PDE-Constrained Equilibrium Problems under Uncertainty: Existence, Optimality Conditions and Regularization
In this paper, we analyze PDE-constrained equilibrium problems under uncertainty. In detail, we discuss a class of risk-neutral generalized Nash equilibrium problems and a class of risk-averse Nash equilibrium problems. For both, the risk-neutral PDE-constrained optimization problems with point...
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|In this paper, we analyze PDE-constrained equilibrium problems under uncertainty.
In detail, we discuss a class of risk-neutral generalized Nash equilibrium problems and a class of risk-averse Nash equilibrium problems.
For both, the risk-neutral PDE-constrained optimization problems with pointwise state constraints and the risk-neutral generalized Nash equilibrium problems,
the existence of solutions and Nash equilibria, respectively, is proved and optimality conditions are derived.
The consideration of inequality conditions on the stochastic state leads in both cases to complications in the derivation of the Lagrange multipliers.
Only by higher regularity of the stochastic state we can resort to the existing optimality theory for convex optimization problems.
The low regularity of the Lagrange multiplier also poses a major challenge for the numerical solvability of these problems. We lay the foundation for a successful numerical treatment of risk-neutral Nash equilibrium problems using Moreau-Yosida regularization by showing that this regularization approach is consistent.
The Moreau-Yosida regularization yields a sequence of parameter-dependent Nash equilibrium problems and the boundary transition in the smoothing parameter shows that the stationary points of the regularized problem converge weakly against a generalized Nash equilibrium of the original problem. Thus, the theory suggests that a numerical method can be built on the Moreau-Yosida regularization.
Based on this, algorithms are proposed to show how to solve risk-neutral PDE-constrained optimization problems with pointwise state bounds and risk-neutral PDE-constrained generalized Nash equilibrium problems.
I n order to model risk preference in the class of risk-averse Nash equilibrium problems, we use coherent risk measures. Since coherent risk measures are generally not smooth, the resulting PDE-constrained Nash equilibrium problem is also not smooth.
Therefore, we smooth the coherent risk measures using an epi-regularization technique.
For both the original Nash equilibrium problem and the smoothed parameter-dependent Nash equilibrium problems, we show the existence of Nash equilibria, and derive optimality conditions.
We provide valuable results for making this smoothing approach suitable for the development of a numerical method by proving that both, a sequence of stationary points and a sequence of Nash equilibria of the epi-regularized problem, have a weakly convergent subsequence whose limit is a Nash equilibrium of the original problem.