Thomas M. Surowiec

First-Order Optimality Conditions for Elliptic Mathematical Programs with Equilibrium Constraints via Variational Analysis

Mathematical programs in which the constraint set is partially defined by the solutions of an elliptic variational inequality, so-called “elliptic MPECs,” are formulated in reflexive Banach spaces. With the goal of deriving explicit first-order optimality conditions amenable to the development of numerical procedures, variational analytic concepts are both applied and further developed. The paper is split into two main parts. The first part concerns the derivation of conditions in which the (lower-level) state constraints are assumed to be polyhedric sets. This part is then completed by two examples, the latter of which involves pointwise bilateral bounds on the gradient of the state. The second part focuses on an important nonpolyhedric example, namely, when the lower-level state constraints are presented by pointwise bounds on the Euclidean norm of the gradient of the state. A formula for the second-order (Mosco) epiderivative of the indicator function for this convex set is derived. This result is then used to demonstrate the (Hadamard) directional differentiability of the solution mapping of the variational inequality, which then leads to the derivation of explicit strong stationarity conditions for this problem.

Several approaches for the derivation of stationarity conditions for elliptic MPECs with upper-level control constraints

The derivation of multiplier-based optimality conditions for elliptic mathematical programs with equilibrium constraints (MPEC) is essential for the characterization of solutions and development of numerical methods. Though much can be said for broad classes of elliptic MPECs in both polyhedric and non-polyhedric settings, the calculation becomes significantly more complicated when additional constraints are imposed on the control. In this paper we develop three derivation methods for constrained MPEC problems: via concepts from variational analysis, via penalization of the control constraints, and via penalization of the lower-level problem with the subsequent regularization of the resulting nonsmoothness. The developed methods and obtained results are then compared and contrasted.

Risk-Averse PDE-Constrained Optimization Using the Conditional Value-At-Risk

Uncertainty is inevitable when solving science and engineering application problems. In the face of uncertainty, it is essential to determine robust and risk-averse solutions. In this work, we consider a class of PDE-constrained optimization problems in which the PDE coefficients and inputs may be uncertain. We introduce two approximations for minimizing the conditional value-at-risk (CVaR) for such PDE-constrained optimization problems. These approximations are based on the primal and dual formulations of CVaR. For the primal problem, we introduce a smooth approximation of CVaR in order to utilize derivative-based optimization algorithms and to take advantage of the convergence properties of quadrature-based discretizations. For this smoothed CVaR, we prove differentiability as well as consistency of our approximation. For the dual problem, we regularize the inner maximization problem, rigorously derive optimality conditions, and demonstrate the consistency of our approximation. Furthermore, we propose a...

On regular coderivatives in parametric equilibria with non-unique multipliers

This paper deals with the computation of regular coderivatives of solution maps associated with a frequently arising class of generalized equations (GEs). The constraint sets are given by (not necessarily convex) inequalities, and we do not assume linear independence of gradients to active constraints. The achieved results enable us to state several versions of sharp necessary optimality conditions in optimization problems with equilibria governed by such GEs. The advantages are illustrated by means of examples.

A bundle-free implicit programming approach for a class of elliptic MPECs in function space

Using a standard first-order optimality condition for nonsmooth optimization problems, a general framework for a descent method is developed. This setting is applied to a class of mathematical programs with equilibrium constraints in function space from which a new algorithm is derived. Global convergence of the algorithm is demonstrated in function space and the results are then illustrated by numerical experiments.

GENERALIZED NASH EQUILIBRIUM PROBLEMS IN BANACH SPACES: THEORY, NIKAIDO-ISODA-BASED PATH-FOLLOWING METHODS, AND APPLICATIONS ∗

Building upon the results in [M. Hintermuller and T. Surowiec, Pac. J. Optim., 9 (2013), pp. 251--273], a class of noncooperative Nash equilibrium problems is presented, in which the feasible set of each player is perturbed by the decisions of their competitors via a convex constraint. In addition, for every vector of decisions, a common “state” variable is given by the solution of an affine linear equation. The resulting problem is therefore a generalized Nash equilibrium problem (GNEP). The existence of an equilibrium for this problem is demonstrated, and first-order optimality conditions are derived under a constraint qualification. An approximation scheme is proposed, which involves the solution of a parameter-dependent sequence of standard Nash equilibrium problems. An associated path-following strategy based on the Nikaido--Isoda function is then proposed. Function-space-based numerics for parabolic GNEPs and a spot-market model are developed.

Elliptic Mathematical Programs with Equilibrium Constraints in Function Space: Optimality Conditions and Numerical Realization

Recent advances in the analytical as well as numerical treatment of classes of elliptic mathematical programs with equilibrium constraints (MPECs) in function space are discussed. In particular, stationarity conditions for control problems with point tracking objectives and subject to the obstacle problem as well as for optimization problems with variational inequality constraints and pointwise constraints on the gradient of the state are derived. For the former problem class including the case of L 2-tracking-type objectives (rather than pointwise ones) a bundle-free solution method as well as adaptive finite element discretizations are introduced. Moreover, the analytical and numerical treatment of shape design problems subject to elliptic variational inequality constraints is highlighted. With respect to problems involving gradient constraints, the paper ends with a fixed-point-Moreau-Yosida-based semismooth Newton solver for a class of nonlinear elliptic quasi-variational inequality problems.