ABSTRACT
This chapter presents the method of Lagrange multipliers to solve constrained optimization problems with equality and inequality constraints, in areas of both mathematics and mathematical finance. It considers the following four cases of necessary and sufficient conditions for optimality: no constraints; only equality constraints; equality and inequality constraints; and only inequality constraints. The subject of concave programming deals with constrained optimization problems, in the sense of maximizing an objective function subject to constrained equality, and inequality constraints. For a linear objective function which is concave or convex, but not strictly concave or strictly convex, the concave programming that satisfies the Karush-Kuhn-Tucker (KKT) conditions will always satisfy the necessary and sufficient conditions for a maximum. The system of equations under the KKT conditions is generally not solved directly, except in a few special cases where a closed-form solution can be derived analytically.
