As others have explained, breaking up debris with missiles is a bad idea. But blasting it with *lasers* looks like a way to go.
> … The laser broom is intended to be used at high enough power to penetrate through the atmosphere with enough remaining power to ablate material from the target.[6] The ablating material imparts a small thrust that lowers its orbital perigee into the upper atmosphere, thereby increasing drag so that its remaining orbital life is short.[7] … The power levels of lasers in this concept are well below the power levels in concepts for more rapidly effective anti-satellite weapons.
https://en.wikipedia.org/wiki/Laser_broom
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