Worst-case compliance for independently constrained uncertain loads

We consider optimization problem formulations for finding the worst compliance of a structure subject to uncertain loading. Typically, there is a least some data available on the direction and magnitude of the loading, and we therefore consider a model in which the loading is the sum of a fixed, known part, and an uncertain variable part. The uncertain loads can vary independently of each other, limited only to reside inside an ellipsoidal region. This leads to a non-convex quadratic optimization problem which can be solved directly using local optimization solvers. As an alternative, we derive a convex optimization problem which yields a load satisfying optimality conditions of the quadratic problem and can be solved efficiently using gradient-based methods. An analytical formulation, where the loads are restricted further to vary synchronously, is also proposed and tested. Several numerical examples are provided to demonstrate the ideas and compare the three different formulations.