We consider a multilevel defense system on a given direction. This is a particular model of terminal discrete optimal control that may be solved by the gradient descent method. The main difficulty is the nondifferentiability of the Lipschitzian functions in the right-hand sides of the equation of motion and their derivatives with respect to all variables, which leads to an ill-posed problem when applying the classical results on differentiability of the terminal function and constructing its gradient from the conjugate system. A method is proposed for the solution of the problem by averaging the right-hand sides in combination with the stochastic gradient projection method. The study develops Germeier’s defense-attack model by allowing for a multilevel defense structure, which in general leads to the synthesis problem for a discrete optimal control system.
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Translated from Prikladnaya Matematika i Informatika, No. 49, 2015, pp. 80–96.
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Reshetov, V.Y., Perevozchikov, A.G. & Lesik, I.A. A Model of Overpowering a Multilevel Defense System by Attack. Comput Math Model 27, 254–269 (2016). https://doi.org/10.1007/s10598-016-9319-0
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DOI: https://doi.org/10.1007/s10598-016-9319-0
Keywords
- discrete optimal control
- objective-function differentiability conditions
- conjugate system
- structure of the objective function gradient
- averaging of the right-hand sides
- structure of the derivatives of the averaged functions
- combined gradient projection method and Polyak method
- randomization of the combined gradient descent method
- stochastic gradient of the average problem
- almost-sure convergence of the randomized procedure