Discrete and continuous methods of mathematical optimization are integrated and treated in this textbook. After an introduction, convex sets are discussed (with an application to necessary optimality conditions under inequality constraints), followed by a more detailed examination of the special case of polyhedra and its connection to linear programming. A comprehensive presentation of the simplex method concludes this section. Next, the convexity of functions (including some relaxations) is investigated and used for a thorough study of optimality criteria as well as Lagrange duality. Finally, there is a look ahead at general algorithms and a brief appendix on affine geometry. In the new edition, the arrangement and presentation of the material covered have been thoroughly revised in line with current BA programs in Mathematics.