I don't understand what you mean by a movable wedge but i am sure that we need the angle the wedge makes with the horizon and then you calculate the force and deceleration on the box moving towards the top and finally you will get the minimum velocity for it to get up the wedge?
perhaps i didn't understand the question please explain it more?
Because all i can think of now is when you push the box some of the kinetic energy will go to the movable wedge and it will start moving...especially since all the surfaces are smooth.
There is one more thing i can think of and that is conservation of energy ...and by that i mean that the kinetic energy and weight(it changes with height) when it is down the wedge should equal the one when it is on top and that means that it should be stationary on top and that means the (weight on top)-(weight on bottom)=kinetic and from there its easy to get the speed...
You can try this the kinetic energy of the wedge and the box at that height is equal to the kinetic energy of the box movement and with the coservation of momentum you get the movement speed of the wedge when the box hits it and then you can calculate the box's initial speed?
I think the last one is the solution and I hope I helped!