The overwhelming majority of modern applications in the natural sciences, engineering, and beyond require both statistical estimation to accurately quantify the behavior of unknown distributed parameters in complex systems as well as a means of making optimal decisions that are resilient to this uncertainty. In this minisymposium, we aim to connect researchers working in optimization of complex systems under uncertainty such as equilibrium problems, differential algebraic equations, and partial differential equations, with statisticians working in variational statistics, infinite-dimensional statistical estimation, and optimum experimental design.
14:00
Risk-Adapted Optimal Experimental Design
Drew P. Kouri | Sandia National Laboratories | United States
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Drew P. Kouri | Sandia National Laboratories | United States
Constructing accurate and conservative statistical models of critical system responses typically requires an enormous amount of experimental data that is often expensive and time consuming to obtain. In this talk, we formulate an optimal experimental design problem for a general class of regression models resulting from the risk quadrangle, such as superquantile regression. We study the large-sample statistics for this class and prove that the resulting sequence of estimators is consistent and asymptotically normal. In addition, we generalize the G- and I-optimality criteria to minimize the risk associated with large prediction variance. We demonstrate our results on an experimental design problem from nondestructive acoustic testing.
14:30
Risk aversion in dynamic systems
Alois Pichler | Chemnitz University of Technology | Germany
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Alois Pichler | Chemnitz University of Technology | Germany
The probabilistic extension of dynamic systems involves the expectation, which is risk neutral.
This talk extends dynamic system, which evolve in time, to the risk averse case. We derive new Hamilton-Jacobi-Bellman equations, which govern risk aversion. The theory of viscosity solutions extends to the new, nonlinear differential equations. Explicit solutions are available in noteworthy situations. They give rise to specific interpretations of risk aversion in dynamic systems, which we describe in more detail.
15:00
Robust Utility Maximization with Drift and Volatility Uncertainty
Kerem Ugurlu | Nazarbayev University | Kazakhstan
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Kerem Ugurlu | Nazarbayev University | Kazakhstan
We give explicit solutions for utility maximization of terminal wealth problem u(X_T ) in the presence of Knightian uncertainty in continuous time [0, T] in a complete market. We assume there is uncertainty on both drift and volatility of the underlying stocks, which induce nonequivalent measures on canonical space of continuous paths Ω. We take that the uncertainty set resides in compact sets that are time dependent. In this framework, we solve the robust optimization problem with logarithmic, power and exponential utility functions, explicitly.
15:30
An Approximation Scheme For Distributionally Robust PDE-Constrained Optimization
Johannes Milz | Technical University of Munich | Germany
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Johannes Milz | Technical University of Munich | Germany
We develop a sampling-free approximation scheme for distributionally robust optimization (DRO) with PDEs. We use moment constraints to define the ambiguity set. To obtain a tractable and an accurate lower-level problem, we approximate nonlinear functions using quadratic Taylor's expansions w.r.t. parameters resulting in an approximate DRO problem. Its objective function is given as the sum of the optimal value functions of a trust-region problem and a semi-definite program. We construct smoothing functions and show global convergence of a homotopy method. Numerical results are presented using the adjoint approach to compute derivatives.