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Automatic Differentiation And Spectral Projected Gradient Methods For Optimal Control Problems
, 1998
"... this paper is to show the application of these canonical formulas to optimal control processes being integrated by the RungeKutta family of numerical methods. There are many papers concerning numerical comparisions between automatic differentiation, finite differences and symbolic differentiation. ..."
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Cited by 12 (5 self)
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this paper is to show the application of these canonical formulas to optimal control processes being integrated by the RungeKutta family of numerical methods. There are many papers concerning numerical comparisions between automatic differentiation, finite differences and symbolic differentiation. See, for example, [1, 2, 6, 7, 21] among others. Another objective is to test the behavior of the spectral projected gradient methods introduced in [5]. These methods combine the classical projected gradient with two recently developed ingredients in optimization: (i) the nonmonotone line search schemes of Grippo, Lampariello and Lucidi ([24]), and (ii) the spectral steplength (introduced by Barzilai and Borwein ([3]) and analyzed by Raydan ([30, 31])). This choice of the steplength requires little computational work and greatly speeds up the convergence of gradient methods. The numerical experiments presented in [5], showing the high performance of these fast and easily implementable methods, motivate us to combine the spectral projected gradient methods with automatic differentiation. Both tools are used in this work for the development of codes for numerical solution of optimal control problems. In Section 2 of this paper, we apply the canonical formulas to the discrete version of the optimal control problem. In Section 3, we give a concise survey about spectral projected gradient algorithms. Section 4 presents some numerical experiments. Some final remarks are presented in Section 5. 2 CANONICAL FORMULAS The basic optimal control problem can be described as follows: Let a process governed by a system of ordinary differential equations be dx(t) dt = f(x(t); u(t); ); T 0 t T f ; (1) where x : [T 0 ; T f ] ! IR nx , u : [T 0 ; T f ] ! U ` IR nu , U compact, and 2 V ...
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"... Modeling and optimization of melting and solidification process A.F. Albou, V.I. Zubov Abstract. An optimal control problem is considered for twophase Stefan problem describing the process of melting and solidification. The problem is solved numerically by variation and finitedifference methods. T ..."
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Modeling and optimization of melting and solidification process A.F. Albou, V.I. Zubov Abstract. An optimal control problem is considered for twophase Stefan problem describing the process of melting and solidification. The problem is solved numerically by variation and finitedifference methods. The results are described and analyzed in detail. Some of them are presented as tables and plots.
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"... Abstract. The optimal control problem of the metal solidification in casting is considered. The process is modeled by a threedimensional twophase initialboundary value problem of the Stefan type. A numerical algorithm for solving the direct problem was presented in the first part of this article, ..."
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Abstract. The optimal control problem of the metal solidification in casting is considered. The process is modeled by a threedimensional twophase initialboundary value problem of the Stefan type. A numerical algorithm for solving the direct problem was presented in the first part of this article, published in [1]. The optimal control problem was solved numerically using the gradient method. The gradient of the cost function was found with the help of conjugate problem. The discreet conjugate problem was posed with the help of Fast Automatic Differentiation technique. Mathematics subject classification: 49J20, 93C20.