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 -