Dual cone of a closed convex cone
$begingroup$
I am reading materials about convex optimizations. It says: If $Z$ is a closed convex cone, then $Z$ is the anti-dual of its anti-dual cone $Z_-$, i.e.,
$$
Z={zin mathbf{R}^{L+1}:z^Tzetaleq0,forallzetain Z_-},
$$
$$
Z_-={zetainmathbf{R}^{L+1}:zeta^Tzleq0,forall zin Z}.
$$
How can I prove this? Why is this property useful?
optimization convex-analysis
asked Dec 11 '18 at 8:10
EggplantEggplant
61
$endgroup$
migrated from stats.stackexchange.com Dec 13 '18 at 20:34
This question came from our site for people interested in statistics, machine learning, data analysis, data mining, and data visualization.
add a comment |
$begingroup$
I am reading materials about convex optimizations. It says: If $Z$ is a closed convex cone, then $Z$ is the anti-dual of its anti-dual cone $Z_-$, i.e.,
$$
Z={zin mathbf{R}^{L+1}:z^Tzetaleq0,forallzetain Z_-},
$$
$$
Z_-={zetainmathbf{R}^{L+1}:zeta^Tzleq0,forall zin Z}.
$$
How can I prove this? Why is this property useful?
optimization convex-analysis
asked Dec 11 '18 at 8:10
EggplantEggplant
61
$endgroup$
migrated from stats.stackexchange.com Dec 13 '18 at 20:34
This question came from our site for people interested in statistics, machine learning, data analysis, data mining, and data visualization.
add a comment |
$begingroup$
I am reading materials about convex optimizations. It says: If $Z$ is a closed convex cone, then $Z$ is the anti-dual of its anti-dual cone $Z_-$, i.e.,
$$
Z={zin mathbf{R}^{L+1}:z^Tzetaleq0,forallzetain Z_-},
$$
$$
Z_-={zetainmathbf{R}^{L+1}:zeta^Tzleq0,forall zin Z}.
$$
How can I prove this? Why is this property useful?
optimization convex-analysis
asked Dec 11 '18 at 8:10
EggplantEggplant
61
$endgroup$
I am reading materials about convex optimizations. It says: If $Z$ is a closed convex cone, then $Z$ is the anti-dual of its anti-dual cone $Z_-$, i.e.,
$$
Z={zin mathbf{R}^{L+1}:z^Tzetaleq0,forallzetain Z_-},
$$
$$
Z_-={zetainmathbf{R}^{L+1}:zeta^Tzleq0,forall zin Z}.
$$
How can I prove this? Why is this property useful?
optimization convex-analysis
optimization convex-analysis
asked Dec 11 '18 at 8:10
EggplantEggplant
61
asked Dec 11 '18 at 8:10
EggplantEggplant
61
asked Dec 11 '18 at 8:10
EggplantEggplant
61
asked Dec 11 '18 at 8:10
EggplantEggplant
61
asked Dec 11 '18 at 8:10
EggplantEggplant
61
61
migrated from stats.stackexchange.com Dec 13 '18 at 20:34
This question came from our site for people interested in statistics, machine learning, data analysis, data mining, and data visualization.
migrated from stats.stackexchange.com Dec 13 '18 at 20:34
This question came from our site for people interested in statistics, machine learning, data analysis, data mining, and data visualization.
add a comment |
add a comment |
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