Lower eyelid bags are composed mainly of "herniated" fatty tissue. Similar to a hernia found in the abdomen, a surgical procedure is typically the best option to resolve the issue. However, a surgical procedure that removes the eyelid bag sound reasonable but commonly removing the eyelid bag will result in hollowing of the eye. In essence, you may be trading one problem for another.
Better techniques seek to restore youthful contours, and preserving the fat and repositioning the fat into the deep groove below the eyelid bag will simultaneously reduce the bag and fill in the groove, thereby providing a smooth contour from the eyelid to the cheek. This is illustrated in these before and after pictures, where a lower blepharoplasty with fat repositioning was performed by Dr. Miller.