Duality, triality and complementary extremum principles in non-convex parametric variational problems with applications

This paper presents a reasonably complete duality theory and a nonlinear dual transformation method for solving the fully nonlinear, non-convex parametric variational problem inf{W(Λu - μ) - F(u)}, and associated nonlinear boundary value problems, where Λ is a nonlinear operator, W is either convex or concave functional of p = Λu, and μ is a given parameter. Detailed mathematical proofs are provided for the complementary extremum principles proposed recently in finite deformation theory. A method for obtaining truly dual variational principles (without a dual gap and involving the dual variable p* of Λu only) in n-dimensional problems is proposed. It is proved that for convex W(p), the critical point of the associated Lagrangian Lμ(u, p*) is a saddle point if and only if the so-called complementary gap function is positive. In this case, the system has only one dual problem. However, if this gap function is negative, the critical point of the Lagrangian is a so-called super-critical point, which is equivalent to the Auchmuty's anomalous critical point in geometrically linear systems. We discover that, in this case, the system may have more than one primal-dual set of problems. The critical point of the Lagrangian either minimizes or maximizes both primal and dual problems. An interesting triality theorem in non-convex systems is proved, which contains a minimax complementary principle and a pair of minimum and maximum complementary principles. Applications in finite deformation theory are illustrated. An open problem left by Hellinger and Reissner is solved completely and a pure complementary energy principle is constructed. It is proved that the dual Euler-Lagrange equation is an algebraic equation, and hence, a general analytic solution for non-convex variational-boundary value problems is obtained. The connection between nonlinear differential equations and algebraic geometry is revealed. Keywords:duality theory; triality; complementary energy; nonlinear variational problem; non-convex optimization; fully nonlinear system; finite deformation theory; partial differential equations; analytic solution; phase transitions; variational inequality