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= Welcome to MWiki =
 
= Welcome to MWiki =
== Theorems of the month ==
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== Theorem of the month ==
=== Definition ===
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The centre method solves every solvable LP in <math>\mathcal{O}(\omega{\vartheta}^{2})</math>.
  
Let <math>f_n^*(z) = f(\eta_nz)</math> <em>sisters</em> of the Taylor series <math>f(z) \in \mathcal{O}(D)</math> centred on 0 on the domain <math>D \subseteq {}^{\omega}\mathbb{C}</math> where <math>m, n \in {}^{\omega}\mathbb{N}^{*}</math> and <math>\eta_n^m := i^{2^{\lceil m/n \rceil}}</math>. Then let <math>\delta_n^*f = (f - f_n^*)/2</math> the <em>halved sister distances</em> of <math>f.</math> For <math>\mu_n^m := m!n!/(m + n)!</math>, <math>\mu</math> and <math>\eta</math> form an calculus, which can be resolved on the level of Taylor series and allows an easy and finite closed representation of integrals and derivatives.<math>\triangle</math>
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== Proof and algorithm ==
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Let <math>z := \grave{m} + n</math> and <math>d \in [0, 1]</math> the density of <math>A</math>. First, normalise and scale <math>{b}^{T}y - {c}^{T}x \le 0, Ax \le b</math> as well as <math>{A}^{T}y \ge c</math>. Let <math>P_r := \{(x, y)^T \in {}^{\omega}\mathbb{R}_{\ge 0}^{z} : {b}^{T}y - {c}^{T}x \le r \in [0, \check{r}], Ax - b \le \underline{r}_m, c - {A}^{T}y \le \underline{r}_n\}</math> have the radius <math>\check{r} := s|\min \; \{b_1, ..., b_m, -c_1, ..., -c_n\}|</math> and the scaling factor <math>s \in [1, 2]</math>. It follows <math>\underline{0}_{z} \in \partial P_{\check{r}}</math>. By the strong duality theorem, the LP min <math>\{ r \in [0, \check{r}] : (x, y)^T \in P_r\}</math> solves the LPs max <math>\{{c}^{T}x : c \in {}^{\omega}\mathbb{R}^{n}, x \in {P}_{\ge 0}\}</math> and min <math>\{{b}^{T}y : y \in {}^{\omega}\mathbb{R}_{\ge 0}^{m}, {A}^{T}y \ge c\}</math>.
  
=== Speedup theorem for integrals ===
 
  
The Taylor series (see below) <math>f(z) \in \mathcal{O}(D)</math> centred on 0 on <math>D \subseteq {}^{\omega}\mathbb{C}</math> gives for <math>\grave{m}, n \in {}^{\omega}\mathbb{N}^*</math><div style="text-align:center;"><math>\int\limits_0^z...\int\limits_0^{\zeta_2}{f(\zeta_1)\text{d}\zeta_1\;...\;\text{d}\zeta_n} = \widehat{n!} f(z\mu_n) z^n.\square</math></div>
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Its solution is the geometric centre <math>g</math> of the polytope <math>P_0</math>. For <math>p_k^* := (\text{min}\,p_k + \text{max}\,p_k)/2</math> and <math>k = 1, ..., \grave{z}</math> approximate <math>g</math> by <math>p_0 := (x_0, y_0, r_0)^T</math> until <math>||\Delta p||_1</math> is sufficiently small. The solution <math>t^o(x^o, y^o, r^o)^T</math> of the two-dimensional LP min <math>\{ r \in [0, \check{r}] : t \in {}^{\omega}\mathbb{R}_{&gt; 0}, t(x_0, y_0)^T \in P_r\}</math> approximates <math>g</math> better and achieves <math>r \le \check{r}/\sqrt{\grave{z}}</math>. Repeat this for <math>t^o(x^o, y^o)^T</math> until <math>g \in P_0</math> is computed in <math>\mathcal{O}({}_z\check{r} {}_e\check{r}dmn)</math> if it exists. Numbers of length <math>\mathcal{O}({\omega})</math> can only be processed in <math>\mathcal{O}(\vartheta)</math> as is generally known.
  
=== Speedup theorem for derivatives ===
 
  
For <math>\mathbb{B}_{\hat{\nu}}(0) \subset  D \subseteq {}^{\omega}\mathbb{C},</math> the Taylor series<div style="text-align:center;"><math>f(z):=f(0) + \sum\limits_{m=1}^{\omega }{\widehat{m!}\,{{f}^{(m)}}(0){z^m}},</math></div><math>b_{\varepsilon n} := \hat{\varepsilon}\,\acute{n}! = 2^j, j, n \in {}^{\omega}\mathbb{N}^{*}, \varepsilon \in ]0, r^n[, {{d}_{\varepsilon k n}}:={{\varepsilon}^{{\hat{n}}}}{e}^{\hat{n}k\tau i}</math> and <math>f</math>'s radius of convergence <math>r \in {}^{\nu}{\mathbb{R}}_{&gt;0}</math> imply<div style="text-align:center;"><math>{{f}^{(n)}}(0)=b_{\varepsilon n}\sum\limits_{k=1}^{n}{\delta_n^* f({{d}_{\varepsilon k n}})}.</math></div>
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Solving all two-dimensional LPs <math>\text{min}_k r_k</math> by bisection methods for <math>r_k \in {}^{\omega}\mathbb{R}_{\ge 0}</math> and <math>k = 1, ..., z</math> in <math>\mathcal{O}({\vartheta}^2)</math> each time determines <math>q \in {}^{\omega}\mathbb{R}^k</math> where <math>q_k := \Delta p_k \Delta r_k/r</math> and <math>r := \text{min}_k \Delta r_k</math>. Let simplified <math>|\Delta p_1| = … = |\Delta p_{z}|</math>. Here min <math>r_z</math> for <math>p^* := p + wq</math> and <math>w \in {}^{\omega}\mathbb{R}_{\ge 0}</math> would be also to solve. If <math>\text{min}_k \Delta r_k r = 0</math> follows, stop, otherwise repeat until min <math>r = 0</math> or min <math>r &gt; 0</math> is sure. If necessary, constraints are temporarily relaxed by the same small modulus.<math>\square</math>
 
 
==== Proof: ====
 
Taylor's theorem<ref name="Remmert">[[w:Reinhold Remmert|<span class="wikipedia">Remmert, Reinhold</span>]]: ''Funktionentheorie 1'' : 3rd, impr. Ed.; 1992; Springer; Berlin; ISBN 9783540552338, p. 165 f.</ref> and the properties of the roots of unity.<math>\square</math>
 
 
 
== Reference ==
 
<references />
 
 
 
== Recommended reading ==
 
  
 +
== Recommended readings ==
 
[https://en.calameo.com/books/003777977258f7b4aa332 Nonstandard Mathematics]
 
[https://en.calameo.com/books/003777977258f7b4aa332 Nonstandard Mathematics]
  
 
[[de:Hauptseite]]
 
[[de:Hauptseite]]

Revision as of 01:55, 1 January 2022

Welcome to MWiki

Theorem of the month

The centre method solves every solvable LP in [math]\displaystyle{ \mathcal{O}(\omega{\vartheta}^{2}) }[/math].

Proof and algorithm

Let [math]\displaystyle{ z := \grave{m} + n }[/math] and [math]\displaystyle{ d \in [0, 1] }[/math] the density of [math]\displaystyle{ A }[/math]. First, normalise and scale [math]\displaystyle{ {b}^{T}y - {c}^{T}x \le 0, Ax \le b }[/math] as well as [math]\displaystyle{ {A}^{T}y \ge c }[/math]. Let [math]\displaystyle{ P_r := \{(x, y)^T \in {}^{\omega}\mathbb{R}_{\ge 0}^{z} : {b}^{T}y - {c}^{T}x \le r \in [0, \check{r}], Ax - b \le \underline{r}_m, c - {A}^{T}y \le \underline{r}_n\} }[/math] have the radius [math]\displaystyle{ \check{r} := s|\min \; \{b_1, ..., b_m, -c_1, ..., -c_n\}| }[/math] and the scaling factor [math]\displaystyle{ s \in [1, 2] }[/math]. It follows [math]\displaystyle{ \underline{0}_{z} \in \partial P_{\check{r}} }[/math]. By the strong duality theorem, the LP min [math]\displaystyle{ \{ r \in [0, \check{r}] : (x, y)^T \in P_r\} }[/math] solves the LPs max [math]\displaystyle{ \{{c}^{T}x : c \in {}^{\omega}\mathbb{R}^{n}, x \in {P}_{\ge 0}\} }[/math] and min [math]\displaystyle{ \{{b}^{T}y : y \in {}^{\omega}\mathbb{R}_{\ge 0}^{m}, {A}^{T}y \ge c\} }[/math].


Its solution is the geometric centre [math]\displaystyle{ g }[/math] of the polytope [math]\displaystyle{ P_0 }[/math]. For [math]\displaystyle{ p_k^* := (\text{min}\,p_k + \text{max}\,p_k)/2 }[/math] and [math]\displaystyle{ k = 1, ..., \grave{z} }[/math] approximate [math]\displaystyle{ g }[/math] by [math]\displaystyle{ p_0 := (x_0, y_0, r_0)^T }[/math] until [math]\displaystyle{ ||\Delta p||_1 }[/math] is sufficiently small. The solution [math]\displaystyle{ t^o(x^o, y^o, r^o)^T }[/math] of the two-dimensional LP min [math]\displaystyle{ \{ r \in [0, \check{r}] : t \in {}^{\omega}\mathbb{R}_{> 0}, t(x_0, y_0)^T \in P_r\} }[/math] approximates [math]\displaystyle{ g }[/math] better and achieves [math]\displaystyle{ r \le \check{r}/\sqrt{\grave{z}} }[/math]. Repeat this for [math]\displaystyle{ t^o(x^o, y^o)^T }[/math] until [math]\displaystyle{ g \in P_0 }[/math] is computed in [math]\displaystyle{ \mathcal{O}({}_z\check{r} {}_e\check{r}dmn) }[/math] if it exists. Numbers of length [math]\displaystyle{ \mathcal{O}({\omega}) }[/math] can only be processed in [math]\displaystyle{ \mathcal{O}(\vartheta) }[/math] as is generally known.


Solving all two-dimensional LPs [math]\displaystyle{ \text{min}_k r_k }[/math] by bisection methods for [math]\displaystyle{ r_k \in {}^{\omega}\mathbb{R}_{\ge 0} }[/math] and [math]\displaystyle{ k = 1, ..., z }[/math] in [math]\displaystyle{ \mathcal{O}({\vartheta}^2) }[/math] each time determines [math]\displaystyle{ q \in {}^{\omega}\mathbb{R}^k }[/math] where [math]\displaystyle{ q_k := \Delta p_k \Delta r_k/r }[/math] and [math]\displaystyle{ r := \text{min}_k \Delta r_k }[/math]. Let simplified [math]\displaystyle{ |\Delta p_1| = … = |\Delta p_{z}| }[/math]. Here min [math]\displaystyle{ r_z }[/math] for [math]\displaystyle{ p^* := p + wq }[/math] and [math]\displaystyle{ w \in {}^{\omega}\mathbb{R}_{\ge 0} }[/math] would be also to solve. If [math]\displaystyle{ \text{min}_k \Delta r_k r = 0 }[/math] follows, stop, otherwise repeat until min [math]\displaystyle{ r = 0 }[/math] or min [math]\displaystyle{ r > 0 }[/math] is sure. If necessary, constraints are temporarily relaxed by the same small modulus.[math]\displaystyle{ \square }[/math]

Recommended readings

Nonstandard Mathematics