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= Welcome to MWiki =
 
= Welcome to MWiki =
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== Theorems of the month ==
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== Theorem of the month ==
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First fundamental theorem of exact differential and integral calculus for line integrals: The function <math>F(z)=\int\limits_{\gamma }{f(\zeta )dB\zeta }</math> where <math>\gamma: [d, x[C \rightarrow A \subseteq {}^{(\omega)}\mathbb{K}, C \subseteq \mathbb{R}, f: A \rightarrow {}^{(\omega)}\mathbb{K}, d \in [a, b[C</math>, and choosing <math>\curvearrowright B \gamma(x) = \gamma(\curvearrowright D x)</math> is exactly <math>B</math>-differentiable, and for all <math>x \in [a, b[C</math> and <math>z = \gamma(x)</math>
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Theorem: The intex method solves every solvable LP in <math>\mathcal{O}({\vartheta}^{3})</math>.
  
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<div style="text-align:center;"><math>F' \curvearrowright B(z) = f(z).</math></div>
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Proof and algorithm: First, we normalise and scale <math>{b}^{T}y - {d}^{T}x \le 0, Ax \le b</math> and <math>{A}^{T}y \ge d</math>. Let the ''height'' <math>h</math> have the initial value <math>{h}_{0} := |\text{min } \{{b}_{1}, ..., {b}_{m}, {-d}_{1}, ..., {-d}_{n}\}|/r</math> for the reduction factor <math>r \in \; ]0, 1[</math>. Let the
  
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LP min <math>\{h \in [0, {h}_{0}] : x \in {}^{\omega}\mathbb{R}_{\ge 0}^{n}, y \in {}^{\omega}\mathbb{R}_{\ge 0}^{m}, {b}^{T}y - {d}^{T}x \le h, Ax - b \le (h, ..., h)^{T} \in {}^{\omega}\mathbb{R}_{\ge 0}^{m}, d - {A}^{T}y \le (h, ..., h)^{T} \in {}^{\omega}\mathbb{R}_{\ge 0}^{n}\}</math> have for <math>\underline{v} := {v}^{T}</math> the feasible interior starting point <math>v := ({\underline{x}, \underline{y}, h)}^{T} \in {}^{\omega}\mathbb{R}_{\ge 0}^{m+n+1}</math>, e.g. <math>({\underline{0}, \underline{0}, {h}_{0})}^{T}</math>.
  
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Proof: <math>dB(F(z))=\int\limits_{t\in [d,x]C}{f(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}D(t)dDt}-\int\limits_{t\in [d,x[C}{f(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}D(t)dDt}=\int\limits_{x}{f(\gamma (t))\frac{\gamma (\curvearrowright Dt)-\gamma (t)}{\curvearrowright Dt-t}dDt}=f(\gamma (x)){{{\gamma }'}_{\curvearrowright }}D(x)dDx=\,f(\gamma (x))(\curvearrowright B\gamma (x)-\gamma (x))=f(z)dBz.\square</math>
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It identifies the mutually dual LPs <math>\{{d}^{T}x : d \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 d\}</math>.
  
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Second fundamental theorem of exact differential and integral calculus for line integrals: According to the conditions from above, we have with <math>\gamma: [a, b[C \rightarrow {}^{(\omega)}\mathbb{K}</math> that
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We successively interpolate all <math>{v}_{k}^{*} := (\text{max } {v}_{k} + \text{min } {v}_{k})/2</math> until all <math>|\Delta{v}_{k}|</math> are sufficiently small. In <math>\mathcal{O}(\omega\vartheta)</math>, we extrapolate then <math>v</math> via <math>{v}^{*}</math> into the boundary of the polytope. The <math>r</math>-fold of the distance exceeding <math>{v}^{*}</math> determines the new starting point <math>v</math>.
  
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If min<math>{}_{k} {h}_{k} t = 0</math> follows from <math>t :=</math> min<math>{}_{k} \Delta{h}_{k}</math>, we end. Then we start over until min <math>h = 0</math> or min <math>h > 0</math> is certain. Since <math>h</math> at least halves itself for each iteration step in <math>\mathcal{O}({\omega\vartheta}^{2})</math>, the strong duality theorem yields the result.<math>\square</math>
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<div style="text-align:center;"><math>F(\gamma (b))-F(\gamma (a))=\int\limits_{\gamma }{{{{{F}'}}_{\curvearrowright }}B(\zeta )dB\zeta }.</math></div>
 
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Proof: <math>F(\gamma (b))-F(\gamma (a))=\sum\limits_{t\in [a,b[C}{F(\curvearrowright B\,\gamma (t))}-F(\gamma (t))=\sum\limits_{t\in [a,b[C}{{{{{F}'}}_{\curvearrowright }}B(\gamma (t))(\curvearrowright B\,\gamma (t)-\gamma (t))}=\int\limits_{t\in [a,b[C}{{{{{F}'}}_{\curvearrowright }}B(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}D(t)dDt}=\int\limits_{\gamma }{{{{{F}'}}_{\curvearrowright }}B(\zeta )dB\zeta }.\square</math>
 
  
 
== Recommended readings ==
 
== Recommended readings ==

Revision as of 19:11, 8 December 2019

Welcome to MWiki

Theorem of the month

Theorem: The intex method solves every solvable LP in [math]\displaystyle{ \mathcal{O}({\vartheta}^{3}) }[/math].

Proof and algorithm: First, we normalise and scale [math]\displaystyle{ {b}^{T}y - {d}^{T}x \le 0, Ax \le b }[/math] and [math]\displaystyle{ {A}^{T}y \ge d }[/math]. Let the height [math]\displaystyle{ h }[/math] have the initial value [math]\displaystyle{ {h}_{0} := |\text{min } \{{b}_{1}, ..., {b}_{m}, {-d}_{1}, ..., {-d}_{n}\}|/r }[/math] for the reduction factor [math]\displaystyle{ r \in \; ]0, 1[ }[/math]. Let the

LP min [math]\displaystyle{ \{h \in [0, {h}_{0}] : x \in {}^{\omega}\mathbb{R}_{\ge 0}^{n}, y \in {}^{\omega}\mathbb{R}_{\ge 0}^{m}, {b}^{T}y - {d}^{T}x \le h, Ax - b \le (h, ..., h)^{T} \in {}^{\omega}\mathbb{R}_{\ge 0}^{m}, d - {A}^{T}y \le (h, ..., h)^{T} \in {}^{\omega}\mathbb{R}_{\ge 0}^{n}\} }[/math] have for [math]\displaystyle{ \underline{v} := {v}^{T} }[/math] the feasible interior starting point [math]\displaystyle{ v := ({\underline{x}, \underline{y}, h)}^{T} \in {}^{\omega}\mathbb{R}_{\ge 0}^{m+n+1} }[/math], e.g. [math]\displaystyle{ ({\underline{0}, \underline{0}, {h}_{0})}^{T} }[/math].

It identifies the mutually dual LPs [math]\displaystyle{ \{{d}^{T}x : d \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 d\} }[/math].

We successively interpolate all [math]\displaystyle{ {v}_{k}^{*} := (\text{max } {v}_{k} + \text{min } {v}_{k})/2 }[/math] until all [math]\displaystyle{ |\Delta{v}_{k}| }[/math] are sufficiently small. In [math]\displaystyle{ \mathcal{O}(\omega\vartheta) }[/math], we extrapolate then [math]\displaystyle{ v }[/math] via [math]\displaystyle{ {v}^{*} }[/math] into the boundary of the polytope. The [math]\displaystyle{ r }[/math]-fold of the distance exceeding [math]\displaystyle{ {v}^{*} }[/math] determines the new starting point [math]\displaystyle{ v }[/math].

If min[math]\displaystyle{ {}_{k} {h}_{k} t = 0 }[/math] follows from [math]\displaystyle{ t := }[/math] min[math]\displaystyle{ {}_{k} \Delta{h}_{k} }[/math], we end. Then we start over until min [math]\displaystyle{ h = 0 }[/math] or min [math]\displaystyle{ h \gt 0 }[/math] is certain. Since [math]\displaystyle{ h }[/math] at least halves itself for each iteration step in [math]\displaystyle{ \mathcal{O}({\omega\vartheta}^{2}) }[/math], the strong duality theorem yields the result.[math]\displaystyle{ \square }[/math]

Recommended readings

Relil - Religion und Lebensweg

Nonstandard Mathematics

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