Duality for non-convex variational principles

Abstract

This paper treats the construction of dual variational principles for non-convex problems using Lagrangian methods. Besides the well-known saddle-point theory, there are dualities involving minimizing the Lagrangian in both variables and dualities where one first maximizes and then minimizes the Lagrangian in each variable. When the primal problem has certain structure, an associated canonical Lagrangian is defined which leads to the appropriate dual variational principle and to a Hamiltonian description. The properties of, and correspondence between, the critical points are analyzed. The last five sections are devoted to examples of these dual principles.