Abstract:
We consider a regularization of the classical Lagrange principle and Pontryagin maximum principle in convex programming, optimal control, and inverse problems. We discuss two basic questions, why a regularization of the classical optimality conditions (COCs) is necessary and what it gives, using the example of the “simplest” problems of constrained infinite-dimensional convex optimization. The so-called regularized COCs considered in the paper are expressed in terms of the regular classical Lagrange and Hamilton-Pontryagin functions and are sequential generalizations of their classical analogs. They (1) “overcome” the possible instability and infeasibility of the COCs, being regularizing algorithms for the solution of optimization problems, (2) are formulated as statements on the existence of bounded minimizing approximate solutions in the sense of Warga in the original problem and preserve the general structure of the COCs, and (3) lead to the COCs “in the limit.” All optimization problems in the paper depend on an additive parameter in the infinite-dimensional equality constraint (the perturbation method). As a result, it is possible to study the connection of regularized COCs with the subdifferential properties of the value functions of the optimization problems.
Citation:
M. I. Sumin, “Regularized Lagrange principle and Pontryagin maximum principle in optimal control and in inverse problems”, Trudy Inst. Mat. i Mekh. UrO RAN, 25, no. 1, 2019, 279–296
\Bibitem{Sum19}
\by M.~I.~Sumin
\paper Regularized Lagrange principle and Pontryagin maximum principle in optimal control and in inverse problems
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2019
\vol 25
\issue 1
\pages 279--296
\mathnet{http://mi.mathnet.ru/timm1616}
\crossref{https://doi.org/10.21538/0134-4889-2019-25-1-279-296}
\elib{https://elibrary.ru/item.asp?id=37051111}
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This publication is cited in the following 17 articles:
M. I. Sumin, “The perturbation method and a regularization of the Lagrange multiplier rule in convex problems for constrained extremum”, Proc. Steklov Inst. Math. (Suppl.), 325, suppl. 1 (2024), S194–S211
V. I. Sumin, M. I. Sumin, “Regulyarizatsiya klassicheskikh uslovii optimalnosti \žadachakh optimizatsii lineinykh raspredelennykh sistem volterrova tipa s potochechnymi fazovymi ogranicheniyami”, Vestnik rossiiskikh universitetov. Matematika, 29:148 (2024), 455–484
V. I. Sumin, M. I. Sumin, “Regulyarizatsiya klassicheskikh uslovii optimalnosti v zadachakh optimizatsii lineinykh sistem volterrova tipa s funktsionalnymi ogranicheniyami”, Vestnik rossiiskikh universitetov. Matematika, 28:143 (2023), 298–325
M. I. Sumin, “O roli mnozhitelei Lagranzha i dvoistvennosti v nekorrektnykh zadachakh na uslovnyi ekstremum. K 60-letiyu metoda regulyarizatsii Tikhonova”, Vestnik rossiiskikh universitetov. Matematika, 28:144 (2023), 414–435
M. I. Sumin, “O nekorrektnykh zadachakh, ekstremalyakh funktsionala Tikhonova i regulyarizovannykh printsipakh Lagranzha”, Vestnik rossiiskikh universitetov. Matematika, 27:137 (2022), 58–79
M. I. Sumin, “O regulyarizatsii klassicheskikh uslovii optimalnosti v vypuklom optimalnom upravlenii”, Materialy Voronezhskoi mezhdunarodnoi zimnei matematicheskoi shkoly «Sovremennye metody teorii funktsii i smezhnye problemy», Voronezh, 28 yanvarya – 2 fevralya 2021 g. Chast 2, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 207, VINITI RAN, M., 2022, 120–143
M. I. Sumin, “Printsip Lagranzha i printsip maksimuma Pontryagina v nekorrektnykh zadachakh optimalnogo upravleniya”, Materialy Voronezhskoi mezhdunarodnoi vesennei matematicheskoi shkoly «Sovremennye metody teorii kraevykh zadach. Pontryaginskie chteniya–XXXII», Voronezh, 3–9 maya 2021 g. Chast 1, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 208, VINITI RAN, M., 2022, 63–78
M. I. Sumin, “Metod vozmuschenii, subdifferentsialy negladkogo analiza i regulyarizatsiya pravila mnozhitelei Lagranzha v nelineinom optimalnom upravlenii”, Tr. IMM UrO RAN, 28, no. 3, 2022, 202–221
V. I. Sumin, M. I. Sumin, “O regulyarizatsii printsipa Lagranzha v zadachakh optimizatsii lineinykh raspredelennykh sistem volterrova tipa s operatornymi ogranicheniyami”, Izv. IMI UdGU, 59 (2022), 85–113
M. I. Sumin, “O regulyarizatsii nedifferentsialnoi teoremy Kuna–Takkera v nelineinoi zadache na uslovnyi ekstremum”, Vestnik rossiiskikh universitetov. Matematika, 27:140 (2022), 351–374
M. I. Sumin, “Regulyarizatsiya printsipa maksimuma Pontryagina v vypukloi zadache optimalnogo granichnogo upravleniya dlya parabolicheskogo uravneniya s operatornym ogranicheniem-ravenstvom”, Tr. IMM UrO RAN, 27, no. 2, 2021, 221–237
M. I. Sumin, “Printsip Lagranzha i ego regulyarizatsiya kak teoreticheskaya osnova ustoichivogo resheniya zadach optimalnogo upravleniya i obratnykh zadach”, Vestnik rossiiskikh universitetov. Matematika, 26:134 (2021), 151–171
V. I. Sumin, M. I. Sumin, “Regulyarizovannye klassicheskie usloviya optimalnosti v iteratsionnoi forme dlya vypuklykh zadach optimizatsii raspredelennykh sistem volterrova tipa”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 31:2 (2021), 265–284
M. I. Sumin, “O regulyarizatsii printsipa Lagranzha i postroenii obobschennykh minimiziruyuschikh posledovatelnostei v vypuklykh zadachakh uslovnoi optimizatsii”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 30:3 (2020), 410–428
F. A. Kuterin, “K voprosu o regulyarizatsii klassicheskikh uslovii optimalnosti v vypukloi zadache optimalnogo upravleniya c fazovymi ogranicheniyami”, Vestnik rossiiskikh universitetov. Matematika, 25:131 (2020), 263–273
M. I. Sumin, “Nedifferentsialnye teoremy Kuna–Takkera v zadachakh na
uslovnyi ekstremum i subdifferentsialy negladkogo analiza”, Vestnik rossiiskikh universitetov. Matematika, 25:131 (2020), 307–330
D. S. Solovjev, I. A. Solovjeva, Yu. V. Litovka, V. A. Nesterov, “Searching method for suboptimal action ensuring acceptable losses in the process quality”, J. Mach. Manuf. Reliab., 49:5 (2020), 429–438
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