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SUPPLEMENTARY OPTIMALITY PROPERTIES OF THE restoration PHASE OF SEQUENTIAL gradient-restoration algorithms FOR OPTIMAL-CONTROL PROBLEMS

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1983年第1期41卷 169-184页

作者： MIELE, A WANG, T Professor of Astronautics and Mathematical Sciences Rice University Houston Texas Graduate Student Department of Mechanical Engineering Rice University Houston Texas

In this paper, sequential gradient-restoration algorithms for optimal control problems are considered, and attention is focused on the restoration phase. It is shown that the Lagrange multipliers associated with the restoration phase not only solve the auxiliary minimization problem of the restoration phase, but are also endowed with a supplementary optimality property: they minimize a special functional, quadratic in the multipliers, subject to the multiplier differential equations and boundary conditions, for given state, control, and parameter.

关键词： Numerical methods computing methods optimal control optimality properties supplementary optimality properties gradient algorithms gradient-restoration algorithms sequential gradient-restoration algorithms restoration algorithms general boundary conditions nondifferential constraints

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SUPPLEMENTARY OPTIMALITY PROPERTIES OF gradient-restoration algorithms FOR OPTIMAL-CONTROL PROBLEMS

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1980年第4期32卷 577-593页

作者： MIELE, A LIU, CT Professor of Astronautics and Mathematical Sciences Graduate Student Department of Mathematical Sciences Rice University Houston Texas

In this paper, sequential gradient-restoration algorithms for optimal control problems are considered, and attention is focused on the gradient phase. It is shown that the Lagrange multipliers associated with the gradient phase not only solve the auxiliary minimization problem of the gradient phase, but are also endowed with a supplementary optimality property: they minimize the error in the optimality conditions, subject to the multiplier differential equations and boundary conditions, for given state, control, and parameter.

关键词： Numerical methods computing methods optimal control optimality properties supplementary optimality properties gradient methods gradient-restoration algorithms sequential gradient-restoration algorithms general boundary conditions nondifferential constraints

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Supplementary Optimality Properties of the Family of gradient-restoration algorithms for Optimal Control Problems

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IFAC Proceedings Volumes 1983年第8期16卷 109-120页

作者： A. Miele T. Wang Departments of Mechanical Engineering and Mathematical Sciences Rice University Houston Texas Department of Mechanical Engineering Rice University Houston Texas

The problem of minimizing a functional subject to differential constraints, nondifferential constraints, initial constraints, and final constraints is considered within the frame of the family of gradient-restoration algorithms for optimal control problems. This family includes sequential gradient-restoration algorithms (SGRA) and combined gradient-restoration algorithms (CGRA). The system of Lagrange multipliers associated with (i) the gradient phase of SGRA, (ii) the restoration phase of SGRA, and (iii) the combined gradient-restoration phase of CGRA is examined. It is shown that, in each case, the Lagrange multipliers are endowed with a suppiementary optimality property: they minimize a special functional, quadratic in the multipliers, subject to the multiplier differential equations and boundary conditions, for given state, control, and parameter. These supplementary optimality properties have considerable computational implications: they allow one to reduce the study of an iteration of (i), (ii), (iii) to a mathematical programming problem involving a finite number of parameters as unknowns.

关键词： Numerical methods computing methods mathematical programming optimal control optimality properties suppiementary optimality properties restoration algorithms gradient algorithms gradient-restoration algorithms sequential gradient-restoration algorithms combined gradient-restoration algorithms general boundary conditions nondifferential constraints

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SEQUENTIAL CONJUGATE-gradient-restoration ALGORITHM FOR OPTIMAL CONTROL PROBLEMS .2. EXAMPLES

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1975年第2期15卷 223-244页

作者： HEIDEMAN, JC LEVY, AV RICE UNIV DEPT MECH & AEROSP ENGNHOUSTONTX 77001 RICE UNIV DEPT MAT SCIHOUSTONTX 77001

In Ref. 1, Heideman and Levy developed the sequential conjugate-gradient-restoration algorithm for minimizing a functional subject to differential constraints and terminal constraints. In this report, several numerical examples are presented, some pertaining to a quadratic functional subject to linear constraints and some pertaining to a nonquadratic functional subject to nonlinear constraints. These examples demonstrate the feasibility as well as the rapid convergence characteristic of the sequential conjugate-gradient-restoration algorithm.

关键词： Optimal control gradient methods conjugate-gradient methods numerical methods computing methods gradient-restoration algorithms

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SEQUENTIAL gradient-restoration ALGORITHM FOR OPTIMAL-CONTROL PROBLEMS WITH GENERAL BOUNDARY-CONDITIONS

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1978年第3期26卷 395-425页

作者： GONZALEZ, S MIELE, A Graduate Student Department of Electrical Engineering Rice University Houston Texas Professor of Astronautics and Mathematical Sciences Rice University Houston Texas

This paper considers the numerical solution of two classes of optimal control problems, called Problem P1 and Problem P2 for easy *** P1 involves a functionalI subject to differential constraints and general boundary conditions. It consists of finding the statex(t), the controlu(t), and the parameter π so that the functionalI is minimized, while the constraints and the boundary conditions are satisfied to a predetermined accuracy. Problem P2 extends Problem P1 to include nondifferential constraints to be satisfied everywhere along the interval of integration. algorithms are developed for both Problem P1 and Problem *** approach taken is a sequence of two-phase cycles, composed of a gradient phase and a restoration phase. The gradient phase involves one iteration and is designed to decrease the value of the functional, while the constraints are satisfied to first order. The restoration phase involves one or more iterations and is designed to force constraint satisfaction to a predetermined accuracy, while the norm squared of the variations of the control, the parameter, and the missing components of the initial state is *** principal property of both algorithms is that they produce a sequence of feasible suboptimal solutions: the functions obtained at the end of each cycle satisfy the constraints to a predetermined accuracy. Therefore, the values of the functionalI corresponding to any two elements of the sequence are *** stepsize of the gradient phase is determined by a one-dimensional search on the augmented functionalJ, while the stepsize of the restoration phase is obtained by a one-dimensional search on the constraint errorP. The gradient stepsize and the restoration stepsize are chosen so that the restoration phase preserves the descent property of the gradient phase. Therefore, the value of the functionalI at the end of any complete gradient-restoration cycle is smaller than the value of the same functional at the beg

关键词： Optimal control numerical methods computing methods gradient methods gradient-restoration algorithms sequential gradient-restoration algorithms general boundary conditions nondifferential constraints bounded control bounded state

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RECENT ADVANCES IN gradient algorithms FOR OPTIMAL CONTROL PROBLEMS

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1975年第5-6期17卷 361-430页

作者： MIELE, A RICE UNIV HOUSTONTX 77001

This paper summarizes recent advances in the area of gradient algorithms for optimal control problems, with particular emphasis on the work performed by the staff of the Aero-Astronautics Group of Rice *** following basic problem is considered: minimize a functionalI which depends on the statex(t), the controlu(t), and the parameter π. Here,I is a scalar,x ann-vector,u anm-vector, and π ap-vector. At the initial point, the state is prescribed. At the final point, the statex and the parameter π are required to satisfyq scalar relations. Along the interval of integration, the state, the control, and the parameter are required to satisfyn scalar differential ***, the sequential gradient-restoration algorithm and the combined gradient-restoration algorithm are presented. The descent properties of these algorithms are studied, and schemes to determine the optimum stepsize are discussed. Both of the above algorithms require the solution of a linear, two-point boundary-value problem at each iteration. Hence, a discussion of integration techniques is ***, a family of gradient-restoration algorithms is introduced. Not only does this family include the previous two algorithms as particular cases, but it allows one to generate several additional algorithms, namely, those with alternate restoration and optional ***, two modifications of the sequential gradient-restoration algorithm are presented in an effort to accelerate terminal convergence. In the first modification, the quadratic constraint imposed on the variations of the control is modified by the inclusion of a positive-definite weighting matrix (the matrix of the second derivatives of the Hamiltonian with respect to the control). The second modification is a conjugate-gradient extension of the sequential gradient-restoration ***, the addition of a nondifferential constraint, to be satisfied everywhere along the interval of integration, is considered. In theory, this

关键词： Survey papers gradient methods numerical methods computing methods calculus of variations optimal control gradient-restoration algorithms boundary-value problems bounded control problems bounded state problems nondifferential constraints

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NUMERICAL-SOLUTION OF MINIMAX PROBLEMS OF OPTIMAL-CONTROL .1.

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1982年第1期38卷 97-109页

作者： MIELE, A MOHANTY, BP VENKATARAMAN, P KUO, YM Present address: School of Systems Science Arkansas Technical University Russellville Arkansas Professor of Astronautics and Mathematical Sciences Senior Research Associate Graduate Student Graduate Student Department of Mechanical Engineering Rice University Houston Texas

This paper contains general transformation techniques useful to convert minimax problems of optimal control into the Mayer-Bolza problem of the calculus of variations [Problem (P)]. We consider two types of minimax problems: minimax problems of Type (Q), in which the minimax function depends on the state and does not depend on the control; and minimax problems of Type (R), in which the minimax function depends on both the state and the control. Both Problem (Q) and Problem (R) can be reduced to Problem (P).For Problem (Q), we exploit the analogy with a bounded-state problem in combination with a transformation of the Jacobson type. This requires the proper augmentation of the state vectorx(t), the control vectoru(t), and the parameter vector π, as well as the proper augmentation of the constraining relations. As a result of the transformation, the unknown minimax value of the performance index becomes a component of the parameter vector being *** Problem (R), we exploit the analogy with a bounded-control problem in combination with a transformation of the Valentine type. This requires the proper augmentation of the control vectoru(t) and the parameter vector π, as well as the proper augmentation of the constraining relations. As a result of the transformation, the unknown minimax value of the performance index becomes a component of the parameter vector being *** a subsequent paper (Part 2), the transformation techniques presented here are employed in conjunction with the sequential gradient-restoration algorithm for solving optimal control problems on a digital computer; both the single-subarc approach and the multiple-subarc approach are discussed.

关键词： Minimax problems minimax function minimax function depending on the state minimax function depending on the control optimal control minimax optimal control numerical methods computing methods transformation techniques gradient-restoration algorithms sequential gradient-restoration algorithms

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NUMERICAL-SOLUTION OF MINIMAX PROBLEMS OF OPTIMAL-CONTROL .2.

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1982年第1期38卷 111-135页

In a previous paper (Part 1), we presented general transformation techniques useful to convert minimax problems of optimal control into the Mayer-Bolza problem of the calculus of variations [Problem (P)]. We considered two types of minimax problems: minimax problems of Type (Q), in which the minimax function depends on the state and does not depend on the control; and minimax problems of Type (R), in which the minimax function depends on both the state and the control. Both Problem (Q) and Problem (R) can be reduced to Problem (P).In this paper, the transformation techniques presented in Part 1 are employed in conjunction with the sequential gradient-restoration algorithm for solving optimal control problems on a digital computer. Both the single-subarc approach and the multiple-subarc approach are employed. Three test problems characterized by known analytical solutions are solved *** is found that the combination of transformation techniques and sequential gradient-restoration algorithm yields numerical solutions which are quite close to the analytical solutions from the point of view of the minimax performance index. The relative differences between the numerical values and the analytical values of the minimax performance index are of order 10−3 if the single-subarc approach is employed. These relative differences are of order 10−4 or better if the multiple-subarc approach is employed.

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