Bauer's maximum principle

The maximum principle of Bauer , also known as the H. Bauer's maximum principle ( English H. Bauer's maximum principle ), is a mathematical theorem that in the transition area between the sub-areas of analysis , the linear optimization and the calculus of variations is based. It comes from a scientific work by the German mathematician Heinz Bauer (1928–2002) from 1960 and is related to both Weierstraß's theorem of minimum and maximum and the fundamental theorem of the calculus of variations . Like these, the maximum principle deals with the fundamental question of the existence of extreme points of certain real-valued functionals and formulates conditions under which these assume their maximum . In addition, the Kerin-Milman theorem can also be understood as a consequence of Bauer's maximum principle.

Formulation of the sentence

Bauer's maximum principle can be stated as follows:

Given a hausdorff shear locally convex topological - vector space and is a non-empty convex compact subset . ${\ displaystyle \ mathbb {R}}$ ${\ displaystyle X}$ ${\ displaystyle K \ subset X}$

Then:

Every convex upper semi- continuous functional (and in particular every linear continuous functional ) assumes its maximum in one of the extreme points of . ${\ displaystyle \ phi \ colon K \ to \ mathbb {R}}$ ${\ displaystyle \ phi \ colon X \ to \ mathbb {R}}$ ${\ displaystyle K}$ ${\ displaystyle K}$

This means: for each such there is an (not necessarily clearly defined) extreme point with ${\ displaystyle \ phi}$ ${\ displaystyle e_ {0} \ in K}$

{\ displaystyle \ phi (e_ {0}) = \ sup _ {x \ in K} {\ phi (x)} = \ max _ {x \ in K} {\ phi (x)}}

.

Significance for linear optimization

Philippe Blanchard and Erwin Brüning comment on this in their Springer textbook Direct Methods of Calculus of Variations (1982):

The statement of the theorem is very important for the determination of the maximum, because it severely restricts the set of potential extreme points of the function. Especially in the case of convex polyhedra, as is often the case in concrete applications, one only needs to look for the extremes of the function in the finite set of extreme points of the polyhedron.

literature

Bernard Beauzamy: Introduction to Banach Spaces and their Geometry. Unchanged reprint of the 1st edition from 1964 (= North-Holland Mathematics Studies . Volume 68 ). North-Holland Publishing Company, Amsterdam [u. a.] 1982, ISBN 0-444-86416-4 ( MR0670943 ).
Heinz Bauer: Minimalization of functions and extreme points. II . In: Archives of Mathematics . tape 11 , 1960, pp. 200-205 ( MR0130390 ).
Philippe Blanchard , Erwin Brüning : Direct methods of the calculus of variations . A textbook. Springer Verlag , Vienna, New York 1982, ISBN 3-211-81692-5 ( MR0687073 ).
Philippe Blanchard, Erwin Brüning: Variational Methods in Mathematical Physics . A unified approach. Translated from the German by Gillian M. Hayes. (= Texts and Monographs in Physics ). Springer Verlag, Berlin 1992 ( MR1230382 ).
Gustave Choquet : Lectures on Analysis / Volume II: Representation Theory . Edited by J. Marsden, T. Lance and S. Gelbart (= Mathematics Lecture Note Series ). WA Benjamin, Inc., New York, Amsterdam 1969 ( MR0250012 ).
DA Edwards: On the representation of certain functionals by measures on the Choquet boundary . In: Université de Grenoble. Annales de l'Institut Fourier . tape 13 , 1963, pp. 111-121 ( MR0147900 ).
Albrecht Pietsch : History of Banach Spaces and Linear Operators . Birkhäuse, Boston, Basel, Berlin 2007, ISBN 0-8176-4367-2 ( MR2300779 ).

Individual evidence

^ ^A ^b Philippe Blanchard, Erwin Brüning: Direct methods of the calculus of variations: A textbook. 1982, p. 30 ff.
^ ^A ^b Philippe Blanchard, Erwin Brüning: Variational Methods in Mathematical Physics. 1992, p. 30 ff.
^ ^A ^b Gustave Choquet: Lectures on Analysis / Volume II. 1969, p. 102 ff.
^ ^A ^b Albrecht Pietsch: History of Banach Spaces and Linear Operators. 2007, p. 231
↑ Bernard Beauzamy: Introduction to Banach Spaces and Their Geometry. 1982, p. 125
↑ Beauzamy, op.cit., P. 123
^ Philippe Blanchard, Erwin Brüning: Direct methods of the calculus of variations: A textbook. 1982, pp. 30-31.

[PB-EB-a01-1] A ^b Philippe Blanchard, Erwin Brüning: Direct methods of the calculus of variations: A textbook. 1982, p. 30 ff.

[PB-EB-b01-2] A ^b Philippe Blanchard, Erwin Brüning: Variational Methods in Mathematical Physics. 1992, p. 30 ff.

[GC-001-3] A ^b Gustave Choquet: Lectures on Analysis / Volume II. 1969, p. 102 ff.

[AP-001-4] A ^b Albrecht Pietsch: History of Banach Spaces and Linear Operators. 2007, p. 231

[BB-001-5] Bernard Beauzamy: Introduction to Banach Spaces and Their Geometry. 1982, p. 125

[BB-002-6] Beauzamy, op.cit., P. 123

[PB-EB-a02-7] Philippe Blanchard, Erwin Brüning: Direct methods of the calculus of variations: A textbook. 1982, pp. 30-31.