This talk will present recent progress related to the theory and applications of the multilevel optimization approach (MG/OPT). The intent of MG/OPT is to use calculations on coarser levels to accelerate the progress of the optimization on the finest level. Significant speedup by using MG/OPT comparing to other existing techniques has shown for a particular type of tessellation problems called centroidal Voronoi tessellations(CVTs) and a variety of other constrained optimization problems.