Pub Date : 2023-02-11DOI: 10.46298/jnsao-2023-10834
Livia M. Betz
Motivated by fatigue damage models, this paper addresses optimal control problems governed by a non-smooth system featuring two non-differentiable mappings. This consists of a coupling between a doubly non-smooth history-dependent evolution and an elliptic PDE. After proving the directional differentiability of the associated solution mapping, an optimality system which is stronger than the one obtained by classical smoothening procedures is derived. If one of the non-differentiable mappings becomes smooth, the optimality conditions are of strong stationary type, i.e., equivalent to the primal necessary optimality condition.
{"title":"Optimal Control of a Viscous Two-Field Damage Model with Fatigue","authors":"Livia M. Betz","doi":"10.46298/jnsao-2023-10834","DOIUrl":"https://doi.org/10.46298/jnsao-2023-10834","url":null,"abstract":"Motivated by fatigue damage models, this paper addresses optimal control problems governed by a non-smooth system featuring two non-differentiable mappings. This consists of a coupling between a doubly non-smooth history-dependent evolution and an elliptic PDE. After proving the directional differentiability of the associated solution mapping, an optimality system which is stronger than the one obtained by classical smoothening procedures is derived. If one of the non-differentiable mappings becomes smooth, the optimality conditions are of strong stationary type, i.e., equivalent to the primal necessary optimality condition.","PeriodicalId":250939,"journal":{"name":"Journal of Nonsmooth Analysis and Optimization","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133399019","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-09DOI: 10.46298/jnsao-2023-10290
A. Marchi
We address composite optimization problems, which consist in minimizing the sum of a smooth and a merely lower semicontinuous function, without any convexity assumptions. Numerical solutions of these problems can be obtained by proximal gradient methods, which often rely on a line search procedure as globalization mechanism. We consider an adaptive nonmonotone proximal gradient scheme based on an averaged merit function and establish asymptotic convergence guarantees under weak assumptions, delivering results on par with the monotone strategy. Global worst-case rates for the iterates and a stationarity measure are also derived. Finally, a numerical example indicates the potential of nonmonotonicity and spectral approximations.
{"title":"Proximal gradient methods beyond monotony","authors":"A. Marchi","doi":"10.46298/jnsao-2023-10290","DOIUrl":"https://doi.org/10.46298/jnsao-2023-10290","url":null,"abstract":"We address composite optimization problems, which consist in minimizing the\u0000sum of a smooth and a merely lower semicontinuous function, without any\u0000convexity assumptions. Numerical solutions of these problems can be obtained by\u0000proximal gradient methods, which often rely on a line search procedure as\u0000globalization mechanism. We consider an adaptive nonmonotone proximal gradient\u0000scheme based on an averaged merit function and establish asymptotic convergence\u0000guarantees under weak assumptions, delivering results on par with the monotone\u0000strategy. Global worst-case rates for the iterates and a stationarity measure\u0000are also derived. Finally, a numerical example indicates the potential of\u0000nonmonotonicity and spectral approximations.","PeriodicalId":250939,"journal":{"name":"Journal of Nonsmooth Analysis and Optimization","volume":"259 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129456191","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-12DOI: 10.46298/jnsao-2022-7269
B. Brogliato, Alexandre Rocca
paper 7269 This article is largely concerned with the time-discretization of descriptor-variable systems coupled to with complementarity constraints. They are named descriptor-variable linear complementarity systems (DVLCS). More speci cally passive DVLCS with minimal state space representation are studied. The Euler implicit discretization of DVLCS is analysed: the one-step non-smooth problem (OSNSP), that is a generalized equation, is shown to be well-posed under some conditions. Then the convergence of the discretized solutions is studied. Several examples illustrate the applicability and the limitations of the developments.
{"title":"Analysis of the implicit Euler time-discretization of passive linear descriptor complementarity systems","authors":"B. Brogliato, Alexandre Rocca","doi":"10.46298/jnsao-2022-7269","DOIUrl":"https://doi.org/10.46298/jnsao-2022-7269","url":null,"abstract":"paper 7269\u0000 This article is largely concerned with the time-discretization of descriptor-variable systems coupled to with complementarity constraints. They are named descriptor-variable linear complementarity systems (DVLCS). More speci cally passive DVLCS with minimal state space representation are studied. The Euler implicit discretization of DVLCS is analysed: the one-step non-smooth problem (OSNSP), that is a generalized equation, is shown to be well-posed under some conditions. Then the convergence of the discretized solutions is studied. Several examples illustrate the applicability and the limitations of the developments.","PeriodicalId":250939,"journal":{"name":"Journal of Nonsmooth Analysis and Optimization","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128742433","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-16DOI: 10.46298/jnsao-2023-10164
Paul Manns, Mirko Hahn, C. Kirches, S. Leyffer, S. Sager
Binary trust-region steepest descent (BTR) and combinatorial integral approximation (CIA) are two recently investigated approaches for the solution of optimization problems with distributed binary-/discrete-valued variables (control functions). We show improved convergence results for BTR by imposing a compactness assumption that is similar to the convergence theory of CIA. As a corollary we conclude that BTR also constitutes a descent algorithm on the continuous relaxation and its iterates converge weakly-$^*$ to stationary points of the latter. We provide computational results that validate our findings. In addition, we observe a regularizing effect of BTR, which we explore by means of a hybridization of CIA and BTR.
{"title":"On Convergence of Binary Trust-Region Steepest Descent","authors":"Paul Manns, Mirko Hahn, C. Kirches, S. Leyffer, S. Sager","doi":"10.46298/jnsao-2023-10164","DOIUrl":"https://doi.org/10.46298/jnsao-2023-10164","url":null,"abstract":"Binary trust-region steepest descent (BTR) and combinatorial integral\u0000approximation (CIA) are two recently investigated approaches for the solution\u0000of optimization problems with distributed binary-/discrete-valued variables\u0000(control functions). We show improved convergence results for BTR by imposing a\u0000compactness assumption that is similar to the convergence theory of CIA. As a\u0000corollary we conclude that BTR also constitutes a descent algorithm on the\u0000continuous relaxation and its iterates converge weakly-$^*$ to stationary\u0000points of the latter. We provide computational results that validate our\u0000findings. In addition, we observe a regularizing effect of BTR, which we\u0000explore by means of a hybridization of CIA and BTR.","PeriodicalId":250939,"journal":{"name":"Journal of Nonsmooth Analysis and Optimization","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132333235","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-19DOI: 10.46298/jnsao-2022-8733
D. Wachsmuth, G. Wachsmuth
We study no-gap second-order optimality conditions for a non-uniformly convex and non-smooth integral functional. The integral functional is extended to the space of measures. The obtained second-order derivatives contain integrals on lower-dimensional manifolds. The proofs utilize the convex pre-conjugate, which is an integral functional on the space of continuous functions. Applications to non-smooth optimal control problems are given.
{"title":"Second-order conditions for non-uniformly convex integrands: quadratic\u0000 growth in $L^1$","authors":"D. Wachsmuth, G. Wachsmuth","doi":"10.46298/jnsao-2022-8733","DOIUrl":"https://doi.org/10.46298/jnsao-2022-8733","url":null,"abstract":"We study no-gap second-order optimality conditions for a non-uniformly convex\u0000and non-smooth integral functional. The integral functional is extended to the\u0000space of measures. The obtained second-order derivatives contain integrals on\u0000lower-dimensional manifolds. The proofs utilize the convex pre-conjugate, which\u0000is an integral functional on the space of continuous functions. Applications to\u0000non-smooth optimal control problems are given.","PeriodicalId":250939,"journal":{"name":"Journal of Nonsmooth Analysis and Optimization","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131907653","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-09-28DOI: 10.46298/jnsao-2022-8537
G. Wachsmuth
We consider a generalized equation governed by a strongly monotone and Lipschitz single-valued mapping and a maximally monotone set-valued mapping in a Hilbert space. We are interested in the sensitivity of solutions w.r.t. perturbations of both mappings. We demonstrate that the directional differentiability of the solution map can be verified by using the directional differentiability of the single-valued operator and of the resolvent of the set-valued mapping. The result is applied to quasi-generalized equations in which we have an additional dependence of the solution within the set-valued part of the equation.
{"title":"From resolvents to generalized equations and quasi-variational\u0000 inequalities: existence and differentiability","authors":"G. Wachsmuth","doi":"10.46298/jnsao-2022-8537","DOIUrl":"https://doi.org/10.46298/jnsao-2022-8537","url":null,"abstract":"We consider a generalized equation governed by a strongly monotone and\u0000Lipschitz single-valued mapping and a maximally monotone set-valued mapping in\u0000a Hilbert space. We are interested in the sensitivity of solutions w.r.t.\u0000perturbations of both mappings. We demonstrate that the directional\u0000differentiability of the solution map can be verified by using the directional\u0000differentiability of the single-valued operator and of the resolvent of the\u0000set-valued mapping. The result is applied to quasi-generalized equations in\u0000which we have an additional dependence of the solution within the set-valued\u0000part of the equation.","PeriodicalId":250939,"journal":{"name":"Journal of Nonsmooth Analysis and Optimization","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126198305","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-04DOI: 10.46298/jnsao-2021-7156
S. Walther
The paper is concerned with an optimal control problem governed by the equations of elasto plasticity with linear kinematic hardening and the inertia term at small strain. The objective is to optimize the displacement field and plastic strain by controlling volume forces. The idea given in [10] is used to transform the state equation into an evolution variational inequality (EVI) involving a certain maximal monotone operator. Results from [27] are then used to analyze the EVI. A regularization is obtained via the Yosida approximation of the maximal monotone operator, this approximation is smoothed further to derive optimality conditions for the smoothed optimal control problem.
{"title":"Optimal Control of Plasticity with Inertia","authors":"S. Walther","doi":"10.46298/jnsao-2021-7156","DOIUrl":"https://doi.org/10.46298/jnsao-2021-7156","url":null,"abstract":"The paper is concerned with an optimal control problem governed by the\u0000equations of elasto plasticity with linear kinematic hardening and the inertia\u0000term at small strain. The objective is to optimize the displacement field and\u0000plastic strain by controlling volume forces. The idea given in [10] is used to\u0000transform the state equation into an evolution variational inequality (EVI)\u0000involving a certain maximal monotone operator. Results from [27] are then used\u0000to analyze the EVI. A regularization is obtained via the Yosida approximation\u0000of the maximal monotone operator, this approximation is smoothed further to\u0000derive optimality conditions for the smoothed optimal control problem.","PeriodicalId":250939,"journal":{"name":"Journal of Nonsmooth Analysis and Optimization","volume":"102 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130194783","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-11-09DOI: 10.46298/jnsao-2021-6903
Felix Harder
It is known in the literature that local minimizers of mathematical programs with complementarity constraints (MPCCs) are so-called M-stationary points, if a weak MPCC-tailored Guignard constraint qualification (called MPCC-GCQ) holds. In this paper we present a new elementary proof for this result. Our proof is significantly simpler than existing proofs and does not rely on deeper technical theory such as calculus rules for limiting normal cones. A crucial ingredient is a proof of a (to the best of our knowledge previously open) conjecture, which was formulated in a Diploma thesis by Schinabeck.
{"title":"A new elementary proof for M-stationarity under MPCC-GCQ for\u0000 mathematical programs with complementarity constraints","authors":"Felix Harder","doi":"10.46298/jnsao-2021-6903","DOIUrl":"https://doi.org/10.46298/jnsao-2021-6903","url":null,"abstract":"It is known in the literature that local minimizers of mathematical programs\u0000with complementarity constraints (MPCCs) are so-called M-stationary points, if\u0000a weak MPCC-tailored Guignard constraint qualification (called MPCC-GCQ) holds.\u0000In this paper we present a new elementary proof for this result. Our proof is\u0000significantly simpler than existing proofs and does not rely on deeper\u0000technical theory such as calculus rules for limiting normal cones. A crucial\u0000ingredient is a proof of a (to the best of our knowledge previously open)\u0000conjecture, which was formulated in a Diploma thesis by Schinabeck.","PeriodicalId":250939,"journal":{"name":"Journal of Nonsmooth Analysis and Optimization","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126104308","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-08-19DOI: 10.46298/jnsao-2021-7215
Mat'uvs Benko, P. Mehlitz
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. Comment: 34 pages
{"title":"On implicit variables in optimization theory","authors":"Mat'uvs Benko, P. Mehlitz","doi":"10.46298/jnsao-2021-7215","DOIUrl":"https://doi.org/10.46298/jnsao-2021-7215","url":null,"abstract":"Implicit variables of a mathematical program are variables which do not need\u0000to be optimized but are used to model feasibility conditions. They frequently\u0000appear in several different problem classes of optimization theory comprising\u0000bilevel programming, evaluated multiobjective optimization, or nonlinear\u0000optimization problems with slack variables. In order to deal with implicit\u0000variables, they are often interpreted as explicit ones. Here, we first point\u0000out that this is a light-headed approach which induces artificial locally\u0000optimal solutions. Afterwards, we derive various Mordukhovich-stationarity-type\u0000necessary optimality conditions which correspond to treating the implicit\u0000variables as explicit ones on the one hand, or using them only implicitly to\u0000model the constraints on the other. A detailed comparison of the obtained\u0000stationarity conditions as well as the associated underlying constraint\u0000qualifications will be provided. Overall, we proceed in a fairly general\u0000setting relying on modern tools of variational analysis. Finally, we apply our\u0000findings to different well-known problem classes of mathematical optimization\u0000in order to visualize the obtained theory.\u0000\u0000 Comment: 34 pages","PeriodicalId":250939,"journal":{"name":"Journal of Nonsmooth Analysis and Optimization","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132141337","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-07-29DOI: 10.46298/jnsao-2021-6673
L. C. Hegerhorst-Schultchen, C. Kirches, M. Steinbach
This work continues an ongoing effort to compare non-smooth optimization problems in abs-normal form to Mathematical Programs with Complementarity Constraints (MPCCs). We study general Nonlinear Programs with equality and inequality constraints in abs-normal form, so-called Abs-Normal NLPs, and their relation to equivalent MPCC reformulations. We introduce the concepts of Abadie's and Guignard's kink qualification and prove relations to MPCC-ACQ and MPCC-GCQ for the counterpart MPCC formulations. Due to non-uniqueness of a specific slack reformulation suggested in [10], the relations are non-trivial. It turns out that constraint qualifications of Abadie type are preserved. We also prove the weaker result that equivalence of Guginard's (and Abadie's) constraint qualifications for all branch problems hold, while the question of GCQ preservation remains open. Finally, we introduce M-stationarity and B-stationarity concepts for abs-normal NLPs and prove first order optimality conditions corresponding to MPCC counterpart formulations.
{"title":"Relations between Abs-Normal NLPs and MPCCs. Part 2: Weak Constraint\u0000 Qualifications","authors":"L. C. Hegerhorst-Schultchen, C. Kirches, M. Steinbach","doi":"10.46298/jnsao-2021-6673","DOIUrl":"https://doi.org/10.46298/jnsao-2021-6673","url":null,"abstract":"This work continues an ongoing effort to compare non-smooth optimization\u0000problems in abs-normal form to Mathematical Programs with Complementarity\u0000Constraints (MPCCs). We study general Nonlinear Programs with equality and\u0000inequality constraints in abs-normal form, so-called Abs-Normal NLPs, and their\u0000relation to equivalent MPCC reformulations. We introduce the concepts of\u0000Abadie's and Guignard's kink qualification and prove relations to MPCC-ACQ and\u0000MPCC-GCQ for the counterpart MPCC formulations. Due to non-uniqueness of a\u0000specific slack reformulation suggested in [10], the relations are non-trivial.\u0000It turns out that constraint qualifications of Abadie type are preserved. We\u0000also prove the weaker result that equivalence of Guginard's (and Abadie's)\u0000constraint qualifications for all branch problems hold, while the question of\u0000GCQ preservation remains open. Finally, we introduce M-stationarity and\u0000B-stationarity concepts for abs-normal NLPs and prove first order optimality\u0000conditions corresponding to MPCC counterpart formulations.\u0000","PeriodicalId":250939,"journal":{"name":"Journal of Nonsmooth Analysis and Optimization","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132775097","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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