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(Fundamental theorems)
(Intex method)
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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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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 ==
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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 the <em>height</em> <math>h</math> have the initial value <math>h_0 := s |\min \; \{b_1, ..., b_m, -d_1, ..., -d_n\}|</math> for the <em>elongation factor</em> <math>s \in \, ]1, 2]</math>.</br>
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The 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 - {c}^{T}x \le h, Ax - b \le (h, ..., h)^T \in {}^{\omega}\mathbb{R}_{\ge 0}^{m}, c - {A}^{T}y \le (h, ..., h)^T \in {}^{\omega}\mathbb{R}_{\ge 0}^{n}\}</math> has <math>k</math> constraints and the feasible starting point <math>(x_0, y_0, h_0/s)^{T} \in {}^{\omega}\mathbb{R}_{\ge 0}^{m+n+1}</math>, e.g. <math>(0, 0, h_0/s)^{T}</math>.</br>
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It identifies the mutually dual 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>.
  
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Let the point <math>p := (x, y, h)^T</math> approximate the subpolytope’s centre of gravity <math>P^*</math> as <math>p_k^* := (\min p_k + \max p_k)/2</math> until <math>{|| \Delta p ||}_{1}</math> is sufficiently small. Here <math>x</math> takes precedence over <math>y</math>. Then extrapolate <math>p</math> via <math>{p}^{*}</math> into <math>\partial P^*</math> as <math>u</math>. Put <math>p := p^* + (u - p^*)/s</math> to shun <math>\partial P^*</math>. Hereon approximate <math>p</math> more deeply again as centre of gravity. After optionally solving all LPs min<math>{}_{k} {h}_{k}</math> by bisection methods for <math>{h}_{k} \in {}^{\omega}\mathbb{R}_{\ge 0}</math> in <math>\mathcal{O}({\vartheta}^{2})</math> each time, <math>v \in {}^{\omega}\mathbb{R}^{k}</math> may be determined such that <math>v_k := \Delta{p}_{k} \Delta{h}_{k}/r</math> and <math>r :=</math> min<math>{}_{k} \Delta{h}_{k}</math>. Simplified let <math>|\Delta{p}_{1}| = ... = |\Delta{p}_{m+n}|</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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Here min <math>{h}_{m+n+1}</math> may be solved for <math>p^* := p + tv</math> where <math>t \in {}^{\omega}\mathbb{R}_{\ge 0}</math> and <math>{v}_{m+n+1} = 0</math>. If min<math>{}_{k} {h}_{k} r = 0</math> follows, end, otherwise start over until min <math>h = 0</math> or min <math>h &gt; 0</math> is certain. If necessary, relax the constraints temporarily by the same small modulus.</br>
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Since almost every iteration step in <math>\mathcal{O}({\omega\vartheta}^{2})</math> halves <math>h</math> at least, the strong duality theorem yields the result.<math>\square</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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<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 20:23, 4 January 2021

Welcome to MWiki

Theorem of the month

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

Proof and algorithm

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 the height [math]\displaystyle{ h }[/math] have the initial value [math]\displaystyle{ h_0 := s |\min \; \{b_1, ..., b_m, -d_1, ..., -d_n\}| }[/math] for the elongation factor [math]\displaystyle{ s \in \, ]1, 2] }[/math].
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 - {c}^{T}x \le h, Ax - b \le (h, ..., h)^T \in {}^{\omega}\mathbb{R}_{\ge 0}^{m}, c - {A}^{T}y \le (h, ..., h)^T \in {}^{\omega}\mathbb{R}_{\ge 0}^{n}\} }[/math] has [math]\displaystyle{ k }[/math] constraints and the feasible starting point [math]\displaystyle{ (x_0, y_0, h_0/s)^{T} \in {}^{\omega}\mathbb{R}_{\ge 0}^{m+n+1} }[/math], e.g. [math]\displaystyle{ (0, 0, h_0/s)^{T} }[/math].
It identifies the mutually dual 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].

Let the point [math]\displaystyle{ p := (x, y, h)^T }[/math] approximate the subpolytope’s centre of gravity [math]\displaystyle{ P^* }[/math] as [math]\displaystyle{ p_k^* := (\min p_k + \max p_k)/2 }[/math] until [math]\displaystyle{ {|| \Delta p ||}_{1} }[/math] is sufficiently small. Here [math]\displaystyle{ x }[/math] takes precedence over [math]\displaystyle{ y }[/math]. Then extrapolate [math]\displaystyle{ p }[/math] via [math]\displaystyle{ {p}^{*} }[/math] into [math]\displaystyle{ \partial P^* }[/math] as [math]\displaystyle{ u }[/math]. Put [math]\displaystyle{ p := p^* + (u - p^*)/s }[/math] to shun [math]\displaystyle{ \partial P^* }[/math]. Hereon approximate [math]\displaystyle{ p }[/math] more deeply again as centre of gravity. After optionally solving all LPs min[math]\displaystyle{ {}_{k} {h}_{k} }[/math] by bisection methods for [math]\displaystyle{ {h}_{k} \in {}^{\omega}\mathbb{R}_{\ge 0} }[/math] in [math]\displaystyle{ \mathcal{O}({\vartheta}^{2}) }[/math] each time, [math]\displaystyle{ v \in {}^{\omega}\mathbb{R}^{k} }[/math] may be determined such that [math]\displaystyle{ v_k := \Delta{p}_{k} \Delta{h}_{k}/r }[/math] and [math]\displaystyle{ r := }[/math] min[math]\displaystyle{ {}_{k} \Delta{h}_{k} }[/math]. Simplified let [math]\displaystyle{ |\Delta{p}_{1}| = ... = |\Delta{p}_{m+n}| }[/math].

Here min [math]\displaystyle{ {h}_{m+n+1} }[/math] may be solved for [math]\displaystyle{ p^* := p + tv }[/math] where [math]\displaystyle{ t \in {}^{\omega}\mathbb{R}_{\ge 0} }[/math] and [math]\displaystyle{ {v}_{m+n+1} = 0 }[/math]. If min[math]\displaystyle{ {}_{k} {h}_{k} r = 0 }[/math] follows, end, otherwise start over until min [math]\displaystyle{ h = 0 }[/math] or min [math]\displaystyle{ h > 0 }[/math] is certain. If necessary, relax the constraints temporarily by the same small modulus.
Since almost every iteration step in [math]\displaystyle{ \mathcal{O}({\omega\vartheta}^{2}) }[/math] halves [math]\displaystyle{ h }[/math] at least, the strong duality theorem yields the result.[math]\displaystyle{ \square }[/math]

Recommended readings

Nonstandard Mathematics

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