TY - GEN A1 - Harder, Felix T1 - Legendre Forms in Reflexive Banach Spaces T2 - Zeitschrift für Analysis und ihre Anwendungen N2 - Legendre forms are used in the literature for second-order sufficient optimality conditions of optimization problems in (reflexive) Banach spaces. We show that if a Legendre form exists on a reflexive Banach space, then this space is already isomorphic to a Hilbert space. KW - Legendre form KW - second-order optimality conditions KW - Hilbertizable space KW - quadratic form KW - coercive bilinear form Y1 - 2018 U6 - https://doi.org/10.4171/ZAA/1619 SN - 1661-4534 SN - 0232-2064 VL - 37 IS - 4 SP - 377 EP - 388 ER - TY - GEN A1 - Harder, Felix A1 - Wachsmuth, Gerd T1 - Optimality Conditions for a Class of Inverse Optimal Control Problems With Partial Differential Equations T2 - Optimization: A Journal of Mathematical Programming and Operations Research N2 - We consider bilevel optimization problems which can be interpreted as inverse optimal control problems. The lower-level problem is an optimal control problem with a parametrized objective function. The upper-level problem is used to identify the parameters of the lower-level problem. Our main focus is the derivation of first-order necessary optimality conditions. We prove C-stationarity of local solutions of the inverse optimal control problem and give a counterexample to show that strong stationarity might be violated at a local minimizer. KW - Inverse optimal control problem KW - optimality condition KW - C-stationarity KW - strong stationarity Y1 - 2018 UR - https://www.tandfonline.com/doi/full/10.1080/02331934.2018.1495205 U6 - https://doi.org/10.1080/02331934.2018.1495205 SN - 0233-1934 SN - 1029-4945 ER - TY - GEN A1 - Wachsmuth, Gerd A1 - Harder, Felix T1 - The limiting normal cone of a complementarity set in Sobolev spaces T2 - Optimization N2 - We investigate the limiting normal cone of the complementarity set associated with non-negative functions in the Sobolev space. By using results from homogenization theory, we provide lower estimates for this limiting normal cone. These estimates are unpleasantly large. KW - Limiting normal cone KW - optimality condition KW - M-stationarity KW - obstacle problem Y1 - 2018 U6 - https://doi.org/10.1080/02331934.2018.1484467 SN - 0233-1934 SN - 1029-4945 VL - 67 IS - 10 SP - 1579 EP - 1603 ER - TY - GEN A1 - Harder, Felix A1 - Wachsmuth, Gerd T1 - Comparison of Optimality Systems for the Optimal Control of the Obstacle Problem T2 - GAMM-Mitteilung N2 - We consider stationarity systems for the optimal control of the obstacle problem. The focus is on the comparison of several different systems which are provided in the literature. We obtain some novel results concerning the relations between these stationarity concepts. KW - obstacle problem KW - complementarity constraints KW - weak stationarity KW - M-stationarity KW - strong stationarity Y1 - 2018 U6 - https://doi.org/10.1002/gamm.201740004 SN - 0936-7195 SN - 1522-2608 VL - 40 IS - 4 SP - 312 EP - 338 ER - TY - GEN A1 - Dempe, Stephan A1 - Harder, Felix A1 - Mehlitz, Patrick A1 - Wachsmuth, Gerd T1 - Solving inverse optimal control problems via value functions to global optimality T2 - Journal of Global Optimization N2 - In this paper, we show how a special class of inverse optimal control problems of elliptic partial differential equations can be solved globally. Using the optimal value function of the underlying parametric optimal control problem, we transfer the overall hierarchical optimization problem into a nonconvex single-level one. Unfortunately, standard regularity conditions like Robinson’s CQ are violated at all the feasible points of this surrogate problem. It is, however, shown that locally optimal solutions of the problem solve a Clarke-stationarity-type system. Moreover, we relax the feasible set of the surrogate problem iteratively by approximating the lower level optimal value function from above by piecewise affine functions. This allows us to compute globally optimal solutions of the original inverse optimal control problem. The global convergence of the resulting algorithm is shown theoretically and illustrated by means of a numerical example. KW - Bilevel optimal control KW - Global optimization KW - Inverse optimal control KW - Optimality conditions KW - Solution algorithm Y1 - 2019 U6 - https://doi.org/10.1007/s10898-019-00758-1 SN - 0925-5001 SN - 1573-2916 VL - 74 IS - 2 SP - 297 EP - 325 ER - TY - GEN A1 - Harder, Felix A1 - Mehlitz, Patrick A1 - Wachsmuth, Gerd T1 - Reformulation of the M-stationarity conditions as a system of discontinuous equations and its solution by a semismooth Newton method T2 - SIAM Journal on Optimization (SIOPT) N2 - We show that the Mordukhovich-stationarity system associated with a mathematical program with complementarity constraints (MPCC) can be equivalently written as a system of discontinuous equations which can be tackled with a semismooth Newton method. It will be demonstrated that the resulting algorithm can be interpreted as an active set strategy for MPCCs. Local fast convergence of the method is guaranteed under validity of an MPCC-tailored version of LICQ and a suitable strong second-order condition. In case of linear-quadratic MPCCs, the LICQ-type constraint qualification can be replaced by a weaker condition which depends on the underlying multipliers. We discuss a suitable globalization strategy for our method. Some numerical results are presented in order to illustrate our theoretical findings. KW - active set method KW - mathematical program with complementarity constraints KW - M-stationarity KW - nonlinear M-stationarity function KW - semismooth Newton method Y1 - 2021 U6 - https://doi.org/10.1137/20m1321413 SN - 1095-7189 SN - 1052-6234 VL - 31 IS - 2 SP - 1459 EP - 1488 ER - TY - GEN A1 - Harder, Felix T1 - A new elementary proof for M-stationarity under MPCC-GCQ for mathematical programs with complementarity constraints T2 - Journal of Nonsmooth Analysis and Optimization N2 - It is known in the literature that local minimizers of mathematical programs with complementarity constraints (MPCCs) are so-called M-stationary points, if a weak MPCC-tailored Guignard constraint qualification (called MPCC-GCQ) holds. In this paper we present a new elementary proof for this result. Our proof is significantly simpler than existing proofs and does not rely on deeper technical theory such as calculus rules for limiting normal cones. A crucial ingredient is a proof of a (to the best of our knowledge previously open) conjecture, which was formulated in a Diploma thesis by Schinabeck. KW - Mathematical program with complementarity constraints KW - Necessary optimality conditions KW - M-stationarity KW - Guignard constraint qualification KW - 90C33 KW - 90C30 Y1 - 2021 U6 - https://doi.org/10.46298/jnsao-2021-6903 SN - 2700-7448 VL - 2021 IS - 2 SP - 1 EP - 7 ER - TY - GEN A1 - Harder, Felix A1 - Wachsmuth, Gerd T1 - M-stationarity for a class of MPCCs in Lebesgue spaces T2 - arXiv N2 - We show that an optimality condition of M-stationarity type holds for minimizers of a class of mathematical programs with complementarity constraints (MPCCs) in Lebesgue spaces. We apply these results also to local minimizers of an inverse optimal control problem (which is an instance of an infinite-dimensional bilevel optimization problem). The multipliers for the M-stationarity system can be constructed via convex combinations of various multipliers to auxiliary, linear problems. However, proving the existence of the multipliers to these auxiliary problems is difficult and only possible in some situations. KW - Mathematical program with complementarity constraints KW - Necessary optimality conditions KW - M-stationarity KW - nverse optimal control Y1 - 2021 UR - https://arxiv.org/pdf/2110.11693.pdf ER - TY - GEN A1 - Harder, Felix T1 - New stationarity conditions between strong and M-stationarity for mathematical programs with complementarity constraints T2 - arXiv N2 - We introduce new first-order necessary conditions for mathematical programs with complementarity constraints (MPCCs), which lie between strong and M-stationarity and have a relatively simple description. We show that they hold for local minimizers under the rather weak constraint qualification MPCC-GCQ. As a generalization, we also get a class of stationarity conditions that lie between strong and C-stationarity and show that they also hold for local minimizers under MPCC-GCQ. We also present similar results for mathematical programs with vanishing constraints (MPVCs), and a very simple and elementary proof of M-stationarity for local minimizers of MPVCs. KW - Mathematical program with complementarity constraints KW - Mathematical program with vanishing constraints KW - Necessary optimality condition KW - M-stationarity KW - Guignard constraint qualification Y1 - 2021 UR - https://arxiv.org/abs/2109.01623 SP - 1 EP - 23 ER - TY - THES A1 - Harder, Felix T1 - On bilevel optimization problems in infinite-dimensional spaces N2 - In this thesis we consider bilevel optimization problems in infinite-dimensional spaces. In particular, we are interested in providing first-order necessary optimality conditions. We consider bilevel optimization problems both in abstract Banach spaces and in some special situations. This includes the optimal control of the obstacle problem, which is a typical bilevel optimization problem in Sobolev spaces, as well as a class of inverse optimal control problems. We obtain optimality conditions for these more specific optimization problems by applying our results from the abstract setting. Our main approach for deriving optimality conditions in the abstract setting utilizes the relaxation of a reformulation of the bilevel optimization problem via the optimal value function. We also introduce the so-called normal-cone-preserving operators and show how this concept can be applied. We also consider other topics that arise in this context. For instance, we investigate the so-called limiting normal cone to a complementarity set in Sobolev spaces. This complementarity set plays a central role in the context of the optimal control of the obstacle problem. The limiting normal cone is a concept which appears in the area of variational analysis and generalizes the usual normal cone from convex analysis. We also investigate in which spaces Legendre forms and Legendre-⋆ forms can exist. We show that if a Legendre-⋆ form exists in a reflexive Banach space or a space with a separable predual space, then this space is already isomorphic to a Hilbert space. We also consider a discretization of a bilevel optimization problem in Lebesgue spaces. We present both theoretical error estimates and numerical experiments. The new results in this thesis are illustrated by examples and counterexamples. In order to present the topics in a self-contained way, we review some known concepts and their basic properties. A particular focus for this is on the definitions and properties from the area of capacity theory. KW - Bilevel optimization KW - Inverse optimal control KW - Obstacle problem KW - Optimal control KW - Optimality conditions Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:co1-opus4-53755 PB - BTU CY - Cottbus ER - TY - GEN A1 - Harder, Felix A1 - Wachsmuth, Gerd T1 - M-stationarity for a class of MPCCs in Lebesgue spaces T2 - Journal of Mathematical Analysis and Applications Y1 - 2022 VL - 512 IS - 2 SP - 1 EP - 28 ER - TY - GEN A1 - Friedemann, Markus A1 - Harder, Felix A1 - Wachsmuth, Gerd T1 - Finding global solutions of some inverse optimal control problems using penalization and semismooth Newton methods T2 - arXiv Y1 - 2022 UR - https://arxiv.org/abs/2203.00325 SP - 1 EP - 45 ER - TY - GEN A1 - Dempe, Stephan A1 - Friedemann, Markus A1 - Harder, Felix A1 - Mehlitz, Patrick A1 - Wachsmuth, Gerd T1 - Bilevel optimal control: theory, algorithms, and applications T2 - arXiv Y1 - 2023 U6 - https://doi.org/10.48550/arXiv.2305.19786 SP - 1 EP - 31 ER - TY - GEN A1 - Friedemann, Markus A1 - Harder, Felix A1 - Wachsmuth, Gerd T1 - Finding global solutions of some inverse optimal control problems using penalization and semismooth Newton methods T2 - Journal of Global Optimization Y1 - 2023 U6 - https://doi.org/10.1007/s10898-023-01288-7 SN - 1573-2916 SN - 0925-5001 VL - 86 IS - 4 SP - 1025 EP - 1061 ER -