Thsi is a fascinating question to consider. Clearly, we are not given a straight answer, so we have to look at what we can infer about the Duke's character from what he says and what he tells us. I would first want to actually look at the last few lines of the poem to give us a massive clue about the Duke's character:
Nay, we'll go
Together down, sir. Notice Neptune, though,
Taming a seahorse, thought a rarity,
Which Claus of Innsbruck cast in bronze for me.
The importance of these lines comes in the way that they are juxtaposed to the viewing of the portrait. Clearly, the Duke is presented as something of an art collector, a gatherer of beauty. The way he calmly turns to refering to a sculpture after talking about how his former Duchess was done away with clearly indicates that, in a sense, to him it doesn't matter whether the duchess is alive or dead. He has "possessed" her through the picture he has of her, and in a sense prefers to have his duchess silent and pretty, a mere object to be admired, than a complicated human who can defy him and his indomitable will with her behaviour. If we read the Duke's monologue carefully, we can see numerable ways in which his former wife's behaviour displeased him. Far better for him, then, to have captured her as a picture, which can be objectified without complaint.