Session: Infinite-dimensional and dynamic aspects in optimization under uncertainty
Chair: Caroline Geiersbach
Cluster: Nonsmooth Optimization
Talk 1: Scenario Approximation for Nonsmooth PDE-Constrained Optimization under Uncertainty
Speaker: Johannes Milz
Abstract: We study statistical guarantees for the scenario approximation method for PDE-constrained optimization with chance constraints. This sample-based technique replaces the original chance constraint with computationally tractable constraints derived from random samples. For example, when a chance constraint requires that a parameterized PDE state constraint be satisfied with high probability, the scenario approximation reformulates it into a standard PDE-constrained optimization problem, where the number of state constraints equals the number of samples. We derive estimates for the sample size needed to ensure, with high confidence, that feasible solutions to the original problem can be obtained through the scenario approximation. We then use these results to establish optimality guarantees for solutions to scenario-based PDE-constrained optimization problems. Our analysis is applicable to both linear and nonlinear PDEs with random inputs.
Talk 2: Risk-adjusted feedback control with PDE constraints
Speaker: Philipp Guth
Abstract: Effective control strategies that are directly applicable to different problem configurations, such as varying initial conditions, are highly desirable--especially in the presence of uncertainty. Unlike open-loop controllers, closed-loop (or feedback) controllers can be constructed independently of the initial condition; hence, this is one reason why they are favourable in the presence of uncertainty. This talk introduces a novel risk-adjusted feedback law specifically designed for risk-averse linear quadratic optimal control problems under uncertainty.
Talk 3: Pontryagin principle for deterministic control of random semilinear parabolic equations with almost sure state constraints
Speaker: Piero Visconti
Abstract: We study a class of optimal control problems governed by random semilinear parabolic equations with almost sure state constraints in the space of continuous functions. We obtain necessary conditions of optimality in the form of a maximum principle with two multipliers, one for the state constraint and one for the cost function, the multiplier for the state constraint takes values in a space of measures. We prove the nontriviality of the multipliers when the state constraint set has nonempty interior. Under a strong stability condition, the multiplier for the cost function can be suppressed.
