Optimization with Uncertain Parameters: A Risk Protection Approach Using Automatic Differentiation

In this essay we propose a risk protection approach for minimizing a nonlinear unconstrained problem with uncertain parameters. By minimizing the average of second order approximations of the original objective function at p sample values of the parameters, we avoid unsatisfactory results due to unfortunate specification of the unknown parameters. We demonstrate that the computational cost of the risk protection approach is acceptable: we give both a theoretical analysis and provide results of computational experiments. In order to obtain accurate derivatives and reduce the computational cost in minimization, the reverse mode of automatic differentiation in ADMAT 2.0 is adopted to calculate gradients and Hessians. We also use ADMAT 2.0 to investigate some interesting sensitivity applications in computational finance such as “the Greeks”, project and real option evaluation, and CVaR minimization.