Regularized classical optimality conditions in iterative form for convex optimization problems for distributed Volterra-type systems

V. I. Sumin   ; M. I. Sumin

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 31 (2021) no. 2, pp. 265-284

Résumé

We consider the regularization of the classical optimality conditions (COCs) — the Lagrange principle and the Pontryagin maximum principle — in a convex optimal control problem with functional constraints of equality and inequality type. The system to be controlled is given by a general linear functional-operator equation of the second kind in the space $L^m_2$, the main operator of the right-hand side of the equation is assumed to be quasinilpotent. The objective functional of the problem is strongly convex. Obtaining regularized COCs in iterative form is based on the use of the iterative dual regularization method. The main purpose of the regularized Lagrange principle and the Pontryagin maximum principle obtained in the work in iterative form is stable generation of minimizing approximate solutions in the sense of J. Warga. Regularized COCs in iterative form are formulated as existence theorems in the original problem of minimizing approximate solutions. They “overcome” the ill-posedness properties of the COCs and are regularizing algorithms for solving optimization problems. As an illustrative example, we consider an optimal control problem associated with a hyperbolic system of first-order differential equations.

Détail

Keywords: convex optimal control, distributed system, functional-operator equation of Volterra type, ill-posedness, iterative regularization, duality, minimizing approximate solution, regularizing operator, Lagrange principle, Pontryagin maximum principle.

Bibliographie

[1] Alekseev V. M., Tikhomirov V. M., Fomin S. V., Optimal Control, Springer, Boston, 1987 | DOI | MR | MR

[2] Vasil'ev F. P., Optimization methods, v. 1, 2, Moscow Center for Continuous Mathematical Education, M., 2011

[3] Sumin M. I., “Regularized parametric Kuhn–Tucker theorem in a Hilbert space”, Computational Mathematics and Mathematical Physics, 51:9 (2011), 1489–1509 | DOI | MR | Zbl

[4] Sumin M. I., “On the stable sequential Kuhn–Tucker theorem and its applications”, Applied Mathematics, 3:10A (2012), 1334–1350 | DOI

[5] Sumin M. I., “Regularized Lagrange principle and Pontryagin maximum principle in optimal control and in inverse problems”, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 25, no. 1, 2019, 279–296 (in Russian) | DOI

[6] Sumin M. I., “On the regularization of the classical optimality conditions in convex optimal control problems”, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 26, no. 2, 2020, 252–269 (in Russian) | DOI

[7] Warga J., Optimal control of differential and functional equations, Academic Press, New York–London, 1972 | MR | Zbl

[8] Tikhonov A. N., Arsenin V. Ya., Solutions of ill-posed problems, Halsted Press, Washington–Winston–New York, 1977 | MR | MR

[9] Sumin M. I., “Duality-based regularization in a linear convex mathematical programming problem”, Computational Mathematics and Mathematical Physics, 47:4 (2007), 579–600 | DOI | MR | Zbl

[10] Sumin V. I., Functional Volterra equations in the theory of optimal control of distributed systems, Nizhny Novgorod University, Nizhny Novgorod, 1992

[11] Sumin V. I., Chernov A. V., “Operators in spaces of measurable functions: the Volterra property and quasinilpotency”, Differential Equations, 34:10 (1998), 1403–1411 | MR | Zbl

[12] Gokhberg I. Ts., Krein M. G., Theory and applications of Volterra operators in Hilbert space, American Mathematical Society, Providence, 1970 | MR | Zbl

[13] Sumin V. I., “Volterra functional-operator equations in the theory of optimal control of distributed systems”, Sov. Math., Dokl., 39:2 (1989), 374–378 | MR | Zbl

[14] Sumin V., “Volterra functional-operator equations in the theory of optimal control of distributed systems”, IFAC-PapersOnLine, 51:32 (2018), 759–764 | DOI

[15] Sumin V. I., “Controlled Volterra functional equations and the contraction mapping principle”, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 25, no. 1, 2019, 262–278 (in Russian) | DOI

[16] Kuterin F. A., Sumin M. I., “The regularized iterative Pontryagin maximum principle in optimal control. II. Optimization of a distributed system”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 27:1 (2017), 26–41 (in Russian) | DOI | MR | Zbl

[17] Ioffe A. D., Tikhomirov V. M., Theory of extremal problems, North-Holland Publishing Company, Amsterdam–New York–Oxford, 1979 | MR | Zbl

[18] Dmitruk A. V., Convex analysis. Elementary introductory course, MAKS Press, M., 2012