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Topic: (Q) transformation of R^n simplex to R^(n-1)
Replies: 2   Last Post: Jul 15, 1996 8:17 AM

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Evgenii Borisovich Rudnyi

Posts: 8
Registered: 12/12/04
(Q) transformation of R^n simplex to R^(n-1)
Posted: Jul 12, 1996 3:44 AM
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I would like to have one-to-one transformation of the open simplex in R^n

x_1 + x_2 + ... + x_n = 1
x_i > 0

to R^(n-1).

This would help me to convert my constraint minimization problem to
the non-constraint one.

For example,

The open simplex in R^2 is a segment (0, 1). The logistic function

y = 1/(exp(-x) + 1) -inf < x < inf, 0 < y < 1

is a tranformation of R^1 (a line) to the open simplex (0, 1) (a
segment) and it is not difficult to write reverse trasformation.

x = log(y/(1 - y)) 0 < y < 1, -inf < x < inf

My problem is that I don't know how it can be generalized to more
dimensional cases.

Evgenii Rudnyi

---
Chemistry Department rudnyi@comp.chem.msu.su rudnyi@mch.chem.msu.su
Moscow State University http://www.chem.msu.su/people/rudnyi/welcome.html
119899 Moscow +(095)939 5452, fax+(095)932 8846, +(095)939 1205
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