TY - GEN A1 - Mehlitz, Patrick A1 - Wachsmuth, Gerd T1 - The Weak Sequential Closure of Decomposable Sets in Lebesgue Spaces and its Application to Variational Geometry T2 - Set-Valued and Variational Analysis N2 - 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. KW - Decomposable set KW - Lebesgue spaces KW - Limiting normal cone KW - Measurability KW - Weak sequential closure Y1 - 2019 U6 - https://doi.org/10.1007/s11228-017-0464-1 SN - 1877-0533 SN - 1877-0541 VL - 27 IS - 1 SP - 265 EP - 294 ER - TY - GEN A1 - Mehlitz, Patrick A1 - Wachsmuth, Gerd T1 - The limiting normal cone to pointwise defined sets in Lebesgue spaces T2 - Set-Valued and Variational Analysis (SVAA) N2 - 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. KW - Decomposable set KW - Lebesgue spaces KW - Limiting normal cone KW - Mathematical program with complementarity constraint KW - Measurability Y1 - 2018 U6 - https://doi.org/10.1007/s11228-016-0393-4 SN - 1877-0533 SN - 1877-0541 VL - 26 IS - 3 SP - 449 EP - 467 ER - TY - GEN A1 - Mehlitz, Patrick A1 - Wachsmuth, Gerd T1 - Weak and strong stationarity in generalized bilevel programming and bilevel optimal control T2 - Optimization N2 - 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. KW - Bilevel programming KW - programming in Banach spaces KW - mathematical program with complementarity constraints KW - stationarity KW - bilevel optimal control Y1 - 2016 U6 - https://doi.org/10.1080/02331934.2015.1122007 SN - 0233-1934 SN - 1029-4945 VL - 65 IS - 5 SP - 907 EP - 935 ER - 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 - Franke, Susanne A1 - Mehlitz, Patrick A1 - Pilecka, Maria T1 - Optimality conditions for the simple convex bilevel programming problem in Banach spaces T2 - Optimization: A Journal of Mathematical Programming and Operations Research N2 - 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. KW - bilevel programming KW - constraint qualifications KW - convex programming KW - mathematical program with complementarity constraints KW - programming in Banach spaces Y1 - 2018 U6 - https://doi.org/10.1080/02331934.2017.1394296 SN - 0233-1934 SN - 1029-4945 VL - 67 IS - 2 SP - 237 EP - 268 ER - TY - GEN A1 - Mehlitz, Patrick T1 - Necessary optimality conditions for a special class of bilevel programming problems with unique lower level solution T2 - Optimization: A Journal of Mathematical Programming and Operations Research N2 - 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. KW - bilevel programming KW - mathematical program with complementarity constraints KW - programming in Banach spaces KW - semidefinite programming Y1 - 2017 U6 - https://doi.org/10.1080/02331934.2017.1349123 SN - 0233-1934 SN - 1029-4945 VL - 66 IS - 10 SP - 1533 EP - 1562 ER - TY - GEN A1 - Mehlitz, Patrick T1 - Bilevel programming problems with simple convex lower level T2 - Optimization: A Journal of Mathematical Programming and Operations Research N2 - 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. KW - bilevel programming KW - optimization in Banach spaces KW - nonsmooth optimization KW - DC-programming KW - optimal control Y1 - 2016 U6 - https://doi.org/10.1080/02331934.2015.1122006 SN - 0233-1934 SN - 1029-4945 VL - 65 IS - 6 SP - 1203 EP - 1227 ER - TY - GEN A1 - Benita, Francisco A1 - Mehlitz, Patrick T1 - Optimal Control Problems with Terminal Complementarity Constraints T2 - SIAM Journal on Optimization (SIOPT) N2 - 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. KW - mathematical program with complementarity constraints KW - optimal control KW - optimality conditions KW - programming in Banach spaces KW - W- KW - M- KW - S-stationarity Y1 - 2018 U6 - https://doi.org/10.1137/16M107637X SN - 1052-6234 SN - 1095-7189 VL - 28 IS - 4 SP - 3079 EP - 3104 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 - Benita, Francisco A1 - Mehlitz, Patrick T1 - Bilevel Optimal Control With Final-State-Dependent Finite-Dimensional Lower Level T2 - SIAM Journal on Optimization (SIOPT) N2 - 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. KW - optimal control KW - bilevel programming KW - Pontryagin maximum principle KW - optimization in Banach spaces KW - nonsmooth optimization KW - calmnes Y1 - 2016 U6 - https://doi.org/10.1137/15M1015984 SN - 1052-6234 SN - 1095-7189 VL - 26 IS - 1 SP - 718 EP - 752 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 - Mehlitz, Patrick T1 - Stationarity conditions and constraint qualifications for mathematical programs with switching constraints T2 - Mathematical Programming N2 - In optimal control, switching structures demanding at most one control to be active at any time instance appear frequently. Discretizing such problems, a so-called mathematical program with switching constraints is obtained. Although these problems are related to other types of disjunctive programs like optimization problems with complementarity or vanishing constraints, their inherent structure makes a separate consideration necessary. Since standard constraint qualifications are likely to fail at the feasible points of switching-constrained optimization problems, stationarity notions which are weaker than the associated Karush–Kuhn–Tucker conditions need to be investigated in order to find applicable necessary optimality conditions. Furthermore, appropriately tailored constraint qualifications need to be formulated. In this paper, we introduce suitable notions of weak, Mordukhovich-, and strong stationarity for mathematical programs with switching constraints and present some associated constraint qualifications. Our findings are exploited to state necessary optimality conditions for (discretized) optimal control problems with switching constraints. Furthermore, we apply our results to optimization problems with either-or-constraints. First, a novel reformulation of such problems using switching constraints is presented. Second, the derived surrogate problem is exploited to obtain necessary optimality conditions for the original program. KW - Constraint qualifications KW - Either-or-constraints KW - Nonlinear programming KW - Optimality conditions KW - Switching constraints Y1 - 2020 U6 - https://doi.org/10.1007/s10107-019-01380-5 SN - 0025-5610 SN - 1436-4646 VL - 181 IS - 1 SP - 149 EP - 186 ER - TY - GEN A1 - Deng, Yu A1 - Mehlitz, Patrick A1 - Prüfert, Uwe T1 - Optimal control in first-order Sobolev spaces with inequality constraints T2 - Computational Optimization and Applications N2 - In this paper, an elliptic optimal control problem with controls from H¹(Ω) which have to satisfy standard box constraints is considered. Thus, Lagrange multipliers associated with the box constraints are, in general, elements of H¹(Ω)* as long as the lower and upper bound belong to H¹(Ω) as well. If these bounds possess less regularity, the overall existence of a Lagrange multiplier is not even guaranteed. In order to avoid the direct solution of a not necessarily available KKT system, a penalty method is suggested which finds the minimizer of the control-constrained problem. Its convergence properties are analyzed. Furthermore, some numerical strategies for the computation of optimal solutions are suggested and illustrated. KW - Control constraints KW - Optimal control KW - Optimality conditions KW - Penalty method KW - Semismooth Newton method Y1 - 2019 U6 - https://doi.org/10.1007/s10589-018-0053-8 SN - 0926-6003 SN - 1573-2894 VL - 72 IS - 3 SP - 797 EP - 826 ER - TY - GEN A1 - Benita, Francisco A1 - Mehlitz, Patrick T1 - Solving optimal control problems with terminal complementarity constraints via Scholtes’ relaxation scheme T2 - Computational Optimization and Applications : (COAP) N2 - We investigate the numerical treatment of optimal control problems of linear ordinary differential equations with terminal complementarity constraints. Therefore, we generalize the well-known relaxation technique of Scholtes to the problem at hand. In principle, any other relaxation approach from finite-dimensional complementarity programming can be adapted in similar fashion. It is shown that the suggested method possesses strong convergence properties under mild assumptions. Finally, some numerical examples are presented. KW - Complementarity-constrained programming KW - Optimal control KW - Relaxation Y1 - 2019 U6 - https://doi.org/10.1007/s10589-018-0050-y SN - 0926-6003 SN - 1573-2894 VL - 72 IS - 2 SP - 413 EP - 430 ER -