Translation does not work for promoted to front page nodes
I'm assuming this question is about D7 ...

The relative URL to display as the front page is specified within admin/config/system/site-information. If you want to be able to specify a value for it that is different for each of the site's languages, make sure you have made it a "multilingual" variable (it should say "This is a multilingual variable.") in the description of that variable on that page.

Refer to the community documentation about Variable translation for details about how to translate text and settings that are stored in Drupal as variables. Here is a quote from that page:


  The Variable translation module, part of the Internationalization (i18n) package, allows you to translate text and settings that are stored in Drupal as variables. These variables include text such as 'site name' and 'site slogan', as well as settings like 'Default front page' and 'Default 404 page'.


The Variable translation module has a dependency on the Variable module.

And these are the specific steps required to make a variable multilangual (mentioned on that community page):


Enable the Variable translation module included with the Internationalization package
Go to Administration > Configuration > Regional and language > Multilingual settings
Click on the Variables tab
Select the variables that will be multilingual
Click Save configuration button


Obviously, in this case in step 4 the variable to be selected is "Default front frontpage", within "Site information".

Finally, this is what remains to be done (another quote from that same page):


  Once you have the correct settings, they'll be marked with "This is a multilingual variable" when you go to the corresponding administration pages (Figure 2). You must switch the site language while in the administration pages to set the variables for each language. A language switcher link will appear at the top of each administrative page that has multilingual variables.


"That's all" what's needed to specify a language specific value for the frontpage of your site ...