Mathematical Programs with Complementarity Constraints in Banach Spaces

Abstract

We consider optimization problems in Banach spaces involving a complementarity constraint defined by a convex cone K. By transferring the local decomposition approach, we define strong stationarity conditions and provide a constraint qualification under which these conditions are necessary for optimality. To apply this technique, we provide a new uniqueness result for Lagrange multipliers in Banach spaces. Under the additional assumption that the cone K is polyhedral, we show that our strong stationarity conditions possess a reasonable strength. Finally, we generalize to the case that K is not a cone and apply the theory to two examples.

Towards M-stationarity for Optimal Control of the Obstacle Problem with Control Constraints

“…If U ad = U and if the range of B is dense in H −1 (Ω), it is shown by [Mignot, 1976] that minimizers of (P) satisfy the system of strong stationarity. The same result was reproduced by [Hintermüller, Surowiec, 2011] by techniques from variational analysis and it was slightly generalized in [G. Wachsmuth, 2015]. The special case U = U ad = H −1 (Ω) can be found in [Outrata, Jarušek, Stará, 2011].…”

Comparison of optimality systems for the optimal control of the obstacle problem

“…Stationarity systems for (19) have been proposed in [15,[36][37][38]. In [15] it was suggested to use the so-called local decomposition approach, which is well known for the finitedimensional problem (12), see [27][28][29][30][31]. This leads to the following stationarity concepts.…”

“…Finally, we mention that often weaker variants similar to (15) are also called C-stationarity, e.g., one only requires ν, p ≤ 0.…”

Optimal Control Problems with Terminal Complementarity Constraints

“…In [40], the author introduces a far more general complementarity constrained optimization problem. He considers the model (MPCC) where the complementarity condition is replaced by…”

“…On the other hand, optimal control problems with variational inequality constraints (which cover complementarity constraints) are studied in a special setting in [19,20] and other contributions by these authors. In [29,40], the authors investigate a very general MPCC in Banach spaces which covers all the above classes of complementarity problems.…”