Basic Issues in Lagrangian Optimization

Abstract

These lecture notes review the basic properties of Lagrange multipliers and constraints in problems of optimization from the perspective of how they influence the setting up of a mathematical model and the solution technique that may be chosen. Conventional problem formulations with equality and inequality constraints are discussed first, and Lagrangian optimality conditions are presented in a general form which accommodates range constraints on the variables without the need for introducing constraint functions for such constraints. Particular attention is paid to the distinction between convex and nonconvex problems and how convexity can be recognized and taken advantage of.

Extended problem statements are then developed in which penalty expressions can be utilized as an alternative to black-and-white constraints. Lagrangian characterizations of optimality for such problems closely resemble the ones for conventional problems and in the presence of convexity take a saddle point form which offers additional computational potential. Extended linear-quadratic programming is explained as a special case.