Theorema bipolare

E Vicipaedia
Salire ad: navigationem, quaerere

Theorema bipolare in mathematica est theorema in explicatione convexa quod condiciones necessarias et satis adhibet ut conus aequet suum bipolare. Theorema bipolare videri potest proprius theorematis Fenchel-Moreauani casus.[1]

Pronuntiatum theorematis[recensere | fontem recensere]

In quaque copia non vacua, C \subset X in nonnullo spatio lineari X, tum conus bipolaris C^{oo} = (C^o)^o datur a

C^{oo} = \operatorname{cl}(\operatorname{co} \{\lambda c: \lambda \geq 0, c \in C\})

ubi \operatorname{co} corticem convexum denotat.[1][2]

Casus proprius[recensere | fontem recensere]

C \subset X est non vacuus conus convexus clausus si et solum si C^{++} = C^{oo} = C cum C^{++} = (C^+)^+, ubi (\cdot)^+ denotat conum dualem positivum.[2][3]

Generatim, si C sit conus convexus, tum conus bipolaris datur a

C^{oo} = \operatorname{cl} C.

Coniunctio cum theoremate Fenchel–Moreauano[recensere | fontem recensere]

Si f(x) = \delta(x|C) = \begin{cases}0 & \text{if } x \in C\\ +\infty & \text{else}\end{cases} sit functio propria coni C, tum coniugatum convexum f^*(x^*) = \delta(x^*|C^o) = \delta^*(x^*|C) = \sup_{x \in C} \langle x^*,x \rangle est functio firmamenti pro C, et f^{**}(x) = \delta(x|C^{oo}). Ergo, C = C^{oo} si et solum si f = f^{**}.[1][3]

Notae[recensere | fontem recensere]

  1. 1.0 1.1 1.2 Borwein, Jonathan; Lewis, Adrian (2006). Convex Analysis and Nonlinear Optimization: Theory and Examples (2 ed.). Springer. ISBN 9780387295701  .
  2. 2.0 2.1 Boyd, Stephen P.; Vandenberghe, Lieven (2004) (pdf). Convex Optimization. Cambridge University Press. pp. 51–53. ISBN 9780521833783  .
  3. 3.0 3.1 Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. pp. 121–125. ISBN 9780691015866  .


mathematica Haec stipula ad mathematicam spectat. Amplifica, si potes!