I am solving an optimization problem and I need to formulate it as a convex optimization problem. Is there any way to write the constraint $$ 1 - e^{z} - \frac{e^{-r}}{1+r} \leq 0 $$ as a convex constraint?
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both $z$ and $r$ are optimizing variables? Or one is a constant? – the_candyman Mar 17 '15 at 14:36
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Yes, both z and r are optimization variables. – alex14204 Mar 17 '15 at 14:40
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Without any additional restrictions on $r$ it is impossible. Even if you said, say, $r>-1$, there is no guarantee. You could perhaps do a change of variables $\tilde{z}=e^z$ and $\tilde{r}=e^r/(1+r)$, but then it's quite likely your problem is not convex in these variables, either. – Michael Grant Mar 17 '15 at 18:30
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Thanks very much for your answers. Actually, variables $r$ and $z$ have the following restrictions: $r > 0$ and $z \leq 0$. Making a change of variables could work but, as you say, this affect the convexity of the objective function and other restrictions. – alex14204 Mar 17 '15 at 20:17
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Your model is simply not convex, then. – Michael Grant Mar 17 '15 at 23:04