Complementarity systems and optimization problems in structural engineering

In the late 1960s complementarity systems have been recognized to be natural modes for some problems in structural mechanics. Complementarity systems are known to arise also as stationarity conditons in optimization problems. This paper, after a short survey of complementarity systems, their connections with variational inequalities and solution algorithms, deals with some recent nonconventional applications in structural engineering. Precisely, a fairly detailed description is provided of the optimum design problem of finding the minimum cost modifications of the supporting profile along a submarine pipeline. In this engineering optimization problem a complementary system is included among the constraints. Mathematically similar problems arise in the area of structural identification and optimization. All these problems are related to analysis problems (unilateral contact, elastoplasticity) which are amenable to complementarity systems or non-linear convex (often quadratic) programming. The present purpose is not to survey the related subjects but to discuss some peculiar aspects of optimization under complementarity constraints.