Nonconvex Structural Optimization Problems

Plastic optimization problems involving convex specific cost functions can be handled by several methods. The earliest technique was based on the uniform energy dissipation principle, a more general approach which could be termed the cost gradiend method was developed later and was extended to multiload, multicomponent systems and to preassigned strength distribution. Some used energy principles in deriving their methods while others used a direct variational method involving slack variables and Lagrangian multipliers. All the foregoing criteria were developed originally as sufficient conditions for various classes of convex cost functions. If the cost function is nonconvex, then the kinematic conditions given by either variational method or by the Prager-Shield approach can be shown to constitute only a necessary condition for a local minimum. In deriving optimal solutions for frames and shells associated with nonconvex cost functions, it has been found that these criteria can still be very useful because they often admit only one (unique) solution. A simple example is presented which demonstrates this method.