On Local and Global Minima in Structural Optimization

Abstract : This paper deals with convexity properties in structural optimization, and with the closely related question of local versus global optima. The problem we investigate is that of minimizing the structural weight subject to constraints on displacements, stresses and natural frequencies. It is assumed that the structure is described by a finite element model, and that the transverse sizes of the elements, e.g. thicknesses of membrane plates, are the design variables. This implies that both the objective function, i.e. the weight, and the structural stiffness matrix depend linearly on the design variables. The constraint functions, however, become nonlinear and they may in the general case give rise to a nonconvex feasible region in the design space. Then there is a risk that a local, but not global, minimum is attained when any of the various existing methods for numerically solving the problem is applied. This fact is illustrated by examples of nonconvex problems. However, there are some special cases where the feasible region always becomes convex, so that, due to the linearity of the objective function, each local optimum is in fact also a global one.