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An approximation scheme for distributionally robust PDE-constrained optimization. (English) Zbl 1493.90120

Summary: We develop a sampling-free approximation scheme for distributionally robust PDE-constrained optimization problems, which are min-max control problems. We define the ambiguity set through moment and entropic constraints. We use second-order Taylor’s expansions of the reduced objective function w.r.t. uncertain parameters, allowing us to compute the expected value of the quadratic function explicitly. The objective function of the approximated min-max problem separates into a trust-region problem and a semidefinite program. We construct smoothing functions for the optimal value functions defined by these problems. We prove the existence of optimal solutions for the distributionally robust control problem, and the approximated and smoothed problems, and show that a worst-case distribution exists. For the numerical solution of the approximated problem, we develop a homotopy method that computes a sequence of stationary points of smoothed problems while decreasing smoothing parameters to zero. The adjoint approach is used to compute derivatives of the smoothing functions. Numerical results for two nonlinear optimization problems are presented.

MSC:

90C17 Robustness in mathematical programming
90C26 Nonconvex programming, global optimization
90C59 Approximation methods and heuristics in mathematical programming
65K05 Numerical mathematical programming methods
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