The only options are digging them out (granted, probably damaging other plant roots in the area in the process), cutting them down completely (repeatedly until they die out), and treating them with systemic herbicide (usually after a cut-back so you're treating stumps). When opting to trim, you must be vigilant to remove any foliage regrowth as soon as it appears, as the resulting photosynthesis will replenish root energy reserves that will fuel further regrowth. By denying the plants a chance to replenish that energy used each time it resprouts, eventually its "battery" will run out and it'll die out completely. This might take one or more years, though.

If using systemic herbicide, choose a formulation which gives instructions for stump/cut wood treatment unless you prefer to spray foliage. If treating foliage, cut the shrubs down first to force regrowth at a more manageable height so you don't have to spray over your head; plus, it will greatly reduce the amount of material you need to treat. Even with a systemic that is designed to kill roots, tough plants like these may take several treatments to finally succumb completely. The resources below provide more detail about what treatments to try, when, and which tends to be the most successful. (Any targeting Autumn Olive can be used for Russian Olive as well.)

• Invasive Shrub Control
• Autumn Olive - Penn State Extension
• Invader of the Month - Maryland Invasive Species Council
• Plant Invaders of Mid-Atlantic Natural Areas: Autumn Olive
  

As with any herbicide, use caution when applying and don't let the spray contact the foliage or bark of desirable plants. Some systemic ingredients, like glyphosate, are fairly inert once bound to the soil so are not going to be taken up by live roots in the area.

Miri