the two tangents as metal bars with vertical springs between them that push them apart and against the two parabolas that are fixed in place.
Fenchel's theorem states that the two problems have the same solution. The points having the minimum vertical separation are also the tangency points for the maximally separated parallel tangents.
^Borwein, Jonathan; Zhu, Qiji (2005). Techniques of Variational Analysis. Springer. pp. 135–137. ISBN978-1-4419-2026-3.
Bauschke, Heinz H.; Combettes, Patrick L. (2017). "Fenchel–Rockafellar Duality". Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer. pp. 247–262. doi:10.1007/978-3-319-48311-5_15. ISBN978-3-319-48310-8.
fenchel, duality, theorem, mathematics, result, theory, convex, functions, named, after, werner, fenchel, proper, convex, function, proper, concave, function, then, regularity, conditions, satisfied, infx, supp, displaystyle, where, convex, conjugate, also, re. In mathematics Fenchel s duality theorem is a result in the theory of convex functions named after Werner Fenchel Let ƒ be a proper convex function on Rn and let g be a proper concave function on Rn Then if regularity conditions are satisfied infx f x g x supp g p f p displaystyle inf x f x g x sup p g p f p where ƒ is the convex conjugate of ƒ also referred to as the Fenchel Legendre transform and g is the concave conjugate of g That is f x sup x x f x x Rn displaystyle f left x right sup left left left langle x x right rangle f left x right right x in mathbb R n right g x inf x x g x x Rn displaystyle g left x right inf left left left langle x x right rangle g left x right right x in mathbb R n right Contents 1 Mathematical theorem 2 One dimensional illustration 3 See also 4 ReferencesMathematical theorem editLet X and Y be Banach spaces f X R displaystyle f X to mathbb R cup infty nbsp and g Y R displaystyle g Y to mathbb R cup infty nbsp be convex functions and A X Y displaystyle A X to Y nbsp be a bounded linear map Then the Fenchel problems p infx X f x g Ax displaystyle p inf x in X f x g Ax nbsp d supy Y f A y g y displaystyle d sup y in Y f A y g y nbsp satisfy weak duality i e p d displaystyle p geq d nbsp Note that f g displaystyle f g nbsp are the convex conjugates of f g respectively and A displaystyle A nbsp is the adjoint operator The perturbation function for this dual problem is given by F x y f x g Ax y displaystyle F x y f x g Ax y nbsp Suppose that f g and A satisfy either f and g are lower semi continuous and 0 core dom g Adom f displaystyle 0 in operatorname core operatorname dom g A operatorname dom f nbsp where core displaystyle operatorname core nbsp is the algebraic interior and dom h displaystyle operatorname dom h nbsp where h is some function is the set z h z lt displaystyle z h z lt infty nbsp or Adom f cont g displaystyle A operatorname dom f cap operatorname cont g neq emptyset nbsp where cont displaystyle operatorname cont nbsp are the points where the function is continuous Then strong duality holds i e p d displaystyle p d nbsp If d R displaystyle d in mathbb R nbsp then supremum is attained 1 One dimensional illustration editIn the following figure the minimization problem on the left side of the equation is illustrated One seeks to vary x such that the vertical distance between the convex and concave curves at x is as small as possible The position of the vertical line in the figure is the approximate optimum nbsp The next figure illustrates the maximization problem on the right hand side of the above equation Tangents are drawn to each of the two curves such that both tangents have the same slope p The problem is to adjust p in such a way that the two tangents are as far away from each other as possible more precisely such that the points where they intersect the y axis are as far from each other as possible Imagine the two tangents as metal bars with vertical springs between them that push them apart and against the two parabolas that are fixed in place nbsp Fenchel s theorem states that the two problems have the same solution The points having the minimum vertical separation are also the tangency points for the maximally separated parallel tangents See also editLegendre transformation Convex conjugate Moreau s theorem Wolfe duality Werner FenchelReferences edit Borwein Jonathan Zhu Qiji 2005 Techniques of Variational Analysis Springer pp 135 137 ISBN 978 1 4419 2026 3 Bauschke Heinz H Combettes Patrick L 2017 Fenchel Rockafellar Duality Convex Analysis and Monotone Operator Theory in Hilbert Spaces Springer pp 247 262 doi 10 1007 978 3 319 48311 5 15 ISBN 978 3 319 48310 8 Rockafellar Ralph Tyrrell 1996 Convex Analysis Princeton University Press p 327 ISBN 0 691 01586 4 Retrieved from https en wikipedia org w index php title Fenchel 27s duality theorem amp oldid 995903306, wikipedia, wiki, book, books, library,