Minimax and Triality Theory in Nonsmooth Variational Problems

Abstract

Dual extremum principles and minimax theory are investigated for nonsmooth variational problems. The critical points of the Lagrangian L(u, p*) for fully nonlinear (both geometrically and physically nonlinear) systems are clarified. We proved that the critical point of L(u, p*) is a saddle point if and only if the Gao-Strang’s 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 set of primal-dual problems. The critical point of the Lagrangian either minimizes or maximizes both primal and dual problems. Application in nonconvex finite deformation theory is illustrated and a pure complementary energy is proposed. It is shown that the dual Euler-Lagrange equation of nonlinear variational boundary value problem is an algebraic equation. An analytic solution is obtained for a non-convex, unilateral variational problem.Key Wordsnonconvex variational problemnonsmooth optimizationduality theorytrialityfinite deformation theorynonlinear complementarity problem.