Minimal solution

Baseline ( english minimal solution ) is a mathematical term , both in approximation theory and in the optimization theory and in related areas of mathematics , such as the functional analysis , the numerical analysis or the calculus of variations , plays a significant role.

The term minimal solution can be found in mathematics - albeit understood in a different sense - also in number theory in connection with Pell's equation and in the theory of differential inequalities in the sense of a solution to certain initial value problems .

definition

The term is used in a broader and a narrower sense.

The term in a broader sense

An arbitrary set , a subset and a numerical function are given . Then there are the following definitions: ${\displaystyle X}$ ${\displaystyle C\subseteq X}$ ${\displaystyle f\colon X\to {\bar {\mathbb {R} }}=\mathbb {R} \cup \{-\infty ,\infty \}}$

• The minimum value of on ${\displaystyle f}$ ${\displaystyle C}$ is called the infimum , and this infimum is set in the case of this . ${\displaystyle \inf _{x\in C}{f(x)}}$ ${\displaystyle C=\emptyset }$ ${\displaystyle =+\infty }$
• The set of minimal solutions from on ${\displaystyle f}$ ${\displaystyle C}$ is understood to be the subset of those elements of which take on the minimum value of on , i.e. the subset . Each of these elements is called a minimal solution of on . ${\displaystyle C}$ ${\displaystyle f}$ ${\displaystyle C}$ ${\displaystyle M(f,C):=\{c\in C\mid f(c)=\inf _{x\in C}{f(x)}\}}$ ${\displaystyle f}$ ${\displaystyle C}$
• If a topological space is and there , then a local minimal solution of on is called if an (open) environment of in exists in such a way that a minimal solution of on is. This term is especially important for the case that is a metric or a normalized space . ${\displaystyle X}$ ${\displaystyle c_{0}\in C\subseteq X}$ ${\displaystyle c_{0}}$ ${\displaystyle f}$ ${\displaystyle C}$ ${\displaystyle U}$ ${\displaystyle c_{0}}$ ${\displaystyle X}$ ${\displaystyle c_{0}}$ ${\displaystyle f}$ ${\displaystyle U\cap C}$ ${\displaystyle X}$
• A maximum value of auf ${\displaystyle f}$ ${\displaystyle C}$ , a maximum solution of auf ${\displaystyle f}$ ${\displaystyle C}$ and a local maximum solution of auf are ${\displaystyle f}$ ${\displaystyle C}$ understood to mean the terms resulting from dualization when the order relation is reversed from to . ${\displaystyle \leq }$ ${\displaystyle {\bar {\mathbb {R} }}}$ ${\displaystyle \geq }$

The term in the narrower sense

Let there be a standardized space (over the body of real or the body of complex numbers ), which should be provided with a norm , as well as a fixed point in space and a subset . ${\displaystyle X}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \|\cdot \|\colon X\to {\mathbb {R} }_{0}^{+}}$ ${\displaystyle a\in X}$ ${\displaystyle V\subseteq X}$

• Here one considers, with regard to the resulting distance function , the associated function and applies the terms defined above in the broader sense. If a minimal solution is then available from on , you have - with regard to and ! - a point of shortest distance , that is, a point in space that assumes this distance infinite and thus fulfills the equation . ${\displaystyle d(x,y)=\|x-y\|\;(x,y\in X)}$ ${\displaystyle a}$ ${\displaystyle d(a,\cdot )\colon X\to {\mathbb {R} }_{0}^{+}\subset {\bar {\mathbb {R} }}}$ ${\displaystyle d(a,\cdot )}$ ${\displaystyle V}$ ${\displaystyle V}$ ${\displaystyle a}$ ${\displaystyle v_{0}\in V}$ ${\displaystyle \|v_{0}-a\|=\inf _{v\in V}{\|v-a\|}}$
• They call this - especially in approximation theory - a minimal solution for respect , (keeping in the context of the distance function assumed here as a given). ${\displaystyle v_{0}}$ ${\displaystyle a}$ ${\displaystyle V}$
• Instead of a minimal solution (in the narrow sense) is not unusual to is also called a best approximation (or best approximation) of respect ${\displaystyle a}$ ${\displaystyle V}$ or a Proximum to in ${\displaystyle a}$ ${\displaystyle V}$ or from a best approximation to / from in ${\displaystyle a}$ ${\displaystyle V}$ . In the theory of topological vector spaces , such a minimal solution (in the narrower sense) is sometimes referred to as a plumb point .
• The concept of the best approximation ( english best approximation ) can be found in the same sense in the general context of metric spaces . If such a point and a fixed point in space as well as a subset are given, then - as above! - a minimal solution from to is called the best approximation of with respect to (or similar). So this is an element that satisfies the equation . ${\displaystyle (X,d)}$ ${\displaystyle a\in X}$ ${\displaystyle V\subseteq X}$ ${\displaystyle d(a,\cdot )}$ ${\displaystyle V}$ ${\displaystyle a}$ ${\displaystyle V}$ ${\displaystyle v_{0}\in V}$ ${\displaystyle d(a,v_{0})=\inf _{v\in V}{d(a,v)}}$
• Some authors also mention the number as the minimum deviation from regarding (or similar). ${\displaystyle d(a,v_{0})}$ ${\displaystyle a}$ ${\displaystyle V}$

sentences

The following sentences are among the results that are often used in connection with questions about minimal solutions.

Minimal solutions in general topology and analysis

The following version of Weierstrass 's theorem of the minimum should be mentioned as a particularly important result :

Let a topological space be given and in it a non-empty compact or sequential compact subset as well as a sub- continuous function . ${\displaystyle X}$ ${\displaystyle K\subseteq X}$ ${\displaystyle f\colon K\to \mathbb {R} }$

Then also offers use of a minimal solution. ${\displaystyle f}$ ${\displaystyle K}$

Minimal solutions in convex optimization

First of all, the following simple sentence should be mentioned, which deals with the relationship between local and global minimal solutions:

A real vector space and a convex subset and a point in space are given . Furthermore, let us be a convex function that may have a local minimal solution. ${\displaystyle X}$ ${\displaystyle C\subseteq X}$ ${\displaystyle x_{0}\in X}$ ${\displaystyle f\colon C\to \mathbb {R} }$ ${\displaystyle x_{0}}$

Then has also quite a minimal solution and the associated minimum value . ${\displaystyle f}$ ${\displaystyle C}$ ${\displaystyle f(x_{0})}$

In addition, a number of other results. Last but not least, the following characterization theorem of convex optimization should be mentioned here:

A real vector space and a convex subset and a point in space are given . Let us also be a convex function. ${\displaystyle X}$ ${\displaystyle C\subseteq X}$ ${\displaystyle x_{0}\in X}$ ${\displaystyle f\colon C\to \mathbb {R} }$

Then there is a minimal solution of on if and only if the inequality is satisfied for all with regard to the right-hand Gâteaux differential . ${\displaystyle x_{0}}$ ${\displaystyle f}$ ${\displaystyle C}$ ${\displaystyle x\in C}$ ${\displaystyle \delta _{+}f(x_{0},x-x_{0})\geq 0}$

The consequence is:

If a convex region is given in Euclidean space and a point in space and a convex differentiable function are given in it , then a minimal solution of on is if and only if the total differential is the zero vector des . ${\displaystyle {\mathbb {R} }^{n}\;(n\in \mathbb {N} )}$ ${\displaystyle U\subseteq {\mathbb {R} }^{n}}$ ${\displaystyle x_{0}\in U}$ ${\displaystyle f\colon U\to \mathbb {R} }$ ${\displaystyle x_{0}}$ ${\displaystyle f}$ ${\displaystyle U}$ ${\displaystyle {\nabla f}(x_{0})}$ ${\displaystyle {\mathbb {R} }^{n}}$

The characterization theorem leads in real Prehilbert spaces (and especially in real Hilbert spaces !) Because of the rich geometric structure given there to a basic approximation theorem , which describes the conditions under which the best approximations are guaranteed there. This approximation theorem is to be formulated as follows:

Let be a real Prähilbert or Hilbert space (with as inner product ) and be given a convex subset and a point in space . ${\displaystyle X}$ ${\displaystyle \langle \cdot ,\cdot \rangle }$ ${\displaystyle C\subseteq X}$ ${\displaystyle x_{0}\in X}$

Under these conditions, a is the (uniquely determined!) Best approximation of with respect to if and only if the inequality is fulfilled for all . ${\displaystyle c_{0}\in C}$ ${\displaystyle x_{0}}$ ${\displaystyle C}$ ${\displaystyle c\in C}$ ${\displaystyle {\langle c_{0}-x_{0},c-c_{0}\rangle }\geq 0}$

With this approximation theorem one directly obtains the following projection theorem :

Let (as before) be a real Prähilbert or Hilbert space and be given a linear subspace and a point in space . ${\displaystyle X}$ ${\displaystyle U\subseteq X}$ ${\displaystyle x_{0}\in X}$

Under these circumstances, is exactly what it is the best approximation of respect , if for all the equation is satisfied. In other words, a is the best approximation of respect if and only if the difference vector to all is normal . ${\displaystyle u_{0}\in U}$ ${\displaystyle x_{0}}$ ${\displaystyle U}$ ${\displaystyle u\in U}$ ${\displaystyle {\langle u_{0}-x_{0},u\rangle }=0}$ ${\displaystyle u_{0}\in U}$ ${\displaystyle x_{0}}$ ${\displaystyle U}$ ${\displaystyle u_{0}-x_{0}}$ ${\displaystyle u\in U}$

Minimal solutions and reflexive Banach spaces

Last but not least, the following two sentences are significant here:

James's Theorem

This sentence goes back to the mathematician Robert Clarke James and says the following:

A Banach space is reflexive if and only if every continuous linear functional on the closed unit sphere has a minimal solution. ${\displaystyle X}$ ${\displaystyle {\overline {B_{X}}}}$

The Shiver-Mazur Theorem

This theorem, attributed to the two mathematicians Juliusz Schauder and Stanisław Mazur , can be represented as follows:

Is a reflexive Banach space and a location is not empty , closed, convex and bounded subset , so every continuous possesses convex function to a minimum solution. ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle V\subseteq X}$ ${\displaystyle f\colon V\to \mathbb {R} }$ ${\displaystyle V}$

Minimal solutions and stability issues

For the question of stability in the context of minimal solutions, there is a general stability theorem , which can be represented as follows:

Let a metric space be given and in it two sequences of non-empty subsets and functions . ${\displaystyle X}$ ${\displaystyle S,S_{1},S_{2},\ldots \subseteq X}$ ${\displaystyle f,f_{1},f_{2},\ldots \colon X\to \mathbb {R} }$

For each there is a minimal solution of on . ${\displaystyle n=1,2,\ldots }$ ${\displaystyle a_{n}}$ ${\displaystyle f_{n}}$ ${\displaystyle S_{n}}$

The following should apply here:

(i) Let them converge continuously to . ${\displaystyle f_{n}}$ ${\displaystyle f}$

(ii) lie as a subset in the upper limit as understood by Kuratowski . ${\displaystyle S}$ ${\displaystyle \varlimsup _{n\rightarrow \infty }{S_{n}}}$

Then every accumulation point of the sequence that lies in is a minimal solution of on . ${\displaystyle (a_{n})_{n=1,2,\ldots }}$ ${\displaystyle S}$ ${\displaystyle f}$ ${\displaystyle S}$

Minimal solutions (in the narrower sense) in linear approximation theory

Here we know an existence and uniqueness theorem , which can be summarized as follows:

Let be a strictly convex normalized space and be given a closed , locally compact and convex subset therein . Then for each point in space with respect to exactly one minimal solution - exactly a best approximation (or plumb)! - . This is especially true if in is a subspace of finite dimension . ${\displaystyle X}$ ${\displaystyle V\subseteq X}$ ${\displaystyle x_{0}\in X}$ ${\displaystyle V}$ ${\displaystyle v_{0}\in V}$ ${\displaystyle V}$ ${\displaystyle X}$

Closely related to this is Haar's uniqueness theorem (presented by the Hungarian mathematician Alfréd Haar in 1917) , which states the following:

Be a compact space and is this the (with the maximum norm provided!) Functional area of on steady ( real- or complex-valued ) functions. ${\displaystyle B}$ ${\displaystyle X=C(B)}$ ${\displaystyle B}$

Here is a subspace of finite dimension and meet the condition that every function not identical to the null function should have at most zeros in . ${\displaystyle V\subseteq X}$ ${\displaystyle n=1,2,\ldots }$ ${\displaystyle V}$ ${\displaystyle v\in V}$ ${\displaystyle n-1}$ ${\displaystyle B}$

Then there is exactly one minimal solution for each function . ${\displaystyle V}$ ${\displaystyle f\in X}$ ${\displaystyle v_{f}\in V}$

Another important theorem in linear approximation theory is Singer's Theorem (named after the mathematician Ivan Singer ) , which provides a characterization of the best approximations and states the following:

Let a real normalized space and the associated dual space of the real-valued continuous linear functionals be assumed , whereby its operator norm should also be denoted by, and a sub-vector space and a point in space are also given. ${\displaystyle (X,\|\cdot \|)}$ ${\displaystyle (X^{*},{\|\cdot \|})}$ ${\displaystyle {\|\cdot \|}}$ ${\displaystyle V\subseteq X}$ ${\displaystyle a\in X}$

Then:

A subspace point is a best approximation of with respect to if and only if there is one that satisfies the following three conditions: ${\displaystyle v_{0}\in V}$ ${\displaystyle a}$ ${\displaystyle V}$ ${\displaystyle u\in X^{*}}$

(1) . ${\displaystyle \|u\|=1}$

(2) for everyone . ${\displaystyle u(v)=0}$ ${\displaystyle v\in V}$

(3) . ${\displaystyle u(a-v_{0})=\|a-v_{0}\|}$

Explanations and Notes

• The above infima always exist because , provided with the usual total order , is a complete lattice . ${\displaystyle {\bar {\mathbb {R} }}}$ ${\displaystyle \leq }$
• For function sequences on metric spaces, the concept of continuous convergence is a tightening of the concept of point-wise convergence .
• By definition, a point belongs to the upper limit as understood by Kuratowski if there is a strictly monotonically growing sequence and a selection sequence with . ${\displaystyle y\in X}$ ${\displaystyle \varlimsup _{n\rightarrow \infty }{S_{n}}}$ ${\displaystyle \mathbb {N} }$ ${\displaystyle n_{1}<n_{2}<\cdots }$ ${\displaystyle y_{n_{k}}\in S_{n_{k}}\;(k=1,2,\ldots )}$ ${\displaystyle \lim _{n\to \infty }y_{n_{k}}=y\ (k\to \infty )}$
• The condition appearing in the uniqueness theorem of Haar is the so-called Haar's condition . A finite-dimensional feature subspace is sufficient in a functional area of this condition is referred to as hair shearing subspace ( English hair subspace ) or hair shear space referred to.
• Haar's uniqueness theorem is also called Kolmogoroff-Haar's theorem by some authors - because of the achievements of the Soviet mathematician Andrej Nikolajewitsch Kolmogoroff in approximation theory .
• For a finite - dimensional (!) Normed space and a closed subset of each point in space has respect a minimum solution in the strict sense, that is, in a best approximation. ${\displaystyle X}$ ${\displaystyle F\subseteq X}$ ${\displaystyle a\in X}$ ${\displaystyle F}$ ${\displaystyle F}$
• For a normalized space (and especially for a normalized feature space) and each fixed point in space is selected the associated function is to always a convex functional and in any case continuously. ${\displaystyle X}$ ${\displaystyle a\in X}$ ${\displaystyle d(a,\cdot )\colon X\to {\mathbb {R} }_{0}^{+}}$ ${\displaystyle d(a,y)=\|a-x\|\;(x\in X)}$
• If the -dimensional Euclidean space is given here and a closed and convex subset and a continuous function are given , then the set is sometimes called the minimum set . It is always closed and in the case that it is convex, it is a convex subset of Euclidean space. ${\displaystyle X={\mathbb {R} }^{n}\;(n\in \mathbb {N} )}$ ${\displaystyle n}$ ${\displaystyle C\subset X}$ ${\displaystyle f\colon C\to \mathbb {R} }$ ${\displaystyle M(f,C)\subseteq C}$ ${\displaystyle {\mathbb {R} }^{n}}$ ${\displaystyle f}$
• In addition to the sentences listed above, there are plenty of other noteworthy results. The approximation theorem for uniformly convex spaces , which is significant for the entire approximation theory, can be used as an important example . The fundamental theorem of the calculus of variations should also be mentioned.

literature

• Lothar Collatz : Functional Analysis and Numerical Mathematics . Unchanged reprint of the 1st edition from 1964 (=  The Basic Teachings of Mathematical Sciences in individual presentations with special consideration of the areas of application . Volume 120 ). 2nd Edition. Springer-Verlag , Berlin, Heidelberg, New York 1968, ISBN 3-540-04135-4 ( MR0165651 ).
• Lothar Collatz, Werner Krabs : Approximation Theory . Chebyshev approximation with applications (=  Teubner study books ). BG Teubner , Stuttgart 1973, ISBN 3-519-02041-6 ( MR0445153 ).
• Klaus Floret : Weakly Compact Sets . Lectures held at SUNY, Buffalo, in Spring 1978 (=  Lecture Notes in Mathematics . Volume 801 ). Springer-Verlag, Berlin 1980, ISBN 3-540-09991-3 ( MR0576235 ).
• Alfréd Haar: Minkowski Geometry and the Approach to Continuous Functions . In: Mathematical Annals . tape 78 , 1917, pp. 294-311 ( [1] ).
• Harro Heuser : Functional Analysis . Theory and application (=  mathematical guidelines ). 4th edition. BG Teubner, Wiesbaden 2006, ISBN 978-3-8351-0026-8 ( MR2380292 ).
• Rainer Hettich , Peter Zencke : Numerical methods of approximation and semi-infinite optimization (=  Teubner study books mathematics ). BG Teubner, Stuttgart 1982, ISBN 3-519-02063-7 ( MR0653476 ).
• Peter Kosmol : Optimization and Approximation (=  De Gruyter Studies ). 2nd Edition. Walter de Gruyter & Co. , Berlin 2010, ISBN 978-3-11-021814-5 ( MR2599674 ).
• Peter Kosmol, Dieter Müller-Wichards : Optimization in Function Spaces . With stability considerations in Orlicz spaces (=  De Gruyter Series in Nonlinear Analysis and Applications . Volume 13 ). Walter de Gruyter & Co., Berlin 2011, ISBN 978-3-11-025020-6 ( MR2760903 ).
• Gottfried Köthe : Topological linear spaces. I. (=  The basic teachings of the mathematical sciences in individual presentations with special consideration of the areas of application . Volume 107 ). 2nd improved edition. Springer Verlag , Berlin, Heidelberg, New York 1966 ( MR0194863 ).
• Jürg T. Marti : Convex Analysis (=  textbooks and monographs from the field of exact sciences, mathematical series . Volume 54 ). Birkhäuser Verlag , Basel, Stuttgart 1977, ISBN 3-7643-0839-7 ( MR0511737 ).
• Günter Meinardus : Approximation of functions and their numerical treatment (=  Springer Tracts in Natural Philosophy . Volume 4 ). Springer Verlag, Berlin, Göttingen, Heidelberg, New York 1964 ( MR0176272 ).
• Arnold Schönhage : Approximation Theory (=  de Gruyter textbook ). Walter de Gruyter & Co., Berlin, New York 1971 ( MR0277960 ).
• Ivan Singer: Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces . Translation of the original Romanian version "Cea mai bună aproximare în spații vectoriale normate prin elemente din subspații vectoriale". Translated by Radu Georgescu (=  The basic teachings of the mathematical sciences in individual representations with special consideration of the areas of application . Volume 171 ). Springer Verlag, Berlin, Heidelberg, New York 1970 ( MR0270044 ).
• A. Wayne Roberts , Dale E. Varberg : Convex Functions (  Pure and Applied Mathematics . Volume 57 ). Academic Press , New York, San Francisco, London 1973 ( MR0442824 ).
• Guido Walz [Red.]: Lexicon of Mathematics in six volumes . First volume. A to Eif. Spectrum Akademischer Verlag , Heidelberg, Berlin 2001, ISBN 3-8274-0303-0 ( MR1839735 ).

See also

• Proximum
• Fundamental theorem of the calculus of variations

Individual evidence

1. ↑ Peter Kosmol: Optimization and Approximation. 2010, p. II (foreword), p. 8 ff., P. 79 ff.
2. ^ Lothar Collatz: Functional Analysis and Numerical Mathematics. 1968, p. 320 ff.
3. ↑ Lothar Collatz, Werner Krabs: Approximation Theory. 1973, p. 12 ff., P. 38 ff.
4. ^ Günter Meinardus: Approximation of functions and their numerical treatment. 1964, p. 1 ff.
5. ↑ Peter Kosmol, Dieter Müller-Wichards: Optimization in Function Spaces. 2011, p. 1 ff., P. 385
6. ↑ Neither the number-theoretical aspect nor that of the theory of differential inequalities is discussed here. A description of the minimal solutions of Pell's equation can be found in the textbook “Introduction to Number Theory” by Peter Bundschuh (Springer 1988). The concept of the minimal solution of a differential inequality is briefly presented in the third volume of the Lexicon of Mathematics in six volumes (Spektrum Akademischer Verlag, Heidelberg & Berlin 2001, p. 425).
7. ↑ ^{a b} Kosmol, op.cit., P. 8
8. ↑ Meinardus, op.cit., P. 63
9. ↑ Kosmol, op.cit., Pp. 98 ff.
10. ↑ Jürg T. Marti: Convex Analysis. 1977, p. 31
11. ^ ^{A b} Guido Walz [Red.]: Lexicon of Mathematics. First volume. 2001, p. 202
12. ^ Arnold Schönhage: Approximation theory. 1971, p. 8 ff., P. 148 ff.
13. ↑ Harro Heuser: functional analysis. 2006, p. 29 ff., P. 572 ff.
14. ↑ ^{a b} Gottfried Köthe: Topological linear spaces. I. 1966, p. 346 ff.
15. ↑ Kosmol, op.cit., Pp. 68 ff.
16. ↑ Ivan Singer: Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces. 1970, p. 377 ff.
17. ↑ Kosmol, op.cit., P. 450
18. ^ A. Wayne Roberts, Dale E. Varberg: Convex Functions. 1973, pp. 122-128, p. 123
19. ↑ Kosmol, op.cit., P. 78
20. ↑ Kosmol, op.cit., P. 79
21. ↑ Kosmol, op. Cit., Pp. 100-101
22. ↑ Kosmol, op.cit., P. 102
23. ↑ Kosmol, op.cit., P. 388 ff.
24. ↑ Kosmol, op.cit., P. 391
25. ↑ Kosmol, op.cit., P. 390
26. ↑ Kosmol, op.cit., P. 71
27. ↑ Kosmol / Müller-Wichards, op.cit., P. 142
28. ↑ Collatz, op.cit., P. 323
29. ↑ Meinardus, op.cit., P. 1
30. ↑ ^{a b} Meinardus, op. Cit., Pp. 15-16
31. ↑ ^{a b} Rainer Hettich, Peter Zencke: Numerical methods of approximation and semi-infinite optimization. 1982, pp. 115-116
32. ↑ Kosmol, op.cit., P. 401
33. ↑ Kosmol / Müller-Wichards, op.cit., P. 109
34. ↑ Kosmol, op.cit., P. 71
35. ↑ Kosmol / Müller-Wichards, op.cit., P. 134
36. ↑ Kosmol, op.cit., P. 69
37. ↑ Kosmol / Müller-Wichards, op.cit., P. 131
38. ↑ Kosmol, op.cit., P. 298
39. ↑ Kosmol / Müller-Wichards, op.cit., P. 12
40. ↑ ^{a b} Marti, op. Cit., Pp. 58-59
41. ↑ Kosmol, op.cit., P. 68
42. ↑ See Hettich / Zencke, op.cit., P. 39! Hettich and Zencke provide the proof only for the case of the space on a compact of continuous real-valued functions. Obviously, however, the issue is more general. ${\displaystyle {\mathbb {R} }^{n}}$
43. ↑ See Marti, op.cit., P. 184! Marti does not mention the convexity condition for the function here . However, this is obviously what is meant. The facts presented here also generally apply in standardized spaces . ${\displaystyle f}$ ${\displaystyle X}$
44. ↑ Schönhage, op.cit., P. 15