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 - TY - GEN A1 - Clason, Christian A1 - Deng, Yu A1 - Mehlitz, Patrick A1 - Prüfert, Uwe T1 - Optimal control problems with control complementarity constraints: existence results, optimality conditions, and a penalty method T2 - Optimization Methods and Software N2 - A special class of optimal control problems with complementarity constraints on the control functions is studied. It is shown that such problems possess optimal solutions whenever the underlying control space is a first-order Sobolev space. After deriving necessary optimality conditions of strong stationarity-type, a penalty method based on the Fischer–Burmeister function is suggested and its theoretical properties are analyzed. Finally, the numerical treatment of the problem is discussed and results of computational experiments are presented. KW - Fischer–Burmeister function KW - mathematical problems with complementarity constraints KW - optimal control KW - optimality conditions KW - penalty method Y1 - 2020 U6 - https://doi.org/10.1080/10556788.2019.1604705 SN - 1055-6788 SN - 1029-4937 VL - 35 IS - 1 SP - 142 EP - 170 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 - Kanzow, Christian A1 - Mehlitz, Patrick A1 - Steck, Daniel T1 - Relaxation schemes for mathematical programmes with switching constraints T2 - Optimization Methods and Software N2 - Switching-constrained optimization problems form a difficult class of mathematical programmes since their feasible set is almost disconnected while standard constraint qualifications are likely to fail at several feasible points. That is why the application of standard methods from nonlinear programming does not seem to be promising in order to solve such problems. In this paper, we adapt several relaxation methods which are well known from the numerical treatment of mathematical programmes with complementarity constraints to the setting of switching-constrained optimization. A detailed convergence analysis is provided for the adapted relaxation schemes of Scholtes as well as Kanzow and Schwartz. While Scholtes' method and the relaxation scheme of Steffensen and Ulbrich only find weakly stationary points in general, it is shown that the adapted relaxation scheme of Kanzow and Schwartz is capable of identifying Mordukhovich-stationary points of switching-constrained programmes under suitable assumptions. Some computational experiments and a numerical comparison of the proposed methods based on examples from logical programming, switching control, and portfolio optimization close the paper. KW - Constraint qualifications KW - mathematical programme with switching constraints KW - relaxation methods KW - global convergence Y1 - 2021 U6 - https://doi.org/10.1080/10556788.2019.1663425 SN - 1055-6788 SN - 1029-4937 VL - 36 IS - 6 SP - 1223 EP - 1258 ER - TY - GEN A1 - Mehlitz, Patrick T1 - On the linear independence constraint qualification in disjunctive programming T2 - Optimization N2 - Mathematical programmes with disjunctive constraints (MPDCs for short) cover several different problem classes from nonlinear optimization including complementarity-, vanishing-, cardinality- and switching-constrained optimization problems. In this paper, we introduce an abstract but reasonable version of the prominent linear independence constraint qualification which applies to MPDCs. Afterwards, we derive first- and second-order optimality conditions for MPDCs under validity of this constraint qualification based on so-called strongly stationary points. Finally, we apply our findings to some popular classes of disjunctive programmes and compare the obtained results to those ones available in the literature. Particularly, new second-order optimality conditions for mathematical programmes with switching constraints are by-products of our approach. KW - Constraint qualifications KW - disjunctive programming KW - linear independence constraint qualification KW - strong stationarity KW - second-order optimality conditions Y1 - 2020 U6 - https://doi.org/10.1080/02331934.2019.1679811 SN - 0233-1934 SN - 1029-4945 VL - Vol. 69 (2020) IS - 10 SP - 2241 EP - 2277 ER - TY - GEN A1 - Deng, Yu A1 - Mehlitz, Patrick A1 - Prüfert, Uwe T1 - On an optimal control problem with gradient constraints T2 - Optimization N2 - Usually, control functions in control-constrained optimal control are chosen from a Lebesgue space. This choice, however, makes it impossible to postulate additional conditions on the control function's slope which is practically relevant in some situations. In order to overcome this disadvantage, a natural assumption would be to demand at least first-order Sobolev regularity for control functions. The present paper is devoted to the study of an elliptic optimal control problem whose control function is chosen from a Sobolev space and has to satisfy additional equality constraints on its weak gradient. Noting that the associated Karush–Kuhn–Tucker conditions do not provide a necessary optimality condition for the underlying optimal control problem in general, one cannot simply solve the problem of interest by considering the system of first-order optimality conditions. Instead a penalization procedure with strong convergence properties for the computational solution is suggested and its computational implementation is studied in detail. Particularly, some essential difficulties arising from the gradient constraints which do not appear in standard optimal control are discussed. KW - Control gradient constraints KW - enforcement phenomena in FEM KW - optimal control KW - optimality conditions KW - penalty method Y1 - 2020 U6 - https://doi.org/10.1080/02331934.2019.1604707 SN - 0233-1934 SN - 1029-4945 VL - 69 IS - 3 SP - 519 EP - 551 ER - TY - GEN A1 - Mehlitz, Patrick A1 - Minchenko, Leonid I. A1 - Zemkoho, Alain B. T1 - A note on partial calmness for bilevel optimization problems with linearly structured lower level T2 - Optimization Letters N2 - Partial calmness is a celebrated but restrictive property of bilevel optimization problems whose presence opens a way to the derivation of Karush–Kuhn–Tucker-type necessary optimality conditions in order to characterize local minimizers. In the past, sufficient conditions for the validity of partial calmness have been investigated. In this regard, the presence of a linearly structured lower level problem has turned out to be beneficial. However, the associated literature suffers from inaccurate results. In this note, we clarify some regarding erroneous statements and visualize the underlying issues with the aid of illustrative counterexamples. KW - Bilevel optimization KW - Linear programming KW - Partial calmness Y1 - 2021 U6 - https://doi.org/10.1007/s11590-020-01636-6 SN - 1862-4480 ER - TY - GEN A1 - Börgens, Eike A1 - Kanzow, Christian A1 - Mehlitz, Patrick A1 - Wachsmuth, Gerd T1 - New Constraint Qualifications for Optimization Problems in Banach Spaces Based on Asymptotic KKT Conditions T2 - SIAM Journal on Optimization N2 - Optimization theory in Banach spaces suffers from a lack of available constraint qualifications. There exist very few constraint qualifications, and these are often violated even in simple applications. This is very much in contrast to finite-dimensional nonlinear programs, where a large number of constraint qualifications is known. Since these constraint qualifications are usually defined using the set of active inequality constraints, it is difficult to extend them to the infinite-dimensional setting. One exception is a recently introduced sequential constraint qualification based on asymptotic KKT conditions. This paper shows that this so-called asymptotic KKT regularity allows suitable extensions to the Banach space setting in order to obtain new constraint qualifications. The relation of these new constraint qualifications to existing ones is discussed in detail. Their usefulness is also shown by several examples as well as an algorithmic application to the class of augmented Lagrangian methods. KW - asymptotic KKT conditions KW - asymptotic KKT regularity KW - constraint qualifications KW - optimization in Banach spaces KW - augmented Lagrangian method Y1 - 2020 U6 - https://doi.org/10.1137/19M1306804 SN - 1095-7189 SN - 1052-6234 VL - 30 IS - 4 SP - 2956 EP - 2982 ER - TY - GEN A1 - Mehlitz, Patrick T1 - Asymptotic stationarity and regularity for nonsmooth optimization problems T2 - Journal of Nonsmooth Analysis and Optimization N2 - Based on the tools of limiting variational analysis, we derive a sequential necessary optimality condition for nonsmooth mathematical programs which holds without any additional assumptions. In order to ensure that stationary points in this new sense are already Mordukhovich-stationary, the presence of a constraint qualification which we call AM-regularity is necessary. We investigate the relationship between AM-regularity and other constraint qualifications from nonsmooth optimization like metric (sub-)regularity of the underlying feasibility mapping. Our findings are applied to optimization problems with geometric and, particularly, disjunctive constraints. This way, it is shown that AM-regularity recovers recently introduced cone-continuity-type constraint qualifications, sometimes referred to as AKKT-regularity, from standard nonlinear and complementarity-constrained optimization. Finally, we discuss some consequences of AM-regularity for the limiting variational calculus. KW - Asymptotic regularity KW - Asymptotic stationarity KW - Constraint qualifications KW - M-stationarity KW - Nonsmooth optimization KW - Variational analysis Y1 - 2020 U6 - https://doi.org/10.46298/jnsao-2020-6575 SN - 2700-7448 VL - 1 ER - TY - GEN A1 - Mehlitz, Patrick T1 - A comparison of solution approaches for the numerical treatment of or-constrained optimization problems T2 - Computational Optimization and Applications N2 - Mathematical programs with or-constraints form a new class of disjunctive optimization problems with inherent practical relevance. In this paper, we provide a comparison of three different solution methods for the numerical treatment of this problem class which are inspired by classical approaches from disjunctive programming. First, we study the replacement of the or-constraints as nonlinear inequality constraints using suitable NCP-functions. Second, we transfer the or-constrained program into a mathematical program with switching or complementarity constraints which can be treated with the aid of well-known relaxation methods. Third, a direct Scholtes-type relaxation of the or-constraints is investigated. A numerical comparison of all these approaches which is based on three essentially different model programs from or-constrained optimization closes the paper. KW - Disjunctive programming KW - Global convergence KW - NCP-functions KW - Or-constrained programming KW - Relaxation methods Y1 - 2020 U6 - https://doi.org/10.1007/s10589-020-00169-z SN - 1573-2894 SN - 0926-6003 VL - 76 IS - 1 SP - 233 EP - 275 ER - TY - CHAP A1 - Mehlitz, Patrick A1 - Wachsmuth, Gerd T1 - Bilevel optimal control: existence results and stationarity conditions T2 - Bilevel Optimization N2 - The mathematical modeling of numerous real-world applications results in hierarchical optimization problems with two decision makers where at least one of them has to solve an optimal control problem of ordinary or partial differential equations. Such models are referred to as bilevel optimal control problems. Here, we first review some different features of bilevel optimal control including important applications, existence results, solution approaches, and optimality conditions. Afterwards, we focus on a specific problem class where parameters appearing in the objective functional of an optimal control problem of partial differential equations have to be reconstructed. After verifying the existence of solutions, necessary optimality conditions are derived by exploiting the optimal value function of the underlying parametric optimal control problem in the context of a relaxation approach. KW - Bilevel optimal control KW - Existence results KW - Inverse optimal control KW - Stationarity conditions Y1 - 2020 SN - 978-3-030-52119-6 SN - 978-3-030-52118-9 U6 - https://doi.org/10.1007/978-3-030-52119-6_16 SP - 451 EP - 484 PB - Springer Nature CY - Schweiz ER - TY - GEN A1 - Clason, Christian A1 - Deng, Yu A1 - Mehlitz, Patrick A1 - Prüfert, Uwe T1 - A penalization scheme for the numerical solution of optimal control problems with control complementarity constraints T2 - PAMM Proceedings in Applied Mathematics and Mechanics N2 - We suggest a simple penalty method for the numerical solution of optimal control problems with control complementarity constraints which is based on the famous Fischer–Burmeister function. The distinct advantage of this approach is the resulting smoothness of the penalty term and the unconstrainedness of the associated penalized surrogate problems. A numerical example from elliptic PDE control is presented as well. Y1 - 2019 U6 - https://doi.org/10.1002/pamm.201900122 SN - 1617-7061 VL - 19 IS - 1 SP - 1 EP - 2 PB - Wiley-VCH GmbH CY - Weinheim ER - TY - GEN A1 - Mehlitz, Patrick T1 - On the Sequential Normal Compactness Condition and its Restrictiveness in Selected Function Spaces T2 - Set-Valued and Variational Analysis N2 - Sequential normal compactness is one of the most important properties in terms of modern variational analysis. It is necessary for the derivation of calculus rules for the computation of generalized normals to set intersections or preimages of sets under transformations. While sequential normal compactness is inherent in finite-dimensional Banach spaces, its presence has to be checked in the infinite-dimensional situation. In this paper, we show that broad classes of sets in Lebesgue and Sobolev spaces which are reasonable in the context of optimal control suffer from an intrinsic lack of sequential normal compactness. KW - Decomposable set KW - Optimal control KW - Sequential normal compactness Y1 - 2019 U6 - https://doi.org/10.1007/s11228-018-0475-6 SN - 1877-0541 SN - 1877-0533 IS - 27 SP - 763 EP - 782 ER -