Can the Gross Value Added be negative? Can the Gross Value Added be negative when calculated using the production approach?
That won't really happen in reality. A sales promotion which runs losses, as in your example, will not mean that the gross value added is negative. The final selling price of the item is the gross value added and that will not typically be negative. So, if an item is sold for 5 dollars then the gross value added is 5 dollars only. No matter if someone in the value chain made losses. 
So, gross value added will never be negative unless and until the final selling price of something is negative.