An Optimization Problem for Predicting the Maximal Effect of Degradation of Mechanical Structures

This paper deals with a nonlinear nonconvex optimization problem that models prediction of degradation in discrete or discretized mechanical structures. The mathematical difficulty lies in equality constraints of the form Σ(i=1)(m) 1/yi A(i) x=b, where A(i) are symmetric and positive semidefinite matrices, b is a vector, and x, y are the vectors of unknowns. The linear objective function to be maximized is (x, y) bar right arrow b(T)x.

In a first step we investigate the problem properties such as existence of solutions and the differentiability of related marginal functions. As a by-product, this gives insight in terms of a mechanical interpretation of the optimization problem. We derive an equivalent convex problem formulation and a convex dual problem, and for dyadic matrices A(i) a quadratic programming problem formulation is developed. A nontrivial numerical example is included, based on the latter formulation.