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 - 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 - Mehlitz, Patrick A1 - Zemkoho, Alain B. T1 - Sufficient Optimality Conditions in Bilevel Programming T2 - Mathematics of Operations Research N2 - This paper is concerned with the derivation of first- and second-order sufficient optimality conditions for optimistic bilevel optimization problems involving smooth functions. First-order sufficient optimality conditions are obtained by estimating the tangent cone to the feasible set of the bilevel program in terms of initial problem data. This is done by exploiting several different reformulations of the hierarchical model as a single-level problem. To obtain second-order sufficient optimality conditions, we exploit the so-called value function reformulation of the bilevel optimization problem, which is then tackled with the aid of second-order directional derivatives. The resulting conditions can be stated in terms of initial problem data in several interesting situations comprising the settings where the lower level is linear or possesses strongly stable solutions. KW - 90c33, 90c46 KW - Bilevel Optimization KW - First-order Sufficient Optimality Conditions KW - Primary: Nonlinear Programming, Optimality Conditions KW - Second-order Directional Derivatives KW - Second-order Sufficient Optimality Conditions KW - Secondary: 49J52, 49J53 KW - Secondary: Complementarity Programming, Nondifferentiable Programming, Parametric Programming Y1 - 2021 U6 - https://doi.org/10.1287/moor.2021.1122 SP - 1 EP - 26 ER - TY - GEN A1 - Deng, Yu A1 - Mehlitz, Patrick A1 - Prüfert, Uwe T1 - Coupled versus decoupled penalization of control complementarity constraints T2 - ESAIM: Control, Optimisation and Calculus of Variations (COCV) N2 - This paper deals with the numerical solution of optimal control problems with control complementarity constraints. For that purpose, we suggest the use of several penalty methods which differ with respect to the handling of the complementarity constraint which is either penalized as a whole with the aid of NCP-functions or decoupled in such a way that non-negativity constraints as well as the equilibrium condition are penalized individually. We first present general global and local convergence results which cover several different penalty schemes before two decoupled methods which are based on a classical ℓ1- and ℓ2-penalty term, respectively, are investigated in more detail. Afterwards, the numerical implementation of these penalty methods is discussed. Based on some examples, where the optimal boundary control of a parabolic partial differential equation is considered, some quantitative properties of the resulting algorithms are compared. KW - Complementarity constraints KW - optimal control KW - parabolic PDE KW - penalty method Y1 - 2021 U6 - https://doi.org/10.1051/cocv/2021022 SN - 1292-8119 SN - 1262-3377 VL - 27 SP - 1 EP - 31 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 - Benko, Matúš A1 - Mehlitz, Patrick T1 - Calmness and Calculus: Two Basic Patterns T2 - Set-Valued and Variational Analysis N2 - We establish two types of estimates for generalized derivatives of set-valued mappings which carry the essence of two basic patterns observed throughout the pile of calculus rules. These estimates also illustrate the role of the essential assumptions that accompany these two patters, namely calmness on the one hand and (fuzzy) inner calmness* on the other. Afterwards, we study the relationship between and sufficient conditions for the various notions of (inner) calmness. The aforementioned estimates are applied in order to recover several prominent calculus rules for tangents and normals as well as generalized derivatives of marginal functions and compositions as well as Cartesian products of set-valued mappings under mild conditions. We believe that our enhanced approach puts the overall generalized calculus into some other light. Some applications of our findings are presented which exemplary address necessary optimality conditions for minimax optimization problems as well as the calculus related to the recently introduced semismoothness* property. KW - Calculus KW - Calmness KW - Generalized differentiation KW - Inner calmness* KW - Set-valued analysis KW - Variation analysis Y1 - 2022 U6 - https://doi.org/10.1007/s11228-021-00589-x SN - 1877-0541 SN - 1877-0533 VL - 30 IS - 1 SP - 81 EP - 117 ER - TY - GEN A1 - Mehlitz, Patrick A1 - Minchenko, Leonid I. T1 - R-Regularity of Set-Valued Mappings Under the Relaxed Constant Positive Linear Dependence Constraint Qualification with Applications to Parametric and Bilevel Optimization T2 - Set-Valued and Variational Analysis N2 - The presence of Lipschitzian properties for solution mappings associated with nonlinear parametric optimization problems is desirable in the context of, e.g., stability analysis or bilevel optimization. An example of such a Lipschitzian property for set-valued mappings, whose graph is the solution set of a system of nonlinear inequalities and equations, is R-regularity. Based on the so-called relaxed constant positive linear dependence constraint qualification, we provide a criterion ensuring the presence of the R-regularity property. In this regard, our analysis generalizes earlier results of that type which exploited the stronger Mangasarian–Fromovitz or constant rank constraint qualification. Afterwards, we apply our findings in order to derive new sufficient conditions which guarantee the presence of R-regularity for solution mappings in parametric optimization. Finally, our results are used to derive an existence criterion for solutions in pessimistic bilevel optimization and a sufficient condition for the presence of the so-called partial calmness property in optimistic bilevel optimization. KW - Bilevel optimization KW - Parametric optimization KW - Partial calmness KW - RCPLD KW - R-regularity KW - 49J53 KW - 90C30 KW - 90C31 Y1 - 2022 U6 - https://doi.org/10.1007/s11228-021-00578-0 SN - 1877-0541 SN - 1877-0533 VL - 30 IS - 1 SP - 179 EP - 205 ER - TY - GEN A1 - Benko, Matúš A1 - Mehlitz, Patrick T1 - On implicit variables in optimization theory T2 - Journal of Nonsmooth Analysis and Optimization N2 - Implicit variables of a mathematical program are variables which do not need to be optimized but are used to model feasibility conditions. They frequently appear in several different problem classes of optimization theory comprising bilevel programming, evaluated multiobjective optimization, or nonlinear optimization problems with slack variables. In order to deal with implicit variables, they are often interpreted as explicit ones. Here, we first point out that this is a light-headed approach which induces artificial locally optimal solutions. Afterwards, we derive various Mordukhovich-stationarity-type necessary optimality conditions which correspond to treating the implicit variables as explicit ones on the one hand, or using them only implicitly to model the constraints on the other. A detailed comparison of the obtained stationarity conditions as well as the associated underlying constraint qualifications will be provided. Overall, we proceed in a fairly general setting relying on modern tools of variational analysis. Finally, we apply our findings to different well-known problem classes of mathematical optimization in order to visualize the obtained theory. KW - Mathematics - Optimization and Control KW - 49J53 KW - 90C30 KW - 90C33 Y1 - 2021 U6 - https://doi.org/10.46298/jnsao-2021-7215 SN - 2700-7448 VL - 2 SP - 7215 ER - TY - GEN A1 - Kruger, Alexander Y. A1 - Mehlitz, Patrick T1 - Optimality conditions, approximate stationarity, and applications – a story beyond Lipschitzness T2 - arXiv N2 - Approximate necessary optimality conditions in terms of Fréchet subgra- dients and normals for a rather general optimization problem with a po- tentially non-Lipschitzian objective function are established with the aid of Ekeland’s variational principle, the fuzzy Fréchet subdifferential sum rule, and a novel notion of lower semicontinuity relative to a set-valued mapping or set. Feasible points satisfying these optimality conditions are referred to as approximately stationary. As applications, we derive a new general version of the extremal principle. Furthermore, we study approximate stationarity conditions for an optimization problem with a composite objective function and geometric constraints, a qualification condition guaranteeing that ap- proximately stationary points of such a problem are M-stationary, and a multiplier-penalty-method which naturally computes approximately station- ary points of the underlying problem. Finally, necessary optimality conditions for an optimal control problem with a non-Lipschitzian sparsity-promoting term in the objective function are established. KW - Approximate stationarity KW - Generalized separation KW - Non-Lipschitzian programming KW - Optimality conditions KW - Sparse control Y1 - 2021 UR - https://arxiv.org/pdf/2110.07268.pdf SP - 1 EP - 47 ER - TY - GEN A1 - Mehlitz, Patrick A1 - Wachsmuth, Gerd T1 - Subdifferentiation of nonconvex sparsity-promoting functionals on Lebesgue spaces T2 - arXiv N2 - Sparsity-promoting terms are incorporated into the objective functions of optimal control problems in order to ensure that optimal controls vanish on large parts of the underlying domain. Typical candidates for those terms are integral functions on Lebesgue spaces based on the ℓp-metric for p∈[0,1) which are nonconvex as well as non-Lipschitz and, thus, variationally challenging. In this paper, we derive exact formulas for the Fréchet, limiting, and singular subdifferential of these functionals. These generalized derivatives can be used for the derivation of necessary optimality conditions for optimal control problems comprising such sparsity-promoting terms. KW - Integral functionals KW - Sparsity-promoting functionals KW - Subdifferentiation KW - Variational analysis Y1 - 2021 UR - https://arxiv.org/abs/2107.09340 SP - 1 EP - 25 ER - TY - THES A1 - Mehlitz, Patrick T1 - On implicit variables and related topics in mathematical optimization N2 - This thesis is concerned with the phenomenon of implicit variables in optimization theory. Roughly speaking, a variable is called implicit whenever it is used to model the feasible set but does not appear in the objective function. At the first glance, such variables seem to be less relevant for the purpose of optimization. First, we provide a theoretical study on optimization problems with implicit variables. Therefore, we rely on a model program which covers several interesting problem classes from optimization theory such as bilevel optimization problems, evaluated multiobjective optimization problems, or optimization problems with cardinality constraints. We start our analysis by clarifying that the interpretation of implicit variables as explicit ones induces additional local minimizers. Afterwards, we study three reasonable stationarity systems of Mordukhovich-stationarity-type for the original problem as well as some comparatively weak associated constraint qualifications. The obtained results are applied to the three example classes mentioned above. Second, we introduce switching- and or-constrained optimization problems. Exploiting the observation that each or-constrained optimization problem can be transferred into a switching-constrained optimization problem with the aid of slack variables, one can interpret or-constrained programs as optimization problems comprising implicit variables. Necessary optimality conditions and constraint qualifications for both problem classes are derived. Furthermore, some approaches for the numerical solution of both problem classes are discussed and results of computational experiments are presented. The shortcomings of implicit variables are highlighted in terms of or-constrained optimization. Third, we study three different scenarios where optimality conditions and constraint qualifications for challenging optimization problems can be constructed while abstaining from the introduction of implicit variables. We start by deriving a generalized version of the linear independence constraint qualification as well as second-order necessary and sufficient optimality conditions for so-called disjunctive optimization problems, which cover several interesting but inherently irregular problem classes like mathematical programs with complementarity, switching, or-, and cardinality constraints. Afterwards, we exploit several different single-level reformulations of standard bilevel optimization problems in order to find first- and second-order sufficient optimality conditions. Finally, we study sequential stationarity and regularity conditions for nonsmooth mathematical problems with generalized equation constraints with the aid of the popular limiting variational analysis. The investigated model problem covers the one we use for the theoretical analysis of implicit variables. KW - Constraint Qualifications KW - Implicit Variables KW - Mathematical Optimization KW - Optimality Conditions KW - Variational Analysis Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:co1-opus4-55291 PB - BTU CY - Cottbus ER - TY - GEN A1 - Jia, Xiaoxi A1 - Kanzow, Christian A1 - Mehlitz, Patrick A1 - Wachsmuth, Gerd T1 - An Augmented Lagrangian Method for Optimization Problems with Structured Geometric Constraints T2 - arXiv N2 - This paper is devoted to the theoretical and numerical investigation of an augmented Lagrangian method for the solution of optimization problems with geometric constraints. Specifically, we study situations where parts of the constraints are nonconvex and possibly complicated, but allow for a fast computation of projections onto this nonconvex set. Typical problem classes which satisfy this requirement are optimization problems with disjunctive constraints (like complementarity or cardinality constraints) as well as optimization problems over sets of matrices which have to satisfy additional rank constraints. The key idea behind our method is to keep these complicated constraints explicitly in the constraints and to penalize only the remaining constraints by an augmented Lagrangian function. The resulting subproblems are then solved with the aid of a problem-tailored nonmonotone projected gradient method. The corresponding convergence theory allows for an inexact solution of these subproblems. Nevertheless, the overall algorithm computes so-called Mordukhovich-stationary points of the original problem under a mild asymptotic regularity condition, which is generally weaker than most of the respective available problem-tailored constraint qualifications. Extensive numerical experiments addressing complementarity- and cardinality-constrained optimization problems as well as a semidefinite reformulation of Maxcut problems visualize the power of our approach. KW - Asymptotic Regularity KW - Augmented Lagrangian Method KW - Complementarity Constraints KW - Cardinality Constraints KW - Maxcut Problem KW - Mordukhovich-Stationarity KW - Non-monotone Projected Gradient Method Y1 - 2021 UR - https://arxiv.org/abs/2105.08317 SP - 1 EP - 49 ER - TY - GEN A1 - Mehlitz, Patrick T1 - Asymptotic regularity for Lipschitzian nonlinear optimization problems with applications to complementarity-constrained and bilevel programming T2 - arXiv N2 - Asymptotic stationarity and regularity conditions turned out to be quite useful to study the qualitative properties of numerical solution methods for standard nonlinear and complementarity-constrained programs. In this paper, we first extend these notions to nonlinear optimization problems with nonsmooth but Lipschitzian data functions in order to find reasonable notions of asymptotic stationarity and regularity in terms of Clarke's and Mordukhovich's subdifferential construction. Particularly, we compare the associated novel asymptotic constraint qualifications with already existing ones. The second part of the paper presents two applications of the obtained theory. On the one hand, we specify our findings for complementarity-constrained optimization problems and recover recent results from the literature which demonstrates the power of the approach. Furthermore, we hint at potential extensions to or- and vanishing-constrained optimization. On the other hand, we demonstrate the usefulness of asymptotic regularity in the context of bilevel optimization. More precisely, we justify a well-known stationarity system for affinely constrained bilevel optimization problems in a novel way. Afterwards, we suggest a solution algorithm for this class of bilevel optimization problems which combines a penalty method with ideas from DC-programming. After a brief convergence analysis, we present results of some numerical experiments. KW - Asymptotic regularity KW - Bilevel optimization KW - Complementarity-constrained optimization KW - DC-optimization KW - Nonsmooth optimization Y1 - 2021 UR - https://arxiv.org/abs/2105.01985 SP - 1 EP - 43 ER - TY - GEN A1 - Kanzow, Christian A1 - Mehlitz, Patrick T1 - Convergence properties of monotone and nonmonotone proximal gradient methods revisited T2 - arXiv N2 - Composite optimization problems, where the sum of a smooth and a merely lower semicontinuous function has to be minimized, are often tackled numerically by means of proximal gradient methods as soon as the lower semicontinuous part of the objective function is of simple enough structure. The available convergence theory associated with these methods requires the derivative of the smooth part of the objective function to be (globally) Lipschitz continuous, and this might be a restrictive assumption in some practically relevant scenarios. In this paper, we readdress this classical topic and provide convergence results for the classical (monotone) proximal gradient method and one of its nonmonotone extensions which are applicable in the absence of (strong) Lipschitz assumptions. KW - Non-Lipschitz Optimization KW - Nonsmooth Optimization KW - Proximal Gradient Method Y1 - 2021 UR - https://arxiv.org/abs/2112.01798 SP - 1 EP - 21 ER - TY - GEN A1 - Mehlitz, Patrick T1 - Asymptotic regularity for Lipschitzian nonlinear optimization problems with applications to complementarity-constrained and bilevel programming T2 - Optimization Y1 - 2023 U6 - https://doi.org/10.1080/02331934.2022.2031190 SN - 1029-4945 VL - 72 IS - 1 SP - 277 EP - 320 ER - TY - GEN A1 - Jia, Xiaoxi A1 - Kanzow, Christian A1 - Mehlitz, Patrick A1 - Wachsmuth, Gerd T1 - An Augmented Lagrangian Method for Optimization Problems with Structured Geometric Constraints T2 - Mathematical Programming Y1 - 2023 U6 - https://doi.org/10.1007/s10107-022-01870-z VL - 199 SP - 1365 EP - 1415 ER - TY - GEN A1 - Kanzow, Christian A1 - Mehlitz, Patrick T1 - Convergence properties of monotone and nonmonotone proximal gradient methods revisited T2 - Journal of Optimization Theory and Applications Y1 - 2022 U6 - https://doi.org/10.1007/s10957-022-02101-3 SN - 1573-2878 SN - 0022-3239 VL - 195 IS - 2 SP - 624 EP - 646 ER - TY - GEN A1 - Kruger, Alexander Y. A1 - Mehlitz, Patrick T1 - Optimality conditions, approximate stationarity, and applications - a story beyond Lipschitzness T2 - Control, Optimisation and Calculus of Variations (ESAIM-COCV) Y1 - 2022 U6 - https://doi.org/10.1051/cocv/2022024 SN - 1262-3377 SN - 1292-8119 VL - 28 ER - TY - GEN A1 - Mehlitz, Patrick A1 - Wachsmuth, Gerd T1 - Subdifferentiation of nonconvex sparsity-promoting functionals on Lebesgue spaces T2 - SIAM Journal on Control and Optimization Y1 - 2022 U6 - https://doi.org/10.1137/21m1435173 SN - 1095-7138 VL - 60 IS - 3 SP - 1819 EP - 1839 ER - TY - GEN A1 - Mehlitz, Patrick T1 - A simple proof of second-order sufficient optimality conditions in nonlinear semidefinite optimization T2 - arXiv Y1 - 2022 UR - https://arxiv.org/abs/2209.12209 SP - 1 EP - 11 ER - TY - GEN A1 - Mehlitz, Patrick A1 - Göttlich, Simone A1 - Schillinger, Thomas T1 - Inverse demand tracking in transportation networks T2 - arXiv Y1 - 2022 UR - https://arxiv.org/abs/2212.11560 SP - 1 EP - 21 ER - TY - GEN A1 - De Marchi, Alberto A1 - Jia, Xiaoxi A1 - Kanzow, Christian A1 - Mehlitz, Patrick T1 - Constrained structured optimization and augmented Lagrangian proximal methods T2 - arXiv Y1 - 2022 UR - https://arxiv.org/abs/2203.05276 SP - 1 EP - 44 ER - TY - GEN A1 - Benko, Matúš A1 - Mehlitz, Patrick T1 - On the directional asymptotic approach in optimization theory Part A: approximate, M- , and mixed-order stationarity T2 - arXiv Y1 - 2022 UR - https://arxiv.org/abs/2204.13932 SP - 1 EP - 40 ER - TY - GEN A1 - Benko, Matúš A1 - Mehlitz, Patrick T1 - On the directional asymptotic approach in optimization theory Part B: constraint qualifications T2 - arXiv Y1 - 2022 UR - https://arxiv.org/abs/2205.00775 SP - 1 EP - 38 ER - TY - GEN A1 - Benko, Matúš A1 - Mehlitz, Patrick T1 - Why second-order sufficient conditions are, in a way, easy -- or -- revisiting calculus for second subderivatives T2 - arXiv Y1 - 2022 UR - https://arxiv.org/abs/2206.03918 SP - 1 EP - 43 ER - TY - GEN A1 - Jolaoso, Lateef O. A1 - Mehlitz, Patrick A1 - Zemkoho, Alain B. T1 - A fresh look at nonsmooth Levenberg-Marquardt methods with applications to bilevel optimization T2 - Optimization Y1 - 2024 U6 - https://doi.org/10.1080/02331934.2024.2313688 VL - 2024 SP - 1 EP - 48 PB - Taylor&Francis ER - TY - GEN A1 - Benko, Matúš A1 - Mehlitz, Patrick T1 - On the directional asymptotic approach in optimization theory T2 - Mathematical Programming Y1 - 2024 U6 - https://doi.org/10.1007/s10107-024-02089-w SN - 1436-4646 SN - 0025-5610 ER - TY - GEN A1 - Hoff, Daniel A1 - Mehlitz, Patrick T1 - Notes on the value function approach to multiobjective bilevel optimization T2 - Optimization Y1 - 2024 U6 - https://doi.org/10.1080/02331934.2024.2323107 VL - 73 IS - 10 SP - 3147 EP - 3183 ER - TY - GEN A1 - Göttlich, Simone A1 - Mehlitz, Patrick A1 - Schillinger, Thomas T1 - Inverse demand tracking in transportation networks T2 - Mathematical Methods of Operations Research Y1 - 2024 U6 - https://doi.org/10.1007/s00186-024-00875-y SN - 1432-5217 SN - 1432-2994 ER - TY - GEN A1 - Hoff, Daniel A1 - Mehlitz, Patrick T1 - Notes on the value function approach to multiobjective bilevel optimization T2 - arXiv Y1 - 2023 UR - https://arxiv.org/abs/2303.15824 SP - 1 EP - 32 ER - TY - GEN A1 - Jia, Xiaoxi A1 - Kanzow, Christian A1 - Mehlitz, Patrick T1 - Convergence analysis of the proximal gradient method in the presence of the Kurdyka-Łojasiewicz property without global Lipschitz assumptions T2 - arXiv Y1 - 2023 U6 - https://doi.org/10.48550/arXiv.2301.05002 SP - 1 EP - 23 ER - TY - GEN A1 - De Marchi, Alberto A1 - Mehlitz, Patrick T1 - Local properties and augmented Lagrangians in fully nonconvex composite optimization T2 - arXiv Y1 - 2023 U6 - https://doi.org/10.48550/arXiv.2309.01980 SP - 1 EP - 42 ER - TY - GEN A1 - Jolaoso, Lateef O. A1 - Mehlitz, Patrick A1 - Zemkoho, Alain B. T1 - A fresh look at nonsmooth Levenberg-Marquardt methods with applications to bilevel optimization T2 - arXiv Y1 - 2023 U6 - https://doi.org/10.48550/arXiv.2305.19870 SP - 1 EP - 41 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 - Fabian, Marián A1 - Kruger, Alexander Y. A1 - Mehlitz, Patrick T1 - Nonsmooth multiplier, sum and intersection rules in non-Lipschitzian settings: decoupling approach revisited T2 - arXiv Y1 - 2023 UR - https://arxiv.org/abs/2305.08484 SP - 1 EP - 40 ER - TY - GEN A1 - Kanzow, Christian A1 - Krämer, Fabius A1 - Mehlitz, Patrick A1 - Wachsmuth, Gerd A1 - Werner, Frank T1 - A nonsmooth augmented Lagrangian method and its application to Poisson denoising and sparse control T2 - arXiv Y1 - 2023 U6 - https://doi.org/10.48550/arXiv.2304.06434 SP - 1 EP - 36 ER - TY - GEN A1 - De Marchi, Alberto A1 - Jia, Xiaoxi A1 - Kanzow, Christian A1 - Mehlitz, Patrick T1 - Constrained composite optimization and augmented Lagrangian methods T2 - Mathematical Programming Y1 - 2023 U6 - https://doi.org/10.1007/s10107-022-01922-4 SN - 0025-5610 SN - 1436-4646 VL - Vol. 201 IS - 1 SP - 863 EP - 896 ER - TY - GEN A1 - Jia, Xiaoxi A1 - Kanzow, Christian A1 - Mehlitz, Patrick T1 - Convergence analysis of the proximal gradient method in the presence of the Kurdyka-Łojasiewicz property without global Lipschitz assumptions T2 - SIAM Journal on Optimization Y1 - 2023 U6 - https://doi.org/10.1137/23m1548293 SN - 1095-7189 SN - 1052-6234 VL - 33 IS - 4 SP - 3038 EP - 3056 ER - TY - GEN A1 - Fabian, Marián A1 - Kruger, Alexander Y. A1 - Mehlitz, Patrick T1 - Fuzzy multiplier, sum and intersection rules in non-Lipschitzian settings: Decoupling approach revisited T2 - Journal of Mathematical Analysis and Applications Y1 - 2024 U6 - https://doi.org/10.1016/j.jmaa.2023.127985 SN - 1096-0813 SN - 0022-247X VL - 532 IS - 2 SP - 1 EP - 39 ER - TY - GEN A1 - Benko, Matúš A1 - Mehlitz, Patrick T1 - Why second-order sufficient conditions are, in a way, easy - or - revisiting calculus for second subderivatives T2 - Journal of Convex Analysis Y1 - 2023 UR - https://www.heldermann.de/JCA/JCA30/JCA302/jca30031.htm VL - 30 IS - 2 SP - 541 EP - 589 ER - TY - GEN A1 - Mehlitz, Patrick T1 - A simple proof of second-order sufficient optimality conditions in nonlinear semidefinite optimization T2 - Optimization Letters Y1 - 2023 U6 - https://doi.org/10.1007/s11590-023-02031-7 SN - 1862-4480 SN - 1862-4472 ER - TY - GEN A1 - Kanzow, Christian A1 - Krämer, Fabius A1 - Mehlitz, Patrick A1 - Wachsmuth, Gerd A1 - Werner, Frank T1 - Variational Poisson denoising via augmented Lagrangian methods T2 - ETNA - Electronic Transactions on Numerical Analysis Y1 - 2025 U6 - https://doi.org/10.1553/etna_vol63s33 VL - 63 SP - 33 EP - 62 ER - TY - GEN A1 - De Marchi, Alberto A1 - Mehlitz, Patrick T1 - Local properties and augmented Lagrangians in fully nonconvex composite optimization T2 - Journal of Nonsmooth Analysis and Optimizaton Y1 - 2024 U6 - https://doi.org/10.46298/jnsao-2024-12235 VL - 5 ER -