Behroozi, Mehdi2021-10-132021-10-132021-08https://hdl.handle.net/11299/224906University of Minnesota M.S. thesis. 2021. Major: Mathematics. Advisor: Gilad Lerman. 1 computer file (PDF); ix, 60 pages.This thesis studies the application of optimization methods in approximating geometric shapes. In particular, it considers the problem of finding maximum volume (axis-aligned) inscribed parallelotopes and boxes in a compact convex set, defined by a finite number of convex inequalities, and presents an optimization approach for solving them. Several optimization models are developed that can be easily generalized to find other inscribed geometric shapes such as triangles, rhombi, and tetrahedrons. To find the largest axis-aligned inscribed rectangles in the higher dimensions, an interior-point method algorithm is presented and analyzed. Finally, a parametrized optimization approach is developed to find the largest (axis-aligned) inscribed rectangles in the two-dimensional space.enApproximation AlgorithmsComputational GeometryGeometric OptimizationInner and Outer ApproximationMaximum Volume Inscribed BoxThesis or Dissertation