@misc{MehlitzWachsmuth, author = {Mehlitz, Patrick and Wachsmuth, Gerd}, title = {The Weak Sequential Closure of Decomposable Sets in Lebesgue Spaces and its Application to Variational Geometry}, series = {Set-Valued and Variational Analysis}, volume = {27}, journal = {Set-Valued and Variational Analysis}, number = {1}, issn = {1877-0533}, doi = {10.1007/s11228-017-0464-1}, pages = {265 -- 294}, abstract = {We provide a precise characterization of the weak sequential closure of nonempty, closed, decomposable sets in Lebesgue spaces. Therefore, we have to distinguish between the purely atomic and the nonatomic regime. In the latter case, we get a convexification effect which is related to Lyapunov's convexity theorem, and in the former case, the weak sequential closure equals the strong closure. The characterization of the weak sequential closure is utilized to compute the limiting normal cone to nonempty, closed, decomposable sets in Lebesgue spaces. Finally, we give an example for the possible nonclosedness of the limiting normal cone in this setting.}, language = {en} } @misc{MehlitzWachsmuth, author = {Mehlitz, Patrick and Wachsmuth, Gerd}, title = {The limiting normal cone to pointwise defined sets in Lebesgue spaces}, series = {Set-Valued and Variational Analysis (SVAA)}, volume = {26}, journal = {Set-Valued and Variational Analysis (SVAA)}, number = {3}, issn = {1877-0533}, doi = {10.1007/s11228-016-0393-4}, pages = {449 -- 467}, abstract = {We consider subsets of Lebesgue spaces which are defined by pointwise constraints. We provide formulas for corresponding variational objects (tangent and normal cones). Our main result shows that the limiting normal cone is always dense in the Clarke normal cone and contains the convex hull of the pointwise limiting normal cone. A crucial assumption for this result is that the underlying measure is non-atomic, and this is satisfied in many important applications (Lebesgue measure on subsets of Rͩ or the surface measure on hypersurfaces in Rͩ). Finally, we apply our findings to an optimization problem with complementarity constraints in Lebesgue spaces.}, language = {en} } @misc{MehlitzWachsmuth, author = {Mehlitz, Patrick and Wachsmuth, Gerd}, title = {Weak and strong stationarity in generalized bilevel programming and bilevel optimal control}, series = {Optimization}, volume = {65}, journal = {Optimization}, number = {5}, issn = {0233-1934}, doi = {10.1080/02331934.2015.1122007}, pages = {907 -- 935}, abstract = {In this article, we consider a general bilevel programming problem in reflexive Banach spaces with a convex lower level problem. In order to derive necessary optimality conditions for the bilevel problem, it is transferred to a mathematical program with complementarity constraints (MPCC). We introduce a notion of weak stationarity and exploit the concept of strong stationarity for MPCCs in reflexive Banach spaces, recently developed by the second author, and we apply these concepts to the reformulated bilevel programming problem. Constraint qualifications are presented, which ensure that local optimal solutions satisfy the weak and strong stationarity conditions. Finally, we discuss a certain bilevel optimal control problem by means of the developed theory. Its weak and strong stationarity conditions of Pontryagin-type and some controllability assumptions ensuring strong stationarity of any local optimal solution are presented.}, language = {en} } @misc{DempeMefoKueMehlitz, author = {Dempe, Stephan and Mefo Kue, Floriane and Mehlitz, Patrick}, title = {Optimality Conditions for Special Semidefinite Bilevel Optimization Problems}, series = {SIAM Journal on Optimization}, volume = {28}, journal = {SIAM Journal on Optimization}, number = {2}, issn = {1052-6234}, doi = {10.1137/16M1099303}, pages = {1564 -- 1587}, abstract = {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.}, language = {en} } @misc{DempeMefoKueMehlitz, author = {Dempe, Stephan and Mefo Kue, Floriane and Mehlitz, Patrick}, title = {Optimality conditions for mixed discrete bilevel optimization problems}, series = {Optimization: A Journal of Mathematical Programming and Operations Research}, volume = {67}, journal = {Optimization: A Journal of Mathematical Programming and Operations Research}, number = {6}, issn = {0233-1934}, doi = {10.1080/02331934.2018.1427092}, pages = {737 -- 756}, abstract = {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.}, language = {en} } @misc{FrankeMehlitzPilecka, author = {Franke, Susanne and Mehlitz, Patrick and Pilecka, Maria}, title = {Optimality conditions for the simple convex bilevel programming problem in Banach spaces}, series = {Optimization: A Journal of Mathematical Programming and Operations Research}, volume = {67}, journal = {Optimization: A Journal of Mathematical Programming and Operations Research}, number = {2}, issn = {0233-1934}, doi = {10.1080/02331934.2017.1394296}, pages = {237 -- 268}, abstract = {The simple convex bilevel programming problem is a convex minimization problem whose feasible set is the solution set of another convex optimization problem. Such problems appear frequently when searching for the projection of a certain point onto the solution set of another program. Due to the nature of the problem, Slater's constraint qualification generally fails to hold at any feasible point. Hence, one has to formulate weaker constraint qualifications or stationarity notions in order to state optimality conditions. In this paper, we use two different single-level reformulations of the problem, the optimal value and the Karush-Kuhn-Tucker approach, to derive optimality conditions for the original program. Since all these considerations are carried out in Banach spaces, the results are not limited to standard optimization problems in Rⁿ. On the road, we introduce and discuss a certain concept of M-stationarity for mathematical programs with complementarity constraints in Banach spaces.}, language = {en} } @misc{Mehlitz, author = {Mehlitz, Patrick}, title = {Necessary optimality conditions for a special class of bilevel programming problems with unique lower level solution}, series = {Optimization: A Journal of Mathematical Programming and Operations Research}, volume = {66}, journal = {Optimization: A Journal of Mathematical Programming and Operations Research}, number = {10}, issn = {0233-1934}, doi = {10.1080/02331934.2017.1349123}, pages = {1533 -- 1562}, abstract = {We consider a bilevel programming problem in Banach spaces whose lower level solution is unique for any choice of the upper level variable. A condition is presented which ensures that the lower level solution mapping is directionally differentiable, and a formula is constructed which can be used to compute this directional derivative. Afterwards, we apply these results in order to obtain first-order necessary optimality conditions for the bilevel programming problem. It is shown that these optimality conditions imply that a certain mathematical program with complementarity constraints in Banach spaces has the optimal solution zero. We state the weak and strong stationarity conditions of this problem as well as corresponding constraint qualifications in order to derive applicable necessary optimality conditions for the original bilevel programming problem. Finally, we use the theory to state new necessary optimality conditions for certain classes of semidefinite bilevel programming problems and present an example in terms of bilevel optimal control.}, language = {en} } @misc{Mehlitz, author = {Mehlitz, Patrick}, title = {Bilevel programming problems with simple convex lower level}, series = {Optimization: A Journal of Mathematical Programming and Operations Research}, volume = {65}, journal = {Optimization: A Journal of Mathematical Programming and Operations Research}, number = {6}, issn = {0233-1934}, doi = {10.1080/02331934.2015.1122006}, pages = {1203 -- 1227}, abstract = {This article is dedicated to the study of bilevel optimal control problems equipped with a fully convex lower level of special structure. In order to construct necessary optimality conditions, we consider a general bilevel programming problem in Banach spaces possessing operator constraints, which is a generalization of the original bilevel optimal control problem. We derive necessary optimality conditions for the latter problem using the lower level optimal value function, ideas from DC-programming and partial penalization. Afterwards, we apply our results to the original optimal control problem to obtain necessary optimality conditions of Pontryagin-type. Along the way, we derive a handy formula, which might be used to compute the subdifferential of the optimal value function which corresponds to the lower level parametric optimal control problem.}, language = {en} } @misc{BenitaMehlitz, author = {Benita, Francisco and Mehlitz, Patrick}, title = {Optimal Control Problems with Terminal Complementarity Constraints}, series = {SIAM Journal on Optimization (SIOPT)}, volume = {28}, journal = {SIAM Journal on Optimization (SIOPT)}, number = {4}, issn = {1052-6234}, doi = {10.1137/16M107637X}, pages = {3079 -- 3104}, abstract = {In this paper, we study an optimal control problem of ordinary differential equations with linear dynamics, affine mixed control-state constraints, and terminal complementarity constraints on the state function. We derive its weak, Mordukhovich, and strong stationarity conditions, and we present constraint qualifications which ensure that these conditions are satisfied at a locally optimal solution of the optimal control problem.}, language = {en} } @misc{BenitaDempeMehlitz, author = {Benita, Francisco and Dempe, Stephan and Mehlitz, Patrick}, title = {Bilevel Optimal Control Problems with Pure State Constraints and Finite-dimensional Lower Level}, series = {SIAM Journal on Optimization (SIOPT)}, volume = {26}, journal = {SIAM Journal on Optimization (SIOPT)}, number = {1}, issn = {1052-6234}, doi = {10.1137/141000889}, pages = {564 -- 588}, abstract = {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.}, language = {en} } @misc{BenitaMehlitz, author = {Benita, Francisco and Mehlitz, Patrick}, title = {Bilevel Optimal Control With Final-State-Dependent Finite-Dimensional Lower Level}, series = {SIAM Journal on Optimization (SIOPT)}, volume = {26}, journal = {SIAM Journal on Optimization (SIOPT)}, number = {1}, issn = {1052-6234}, doi = {10.1137/15M1015984}, pages = {718 -- 752}, abstract = {In this paper we discuss special bilevel optimal control problems where the upper level problem is an optimal control problem of ODEs with control and terminal constraints and the lower level problem is a finite-dimensional parametric optimization problem where the parameter is the final state of the state variable of the upper level. We tackle this problem using tools from nonsmooth analysis, optimization in Banach spaces, and bilevel programming to derive necessary optimality conditions of linearized Pontryagin-type.}, language = {en} }