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How do you enforce positivity constraints in non-linear optimization (e.g. a constraint $x > 0$)? I remember there being a good reason for why most models use non-negativity constraints.

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How about the KKT conditions? en.wikipedia.org/wiki/… –  matt Jan 18 '12 at 11:43
    
@matt:They handle non-negative and equality constraints. –  Jacob Jan 18 '12 at 12:03
    
:You could use "Log Barrier methods". I will post an answer with more detail. –  matt Jan 19 '12 at 10:12

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Consider the following: $$\begin{array}{rll} \min_x& f(x)&\\ \text{subject to}& g_i(x)\leq0 &\text{for each }i\\ & h_i(x)=0 &\text{for each }j \\ \end{array}$$

For $\alpha>0$ we define the log barrier penalty function, $P_\alpha$, to be:

$$ P_\alpha(x)=f(x)-\frac1\alpha\sum_i\log(-g(_i(x))+\alpha\sum_jh_j(x)^2 $$

where $x$ must be strictly feasible, i.e. $g_i(x)<0$ for each $i$, in order for the log term to be defined.

We seek to minimise $P_\alpha(x)$. The idea is that the boundary of the feasible region (i.e $g_i(x)=0$) acts a a barrier for $x$ close to $0$.

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