Abstract
This paper addresses the nonconvex optimal control (OC) problem with the cost functional and inequality constraint given by the functionals of Bolza. All the functions in the statement of the problem are state-DC, i.e. presented by a difference of the state-convex functions. Meanwhile, the control system is state-linear. Further, with the help of the Exact Penalization Theory we propose the state-DC form of the penalized cost functional and, using the linearization with respect to the basic nonconvexity of the penalized problem, we study the linearized OC problem.
On this basis, we develop a general scheme of the special Local Search Method with a varying penalty parameter. Finally, we address the convergence of the proposed scheme.
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References
Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Interscience, New York (1976)
Marchuk, G.I.: Mathematical Modeling in the Environmental Problem. Nauka, Moscow (1982). (in Russian)
Chernousko, F.L., Ananievski, I.M., Reshmin, S.A.: Control of Nonlinear Dynamical Systems: Methods and Applications. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-70784-4
Chernousko, F.L., Banichuk, N.V.: Variational Problems of Mechanics and Control. Numerical Methods. Nauka, Moscow (1973). (in Russian)
Kurzhanski, A.B., Varaiya, P.: Dynamics and Control of Trajectory Tubes: Theory and Computation. Birkhauser, Boston (2014)
Kurzhanski, A.B.: Control and Observation Under Conditions of Uncertainty. Nauka, Moscow (1977). (in Russian)
Vasil’ev, F.P.: Optimization Methods. Factorial Press, Moscow (2002). (in Russian)
Fedorenko, R.P.: Approximate Solution of Optimal Control Problems. Nauka, Moscow (1978). (in Russian)
Gabasov, R., Kirillova, F.M.: Maximum’s Principle in the Optimal Control Theory. Nauka i Technika, Minsk (1974). (in Russian)
Vasiliev, O.V.: Optimization Methods. Word Federation Publishing Company, Atlanta (1996)
Srochko, V.A.: Iterative Solution of Optimal Control Problems. Fizmatlit, Moscow (2000). (in Russian)
Hiriart-Urruty, J.-B., Lemarechal, C.: Convex Analysis and Minimization Algorithms. Springer, Heidelberg (1993). https://doi.org/10.1007/978-3-662-02796-7
Hiriart-Urruty, J.-B.: Generalized differentiability, duality and optimization for problem dealing with difference of convex functions. In: Ponstein, J. (ed.) Convexity and Duality in Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 256, pp. 37–69. Springer, Heidelberg (1985). https://doi.org/10.1007/978-3-642-45610-7_3
Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches. Springer, Heidelberg (1996). https://doi.org/10.1007/978-3-662-02598-7
Rockafellar, R.: Convex Analysis. Princeton University Press, Princeton (1970)
Eremin, I.: The penalty method in convex programming. Soviet Math. Dokl. 8, 459–462 (1966)
Zangwill, W.: Non-linear programming via penalty functions. Manage. Sci. 13(5), 344–358 (1967)
Zaslavski, A.J.: Exact penalty property in optimization with mixed constraints via variational analysis. SIAM J. Optim. 23(1), 170–187 (2013)
Burke, J.: An exact penalization viewpoint of constrained optimization. SIAM J. Control Optim. 29(4), 968–998 (1991)
Dolgopolik, M.V.: A unifying theory of exactness of linear penalty functions. Optimization 65(6), 1167–1202 (2016)
Dolgopolik, M.V., Fominyh, A.V.: Exact penalty functions for optimal control problems I: Main theorem and free-endpoint problems. Optim. Control Appl. Meth. 40, 1018–1044 (2019)
Izmailov, A.F., Solodov, M.V.: Newton-Type Methods for Optimization and Variational Problems. SSORFE. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-04247-3
Strekalovsky, A.S.: On solving optimization problems with hidden nonconvex structures. In: Themistocles, M., Floudas, C.A., Butenko, S. (eds.) Optimization in Science and Engineering, pp. 465–502. Springer, New York (2014). https://doi.org/10.1007/978-3-642-45610-7_3
Strekalovsky, A.S.: Global optimality conditions for optimal control problems with functions of A.D. Alexandrov. J. Optim. Theory Appl. 159, 297–321 (2013)
Strekalovsky, A.S.: Maximizing a state convex Lagrange functional in optimal control. Autom. Remote Control 73(6), 949–961 (2012)
Strekalovsky, A.S.: Elements of Nonconvex Optimization. Nauka, Novosibirsk (2003). (in Russian)
Strekalovsky, A.S.: Global optimality conditions and exact penalization. Optim. Lett. 13(2), 597–615 (2019). https://doi.org/10.1007/s11590-017-1214-x
Strekalovsky, A.S.: Global optimality conditions in nonconvex optimization. J. Optim. Theory Appl. 173(3), 770–792 (2017)
Strekalovsky, A.S.: New global optimality conditions in a problem with DC constraints. Trudy Inst. Mat. Mekh. UrO RAN 25, 245–261 (2019). (in Russian)
Strekalovsky, A.S., Yanulevich, M.V.: On global search in nonconvex optimal control problems. J. Global Optim. 65(1), 119–135 (2016). https://doi.org/10.1007/s10898-015-0321-4
Strekalovsky, A.S., Yanulevich, M.V.: Global search in the optimal control problem with a terminal objective functional represented as a difference of two convex functions. Comput. Math. Math. Phys. 48(7), 1119–1132 (2008)
Strekalovsky, A.S.: Local search for nonsmooth DC optimization with DC equality and inequality constraints. In: Bagirov, A.M., Gaudioso, M., Karmitsa, N., Mäkelä, M.M., Taheri, S. (eds.) Numerical Nonsmooth Optimization. SSORFE, pp. 229–261. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-34910-3_7
Strekalovsky, A.S., Minarchenko, I.M.: A local search method for optimization problem with DC inequality constraints. Appl. Math. Model. 58, 229–244 (2018)
Strekalovsky, A.S.: On local search in DC optimization problems. Appl. Math. Comput. 255, 73–83 (2015)
Acknowledgement
The research was funded by the Ministry of Education and Science of the Russian Federation within the framework of the project “Theoretical foundations, methods and high-performance algorithms for continuous and discrete optimization to support interdisciplinary research” (No. of state registration: 121041300065-9).
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Strekalovsky, A.S. (2021). A Local Search Scheme for the Inequality-Constrained Optimal Control Problem. In: Pardalos, P., Khachay, M., Kazakov, A. (eds) Mathematical Optimization Theory and Operations Research. MOTOR 2021. Lecture Notes in Computer Science(), vol 12755. Springer, Cham. https://doi.org/10.1007/978-3-030-77876-7_2
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