The course will cover linear, convex, and parametric optimization. In each of these areas, the role of duality will be emphasized as it informs the design of efficient algorithms and provides a rigorous basis for determining optimality. Various versions of the Simplex Method for linear programming will be presented. The dangers of degeneracy and ways to avoid it will be explained. Also, both the worst-case and average-case efficiency of the algorithms will be described. Finally, an efficient algorithm for parametrically solving multi-objective optimization problems will be presented, analyzed, and proposed as a new algorithm for sparse regression.