@misc{Mehlitz, author = {Mehlitz, Patrick}, title = {Asymptotic regularity for Lipschitzian nonlinear optimization problems with applications to complementarity-constrained and bilevel programming}, series = {arXiv}, journal = {arXiv}, pages = {1 -- 43}, abstract = {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.}, language = {en} } @misc{KanzowMehlitz, author = {Kanzow, Christian and Mehlitz, Patrick}, title = {Convergence properties of monotone and nonmonotone proximal gradient methods revisited}, series = {arXiv}, journal = {arXiv}, pages = {1 -- 21}, abstract = {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.}, language = {en} } @misc{Mehlitz, author = {Mehlitz, Patrick}, title = {Asymptotic regularity for Lipschitzian nonlinear optimization problems with applications to complementarity-constrained and bilevel programming}, series = {Optimization}, volume = {72}, journal = {Optimization}, number = {1}, issn = {1029-4945}, doi = {10.1080/02331934.2022.2031190}, pages = {277 -- 320}, language = {en} } @misc{JiaKanzowMehlitzetal., author = {Jia, Xiaoxi and Kanzow, Christian and Mehlitz, Patrick and Wachsmuth, Gerd}, title = {An Augmented Lagrangian Method for Optimization Problems with Structured Geometric Constraints}, series = {Mathematical Programming}, volume = {199}, journal = {Mathematical Programming}, doi = {10.1007/s10107-022-01870-z}, pages = {1365 -- 1415}, language = {en} } @misc{KanzowMehlitz, author = {Kanzow, Christian and Mehlitz, Patrick}, title = {Convergence properties of monotone and nonmonotone proximal gradient methods revisited}, series = {Journal of Optimization Theory and Applications}, volume = {195}, journal = {Journal of Optimization Theory and Applications}, number = {2}, issn = {1573-2878}, doi = {10.1007/s10957-022-02101-3}, pages = {624 -- 646}, language = {en} } @misc{KrugerMehlitz, author = {Kruger, Alexander Y. and Mehlitz, Patrick}, title = {Optimality conditions, approximate stationarity, and applications - a story beyond Lipschitzness}, series = {Control, Optimisation and Calculus of Variations (ESAIM-COCV)}, volume = {28}, journal = {Control, Optimisation and Calculus of Variations (ESAIM-COCV)}, issn = {1262-3377}, doi = {10.1051/cocv/2022024}, pages = {35}, language = {en} } @misc{MehlitzWachsmuth, author = {Mehlitz, Patrick and Wachsmuth, Gerd}, title = {Subdifferentiation of nonconvex sparsity-promoting functionals on Lebesgue spaces}, series = {SIAM Journal on Control and Optimization}, volume = {60}, journal = {SIAM Journal on Control and Optimization}, number = {3}, issn = {1095-7138}, doi = {10.1137/21m1435173}, pages = {1819 -- 1839}, language = {en} } @misc{Mehlitz, author = {Mehlitz, Patrick}, title = {A simple proof of second-order sufficient optimality conditions in nonlinear semidefinite optimization}, series = {arXiv}, journal = {arXiv}, pages = {1 -- 11}, language = {en} } @misc{MehlitzGoettlichSchillinger, author = {Mehlitz, Patrick and G{\"o}ttlich, Simone and Schillinger, Thomas}, title = {Inverse demand tracking in transportation networks}, series = {arXiv}, journal = {arXiv}, pages = {1 -- 21}, language = {en} } @misc{DeMarchiJiaKanzowetal., author = {De Marchi, Alberto and Jia, Xiaoxi and Kanzow, Christian and Mehlitz, Patrick}, title = {Constrained structured optimization and augmented Lagrangian proximal methods}, series = {arXiv}, journal = {arXiv}, pages = {1 -- 44}, language = {en} } @misc{BenkoMehlitz, author = {Benko, Mat{\´u}š and Mehlitz, Patrick}, title = {On the directional asymptotic approach in optimization theory Part A: approximate, M- , and mixed-order stationarity}, series = {arXiv}, journal = {arXiv}, pages = {1 -- 40}, language = {en} }