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  • 1
    ISSN: 1573-2878
    Keywords: Mathematical programming ; function minimization ; method of dual matrices ; computing methods ; numerical methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In Ref. 2, four algorithms of dual matrices for function minimization were introduced. These algorithms are characterized by the simultaneous use of two matrices and by the property that the one-dimensional search for the optimal stepsize is not needed for convergence. For a quadratic function, these algorithms lead to the solution in at mostn+1 iterations, wheren is the number of variables in the function. Since the one-dimensional search is not needed, the total number of gradient evaluations for convergence is at mostn+2. In this paper, the above-mentioned algorithms are tested numerically by using five nonquadratic functions. In order to investigate the effects of the stepsize on the performances of these algorithms, four schemes for the stepsize factor are employed, two corresponding to small-step processes and two corresponding to large-step processes. The numerical results show that, in spite of the wide range employed in the choice of the stepsize factor, all algorithms exhibit satisfactory convergence properties and compare favorably with the corresponding quadratically convergent algorithms using one-dimensional searches for optimal stepsizes.
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  • 2
    ISSN: 1573-2878
    Keywords: Two-point boundary-value problems ; differential equations ; Newton-Raphson methods ; computing methods ; numerical methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A method based on matching a zero of the right-hand side of the differential equations, in a two-point boundary-value problem, to the boundary conditions is suggested. Effectiveness of the procedure is tested on three nonlinear, two-point boundary-value problems.
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  • 3
    ISSN: 1573-2878
    Keywords: Calculus of variations ; optimal control ; computing methods ; numerical methods ; boundary-value problems ; modified quasilinearization algorithm ; nondifferential constraints
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper considers the numerical solution of optimal control problems involving a functionalI subject to differential constraints, nondifferential constraints, and terminal constraints. The problem is to find the statex(t), the controlu(t), and the parameter π so that the functional is minimized, while the constraints are satisfied to a predetermined accuracy. A modified quasilinearization algorithm is developed. Its main property is the descent property in the performance indexR, the cumulative error in the constraints and the optimality conditions. Modified quasilinearization differs from ordinary quasilinearization because of the inclusion of the scaling factor (or stepsize) α in the system of variations. The stepsize is determined by a one-dimensional search on the performance indexR. Since the first variation δR is negative, the decrease inR is guaranteed if α is sufficiently small. Convergence to the solution is achieved whenR becomes smaller than some preselected value. In order to start the algorithm, some nominal functionsx(t),u(t), π and nominal multipliers λ(t), ρ(t), μ must be chosen. In a real problem, the selection of the nominal functions can be made on the basis of physical considerations. Concerning the nominal multipliers, no useful guidelines have been available thus far. In this paper, an auxiliary minimization algorithm for selecting the multipliers optimally is presented: the performance indexR is minimized with respect to λ(t), ρ(t), μ. Since the functionalR is quadratically dependent on the multipliers, the resulting variational problem is governed by optimality conditions which are linear and, therefore, can be solved without difficulty. To facilitate the numerical solution on digital computers, the actual time θ is replaced by the normalized timet, defined in such a way that the extremal arc has a normalized time length Δt=1. In this way, variable-time terminal conditions are transformed into fixed-time terminal conditions. The actual time τ at which the terminal boundary is reached is regarded to be a component of the parameter π being optimized. The present general formulation differs from that of Ref. 3 because of the inclusion of the nondifferential constraints to be satisfied everywhere over the interval 0⩽t⩽1. Its importance lies in that (i) many optimization problems arise directly in the form considered here, (ii) there are problems involving state equality constraints which can be reduced to the present scheme through suitable transformations, and (iii) there are some problems involving inequality constraints which can be reduced to the present scheme through the introduction of auxiliary variables. Numerical examples are presented for the free-final-time case. These examples demonstrate the feasibility as well as the rapidity of convergence of the technique developed in this paper.
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  • 4
    ISSN: 1573-2878
    Keywords: Time-optimal control ; decomposition methods ; two-point boundary-value problems ; trajectories ; numerical methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A decomposition technique is presented for minimum-time trajectories which are characterized by intermediate constraints and discontinuities. The optimization of such multiple are trajectories is usually a formidable task. One optimization method, trajectory decomposition, breaks the original trajectory at points of discontinuity into separate arcs and then optimizes each are subject to prescribed boundary conditions. This constitutes a first level of control. Each first-level solution is evaluated by a second-level controller, which iteratively specifies new are boundary conditions in order to achieve an optimum solution. Unfortunately, this two-level method cannot be applied directly to minimum-time trajectories. The two-level trajectory decomposition method is extended here to a three-level technique for treating the minimum-time trajectory. The first level again optimizes each are for specified intervention parameters. The new second level, the time interface controller, exploits certain homogeneity properties to satisfy time transversality conditions at all boundaries and to couple the first-level solution arcs in time. The third level, the state interface controller, satisfies state transversality conditions at the arc junctions iteratively while driving the trajectory to its optimum. The new three-level procedure represents a feasible decomposition because each solution trajectory in the iterative sequence is physically realizable. The technique also offers a decentralization of control effort and reduction of initial-value sensitivities. An example problem is formulated.
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  • 5
    ISSN: 1573-2878
    Keywords: Stochastic control problems ; numerical methods ; perturbation methods ; suboptimal control ; closed-loop control
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A numerical technique is described for solving approximately certain small-noise stochastic control problems. The method uses quantities computable from the optimal solution to the corresponding deterministic control problem. Numerical results are given for a two-dimensional linear regular problem with saturation and a time-optimal problem.
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  • 6
    ISSN: 1573-2878
    Keywords: Optimal control theory ; functional differential equations ; numerical methods ; hereditary processes ; contraction mapping principle
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This note considers the optimal control of a system represented by nonlinear differential-integral equations with a general cost function. A second-order iterative method of solution based on the fixed-point contraction mapping principle is proposed.
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  • 7
    ISSN: 1573-2878
    Keywords: Differential games ; closed-loop controls ; numerical methods ; optimal strategies ; zero-sum games
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper presents a method for generating nearoptimal closed-loop solutions to zero-sum perfect information differential games with and without the final time explicitly specified, and with and without control constraints. This near-optimal closed-loop solution is generated by periodically updating the solution to the two-point boundary-value problem obtained by the application of the necessary conditions for a saddle-point solution. The resulting updated open-loop control is then used between updating intervals. Three examples are presented to illustrate the application of this method.
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  • 8
    ISSN: 1573-2878
    Keywords: Calculus of variations ; optimal control ; computing methods ; numerical methods ; gradient methods ; seqential gradient-restoration algorithm ; restoration algorithm ; boundary-value problems ; bounded control problems ; bounded state problems ; nondifferential constraints
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper considers the numerical solution of optimal control problems involving a functionalI subject to differential constraints, nondifferential constraints, and terminal constraints. The problem is to find the statex(t), the controlu(t), and the parameter π so that the functional is minimized, while the constraints are satisfied to a predetermined accuracy. The approach taken is a sequence of two-phase processes or cycles, composed of a gradient phase and a restoration phase. The gradient phase involves a single iteration and is designed to decrease the functional, while the constraints are satisfied to first order. The restoration phase involves one or several iterations and is designed to restore the constraints to a predetermined accuracy, while the norm of the variations of the control and the parameter is minimized. The principal property of the algorithm is that it produces a sequence of feasible suboptimal solutions: the functionsx(t),u(t), π obtained at the end of each cycle satisfy the constraints to a predetermined accuracy. Therefore, the functionals of any two elements of the sequence are comparable. The stepsize of the gradient phase is determined by a one-dimensional search on the augmented functionalJ, and the stepsize of the restoration phase by a one-dimensional search on the constraint errorP. If α g is the gradient stepsize and α r is the restoration stepsize, the gradient corrections are ofO(α g ) and the restoration corrections are ofO(α r α g 2). Therefore, for α g sufficiently small, the restoration phase preserves the descent property of the gradient phase: the functionalÎ at the end of any complete gradient-restoration cycle is smaller than the functionalI at the beginning of the cycle. To facilitate the numerical solution on digital computers, the actual time ϑ is replaced by the normalized timet, defined in such a way that the extremal arc has a normalized time length Δt=1. In this way, variable-time terminal conditions are transformed into fixed-time terminal conditions. The actual time τ at which the terminal boundary is reached is regarded to be a component of the parameter π being optimized. The present general formulation differs from that of Ref. 4 because of the inclusion of the nondifferential constraints to be satisfied everywhere over the interval 0 ≤t ≤ 1. Its importance lies in that (i) many optimization problems arise directly in the form considered here, (ii) problems involving state equality constraints can be reduced to the present scheme through suitable transformations, and (iii) problems involving inequality constraints can be reduced to the present scheme through suitable transformations. The latter statement applies, for instance, to the following situations: (a) problems with bounded control, (b) problems with bounded state, (c) problems with bounded time rate of change of the state, and (d) problems where some bound is imposed on an arbitrarily prescribed function of the parameter, the control, the state, and the time rate of change of the state. Numerical examples are presented for both the fixed-final-time case and the free-final-time case. These examples demonstrate the feasibility as well as the rapidity of convergence of the technique developed in this paper.
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  • 9
    Electronic Resource
    Electronic Resource
    Springer
    ISSN: 1573-2878
    Keywords: Mathematical programming ; function minimization ; method of dual matrices ; computing methods ; numerical methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, the method of dual matrices for the minimization of functions is introduced. The method, which is developed on the model of a quadratic function, is characterized by two matrices at each iteration. One matrix is such that a linearly independent set of directions can be generated, regardless of the stepsize employed. The other matrix is such that, at the point where the first matrix fails to yield a gradient linearly independent of all the previous gradients, it generates a displacement leading to the minimal point. Thus, the one-dimensional search is bypassed. For a quadratic function, it is proved that the minimal point is obtained in at mostn + 1 iterations, wheren is the number of variables in the function. Since the one-dimensional search is not needed, the total number of gradient evaluations for convergence is at mostn + 2. Three algorithms of the method are presented. A reverse algorithm, which permits the use of only one matrix, is also given. Considerations pertaining to the applications of this method to the minimization of a quadratic function and a nonquadratic function are given. It is believed that, since the one-dimensional search can be bypassed, a considerable amount of computational saving can be achieved.
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  • 10
    ISSN: 1573-2878
    Keywords: Nonlinear programming ; numerical methods ; unconstrained minimization ; function minimization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper is concerned with the problem of investigating the properties and comparing the methods of nonlinear programming. The steepest-descent method, the method of Davidon, the method of conjugate gradients, and other methods are investigated for the class of essentially nonlinear valley functions.
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