Some Useful Bijections

In constrained optimization problems with very simple (constant) constraints it is sometimes useful to simply use a global optimization algorithm with an appropriate one-to-one transformation of the parameters. Suppose we want to optimize an objective function f(θ) where θA. If there exists a continuous mapping ϕ:A such that for all x,yA, ϕ(x)=ϕ(y) if and only if x=y, then it is equivalent to optimize f(ϕ(x)) over . If x¯ is the resulting optimum, then we can apply the inverse transformation to obtain θ¯=ϕ 1(x¯).

Below is a table of useful transformations of this type. Most of them are not very difficult to derive, but it seems useful to have a list of them in one place. A denotes the constraint set. They can be scaled as needed for other intervals.

Useful Bijections
[0,)e xln(x)
(,0]e xln(x)
[0,1]11+e xln(1θ1)
[1,1]21+e x1ln(2θ+11)

Note that the mapping to [0,1] is the Sigmoid Function.

Finally, a multidimensional transformation is useful when the parameters represent probabilities. Suppose θA where A={(θ 1,θ 2,,θ n)|0θ i1and iθ i=1}. Here, A is the standard n1 simplex. The corresponding mapping is ϕ: n1A where ϕ i(x)=e x i1+ j=1 n1e x j for 1in1 and ϕ n(x)=11+ j=1 n1e x j. This is the same mapping that arises in the multinomial logit and conditional logit regression models.