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Convex optimization
In mathematics and optimization theory, a typical convex optimization problem is to minimize f(x), where f is a convex function, and x is point, of a vector space X, subject to
- g1(x) = 0,...,gm(x) = 0,
where the gi(x) are convex functions.
The Lagrange function for the problem (see Lagrange multipliers for the smooth function case) is
- L = l0f(x) + l1g1(x) + ... + lmgm(x).
See Kuhn-Tucker theorem . Then we have
- L(xopt,l0_opt,l1_opt,...,lm_opt) = minx L(l0_opt,l1_opt,...,lm_opt),
- l0_opt ≥ 0,l1_opt ≥ 0,...,lm_opt ≥ 0
- l1,optg1(xopt) = 0, ... , lm,optgm(xopt) = 0
If l0,opt ≠ 0, so 1)-3) enough to find xopt.
l0,opt ≠ 0, if there exists x, so that
- g1(x) < 0,...,gm(x) < 0.
Last updated: 10-18-2005 17:52:33
10-26-2009 08:16:03
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The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details