Conventional slope stability analysis uses various Limit Equilibrium (LE) methods to determine the minimum factor of safety and its associated critical failure mechanism. These methods frequently assume that collapse will follow a particular assumed geometry, which are effective for simple geotechnical problems, yet may encounter difficulties when considering complex problems. For such complex problems, one effective solution lies in the use of Limit Analysis (LA) based on the upper-bound of plasticity in conjunction with a discretization procedure known as Discontinuity Layout Optimization (DLO), which can be an effective means of determining a critical failure mechanism without the limitation of an assumed slip surface. This paper compares the use of LE (Spencer Method with  dynamic programming optimization) and LA for several well-known examples of complex slopes. It is shown LA generally provides slightly lower factors of safety than rigorous LE and manages to handle complexities more effectively without assuming the geometry of the slip surface. It is demonstrated that using the DLO with LA can be applied to various extreme value problems beyond classical slope problems.