According to the English dictionaries, 
`to engage in`, as an intransitive verb, means to `begin an activity` (Merriam-Webster), which is sensible for "to engage in the discussion with someone on some topic". 

However, a search in `books.google.com` suggests many authors use the following expression `engage the discussion`. 

For example,
"To engage the discussion about the impact of such a pervasive epidemic as HIV/AIDS from a perspective ..." 
<https://books.google.com/books?isbn=1134656556>

I cannot find an appropriate definition of this usage (maybe except definition 7? see below), as most examples suggest the object should be a person, not a thing

So what is "to engage the discussion" ? Or is it a misuse of the verb `to engage`

(Merriam-Webster)

> 
> **engage**, transitive verb
> 
> 
> 1 : to offer (something, such as one's life or word) as backing to a cause or aim : to expose to risk for the attainment or support of some end
> 
> 
> 2 : to attract and hold by influence or power  
> 
>  : to interlock with : mesh; also : to cause (mechanical parts) to mesh
> 
> 
> 3 : to bind (someone, such as oneself) to do something; especially : to bind by a pledge to marry
> 
> 
> 4 a : to provide occupation for : involve  
> 
>  b : to arrange to obtain the use or services of : hire
> 
> 
> 5 a : to hold the attention of : engross  
> 
>  b : to induce to participate
> 
> 
> 6 a : to enter into contest or battle with  
> 
>  b : to bring together or interlock (weapons)
> 
> 
> 7 : to deal with especially at length
> 
> 
>