TY - GEN A1 - Dempe, Stephan A1 - Mefo Kue, Floriane A1 - Mehlitz, Patrick T1 - Optimality Conditions for Special Semidefinite Bilevel Optimization Problems T2 - SIAM Journal on Optimization N2 - In this paper, we consider an optimistic bilevel programming problem whose lower level is a semidefinite programming problem. Two main approaches, namely, the optimal value reformulation and the Karush--Kuhn--Tucker reformulation, are considered in order to transform the original problem into a single-level programming problem. Afterwards, the relationship between the original problem and its substitute is studied in each case and some necessary optimality conditions are derived as well. Therefore, among others, we exploit some calmness-type constraint qualifications studied in the general framework of finite-dimensional Hilbert spaces. KW - bilevel programming KW - semidefinite programming KW - optimality conditions Y1 - 2018 U6 - https://doi.org/10.1137/16M1099303 SN - 1052-6234 SN - 1095-7189 VL - 28 IS - 2 SP - 1564 EP - 1587 ER - TY - GEN A1 - Dempe, Stephan A1 - Mefo Kue, Floriane A1 - Mehlitz, Patrick T1 - Optimality conditions for mixed discrete bilevel optimization problems T2 - Optimization: A Journal of Mathematical Programming and Operations Research N2 - In this article, we consider bilevel optimization problems with discrete lower level and continuous upper level problems. Taking into account both approaches (optimistic and pessimistic) which have been developed in the literature to deal with this type of problem, we derive some conditions for the existence of solutions. In the case where the lower level is a parametric linear problem, the bilevel problem is transformed into a continuous one. After that, we are able to discuss local optimality conditions using tools of variational analysis for each of the different approaches. Finally, we consider a simple application of our results namely the bilevel programming problem with the minimum spanning tree problem in the lower level. KW - bilevel programming KW - discrete parametric optimization KW - optimality conditions Y1 - 2018 U6 - https://doi.org/10.1080/02331934.2018.1427092 SN - 0233-1934 SN - 1029-4945 VL - 67 IS - 6 SP - 737 EP - 756 ER - TY - GEN A1 - Benita, Francisco A1 - Dempe, Stephan A1 - Mehlitz, Patrick T1 - Bilevel Optimal Control Problems with Pure State Constraints and Finite-dimensional Lower Level T2 - SIAM Journal on Optimization (SIOPT) N2 - This paper focuses on the development of optimality conditions for a bilevel optimal control problem with pure state constraints in the upper level and a finite-dimensional parametric optimization problem in the lower level. After transforming the problem into an equivalent single-level problem, we concentrate on the derivation of a necessary optimality condition of Pontryagin type. We point out some major difficulties arising from the bilevel structure of the original problem and its pure state constraints in the upper level leading to a degenerated maximum principle in the absence of constraint qualifications. Hence, we use a partial penalization approach and a well-known regularity condition for optimal control problems with pure state constraints to ensure the nondegeneracy of the derived maximum principle. Finally, we illustrate the applicability of the derived theory by means of a small example. KW - bilevel optimization KW - optimal control KW - pure state constraints KW - optimality conditions KW - partial calmness Y1 - 2016 U6 - https://doi.org/10.1137/141000889 SN - 1052-6234 SN - 1095-7189 VL - 26 IS - 1 SP - 564 EP - 588 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 - Dempe, Stephan A1 - Mehlitz, Patrick T1 - Semivectorial bilevel programming versus scalar bilevel programming T2 - Optimization N2 - We consider an optimistic semivectorial bilevel programming problem in Banach spaces. The associated lower level multicriteria optimization problem is assumed to be convex w.r.t. its decision variable. This property implies that all its weakly efficient points can be computed applying the weighted-sum-scalarization technique. Consequently, it is possible to replace the overall semivectorial bilevel programming problem by means of a standard bilevel programming problem whose upper level variables comprise the set of suitable scalarization parameters for the lower level problem. In this note, we consider the relationship between this surrogate bilevel programming problem and the original semivectorial bilevel programming problem. As it will be shown, this is a delicate issue as long as locally optimal solutions are investigated. The obtained theory is applied in order to derive existence results for semivectorial bilevel programming problems with not necessarily finite-dimensional lower level decision variables. Some regarding examples from bilevel optimal control are presented. KW - Bilevel programming KW - existence theory KW - multiobjective optimization KW - optimal control Y1 - 2020 U6 - https://doi.org/10.1080/02331934.2019.1625900 SN - 0233-1934 SN - 1029-4945 VL - 69 IS - 4 SP - 657 EP - 679 ER - TY - GEN A1 - Dempe, Stephan A1 - Mehlitz, Patrick T1 - Lipschitz continuity of the optimal value function in parametric optimization T2 - Journal of Global Optimization N2 - We study generalized parametric optimization problems in Banach spaces, given by continuously Fréchet differentiable mappings and some abstract constraints, in terms of local Lipschitz continuity of the optimal value function. Therefore, we make use of the well-known regularity condition by Kurcyusz, Robinson and Zowe, an inner semicontinuity property of the solution set mapping and some earlier results by Mordukhovich, Nam and Yen. The main theorem presents a handy formula which can be used in order to approximate the Clarke subdifferential of the optimal value function, provided that the conditions mentioned above are satisfied and hence the optimal value function is locally Lipschitz continuous. Throughout the paper we avoid any compactness assumptions. KW - Parametric optimization KW - Variational analysis KW - Banach space KW - Optimal value function KW - Lipschitz continuity KW - Inner semicontinuity KW - Clarke subdifferential Y1 - 2015 U6 - https://doi.org/10.1007/s10898-014-0169-z SN - 1573-2916 SN - 0925-5001 VL - 61 IS - 2 SP - 363 EP - 377 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 -