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  • 1
    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.
    Type of Medium: Electronic Resource
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  • 2
    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.
    Type of Medium: Electronic Resource
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  • 3
    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.
    Type of Medium: Electronic Resource
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  • 4
    ISSN: 1573-2878
    Keywords: Augmented penalty function ; penalty function methods ; method of multipliers ; numerical methods ; state variable constraints
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Three augmented penalty function algorithms are tested and compared with an ordinary penalty function algorithm for two demonstration optimal control problems. Although the augmented penalty function is quite helpful in solving control problems with terminal state constraints, the convergence can be improved significantly by providing systematic increases in the penalty constant.
    Type of Medium: Electronic Resource
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  • 5
    ISSN: 1573-2878
    Keywords: Nonlinear programming ; unidimensional search ; numerical methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract If the functionf to be minimized is not unimodal, the Fibonacci search and the golden section search can determine final search intervals where the functionf takes greater values than at the borders of the first search interval. This can be avoided by small modifications for sequential search procedures where, in each iteration step,f is evaluated at two interior points of the present search interval. The properties of the point of concentration of the search intervals are given.
    Type of Medium: Electronic Resource
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  • 6
    ISSN: 1573-2878
    Keywords: Mathematical programming ; nonlinear programming ; inequality constraints ; numerical methods ; descent methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper is concerned with first-order methods of feasible directions. Pironneau and Polak have recently proved theorems which show that three of these methods have a linear rate of convergence for certain convex problems in which the objective functions have positive definite Hessians near the solutions. In the present note, it is shown that these theorems on rate of convergence can be extended to larger classes of problems. These larger classes are determined in part by certain second-order sufficiency conditions, and they include many nonconvex problems. The arguments used here are based on the finite-dimensional version of Hestenes' indirect sufficiency method.
    Type of Medium: Electronic Resource
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  • 7
    Electronic Resource
    Electronic Resource
    Springer
    ISSN: 1573-2878
    Keywords: Mathematical programming ; nonlinear programming ; inequality constraints ; numerical methods ; descent methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The method of centers is a well-known method for solving nonlinear programming problems having inequality constraints. Pironneau and Polak have recently presented a new version of this method. In the new method, the direction of search is obtained, at each iteration, by solving a convex quadratic programming problem. This direction finding subprocedure is essentially insensitive to the dimension of the space on which the problem is defined. Moreover, the method of Pironneau and Polak is known to converge linearly for finite-dimensional convex programs for which the objective function has a positive-definite Hessian near the solution (and for which the functions involved are twice continuously differentiable). In the present paper, the method and a completely implementable version of it are shown to converge linearly for a very general class of finite-dimensional problems; the class is determined by a second-order sufficiency condition and includes both convex and nonconvex problems. The arguments employed here are based on the indirect sufficiency method of Hestenes. Furthermore, the arguments can be modified to prove linear convergence for a certain class of infinite-dimensional convex problems, thus providing an answer to a conjecture made by Pironneau and Polak.
    Type of Medium: Electronic Resource
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  • 8
    ISSN: 1573-2878
    Keywords: Nonlinear programming ; unidemensional search ; numerical methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A correction of the procedure of Ref. 1 is given.
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  • 9
    ISSN: 1573-2878
    Keywords: Fredholm integral equations ; iterative methods ; Neumann series ; numerical methods ; transport theory
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The present paper extends the synthetic method of transport theory to a large class of integral equations. Convergence and divergence properties of the algorithm are studied analytically, and numerical examples are presented which demonstrate the expected theoretical behavior. It is shown that, in some instances, the computational advantage over the familiar Neumann approach is substantial.
    Type of Medium: Electronic Resource
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  • 10
    ISSN: 1573-2878
    Keywords: Integral equations ; numerical methods ; differential equations ; numerical integration
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract An initial-value method of Bownds for solving Volterra integral equations is reexamined using a variable-step integrator to solve the differential equations. It is shown that such equations may be easily solved to an accuracy ofO(10−8), the error depending essentially on that incurred in truncating expansions of the kernel to a degenerate one.
    Type of Medium: Electronic Resource
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