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 -