That was what I was looking for. Thanks.
But I encountered another issue.
I am using jQuery to trigger a click event in the inputText (when some radio button is selected) and I am using what you suggested to open the suggestion box.
The problem is that when clicking the radion button first (without first clicking the inputText) I am getting a JS error:
document.getElementById("popup:autocompleteUserInputSuggestion").component is undefined
Is there something else I have to do ?