Optimization with Variable Sets of Constraints and an Application to Truss Design

We discuss the minimization of a continuous function on a subset of Rⁿ subject to a finite set of continuous constraints. At each point, a given set-valued map determines the subset of constraints considered at this point. Such problems arise e.g. in the design of engineering structures.After a brief discussion on the existence of solutions, the numerical treatment of the problem is considered. It is briefly motivated why standard approaches generally fail. A method is proposed approximating the original problem by a standard one depending on a parameter. It is proved that by choosing this parameter large enough, each solution to the approximating problem is a solution to the original one. In many applications, an upper bound for this parameter can be computed, thus yielding the equivalence of the original problem to a standard optimization problem.The proposed method is applied to the problem of optimally designing a loaded truss subject to local buckling conditions. To our knowledge this problem has not been solved before. A numerical example of reasonable size shows the proposed methodology to work well.