In all the years I have been gardening, since 1952, I have not found anything that will do more then slow the growth a bit. The people I worked for had me spray many different "weed" killers with no success whatsoever.
I have learned to live with them, knowing that they often provide the bees with a pollen source when there are few other pollen sources around.

I don't have a real problem with them; they crop up here and there in some shady spots, and only rarely in the lawn. I enjoy them when they bloom. Then, I dig them up before they set seed being careful to get the white node just under the soil line.

Persistent digging might be your only control.

Mark your calendar for early next spring...

When the new violet leaves first emerge (whenever THAT is in your climate) you can attack this beast by going after it with an ESTER-based herbicide.
But you can only use ester (alcohol-based) herbicides... when it's still generally COOL outside, or you stand the risk of volitolization, especially in temps exceeding 80F or so.

Examples of ester herbicides could include 'Speed Zone', 'Q4'(sold at Lesco), and 'Quiksilver'.

Lesco /John Deere also sells a "marriage" product at their stores, called 'Momentum FX', which is sort of a mix between (water-base) amine and ester selective herbicides, so it's significantly safer to use in WARMER temps !
Homeowners could make a gallon of this last FOREVER.
Just don't let it FREEZE.

Violets are so difficult to control because of the VERY thick root mass it has...almost looks like a "knot", when you pull it up !
So even with the "good stuff"...you'll likely need repeat applications!

Once the violets are beginning to die....Don't forget to sow grass seed !

Many folks spray weed killer and walk away, only to have OTHER weeds slide in... to watch the funeral !
(:
(rim shot, please !)

Remember...the #1 defense against weeds in a lawn, is to have a THICK, HEALTHY, stand of turf, to fend them off in the 1st place !

Is the monetary expense of treating the violets worth it?
Is the time you spend spraying them worth that effort?
What real harm do they do that requires they be eradicated?