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  • nonlinear programming  (101)
  • dynamic programming  (60)
  • 1
    ISSN: 1573-2878
    Keywords: Optimal control ; dynamic programming ; interior-point methods ; sequential quadratic programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A dynamic programming method is presented for solving constrained, discrete-time, optimal control problems. The method is based on an efficient algorithm for solving the subproblems of sequential quadratic programming. By using an interior-point method to accommodate inequality constraints, a modification of an existing algorithm for equality constrained problems can be used iteratively to solve the subproblems. Two test problems and two application problems are presented. The application examples include a rest-to-rest maneuver of a flexible structure and a constrained brachistochrone problem.
    Type of Medium: Electronic Resource
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  • 2
    ISSN: 1573-2878
    Keywords: Multicharacteristic inspection ; sequencing ; dynamic programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We address the problem of sequentially inspecting the dependent characteristics of a product, where the dependency is expressed in terms of the joint probabilities of the fitness of the characteristics. We show that, even when the inspection has classification errors, the joint probability mass function of the observed fitness of the characteristics is independent of the sequence of inspection. Using this result, a dynamic programming approach is presented for finding the optimal sequence that minimizes the expected total cost of inspection. Previously reported policies for independent characteristics are shown to be special cases of the results presented here.
    Type of Medium: Electronic Resource
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  • 3
    ISSN: 1573-2878
    Keywords: nonlinear programming ; path-following methods ; differential equations ; projective algorithms
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A new method for linearly constrained nonlinear programming is proposed. This method follows affine scaling paths defined by systems of ordinary differential equations and it is fully parallelizable. The convergence of the method is proved for a nondegenerate problem with pseudoconvex objective function. In practice, the algorithm works also under more general assumptions on the objective function. Numerical results obtained with this computational method on several test problems are shown.
    Type of Medium: Electronic Resource
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  • 4
    Electronic Resource
    Electronic Resource
    Springer
    ISSN: 1573-2878
    Keywords: Cauchy method ; steepest descent ; convex programming ; nonlinear programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We discuss two issues related to the Cauchy algorithm. First, we use anArmijo search with constant α≥0.5 and show that the sequence isFejer convergent to the optimal set, and hence convergent. Second, we useexact line searches and show an example in which the sequence fails toconverge.
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  • 5
    ISSN: 1573-2878
    Keywords: Duality theory ; linear systems ; nonlinear programming ; convex programming ; distributed control systems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A class of singular control problems involving amplitude constraints on the controls is examined. IfL ∞ is the space of control functionsU, the control constraint setS can be identified with the unit ball inL ∞. Now, for anyn ∈ (1, ∞), an analogous problem may be set up withL n forU and the unit ball inL n forS. This modified problem is necessarily nonsingular for controllable systems. It is shown that, by takingn sufficiently large, the solution to the modified problem also solves the original problem arbitrarily closely (in a sense made precise). Behavior asn → ∞ is investigated.
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  • 6
    ISSN: 1573-2878
    Keywords: Engineering design ; nonlinear programming ; convex programming ; optimization theorems ; approximation of functions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A possible mathematical formulation of the practical problem of computer-aided design of electrical circuits (for example) and systems and engineering designs in general, subject to tolerances onk independent parameters, is proposed. An automated scheme is suggested, starting from arbitrary initial acceptable or unacceptable designs and culminating in designs which, under reasonable restrictions, are acceptable in the worst-case sense. It is proved, in particular, that, if the region of points in the parameter space for which designs are both feasible and acceptable satisfies a certain condition (less restrictive than convexity), then no more than 2 k points, the vertices of the tolerance region, need to be considered during optimization.
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  • 7
    ISSN: 1573-2878
    Keywords: Mathematical programming ; nonlinear programming ; penalty function methods ; gradient projection methods ; convergence analysis
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A new programming algorithm for nonlinear constrained optimization problems is proposed. The method is based on the penalty function approach and thereby circumyents the necessity to maintain feasibility at each iteration, but it also behaves much like the gradient projection method. Although only first-order information is used, the algorithm converges asymptotically at a rate which is independent of the magnitude of the penalty term; hence, unlike the simple gradient method, the asymptotic rate of the proposed method is not affected by the ill-conditioning associated with the introduction of the penalty term. It is shown that the asymptotic rate of convergence of the proposed method is identical with that of the gradient projection method.
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  • 8
    ISSN: 1573-2878
    Keywords: Structural optimization ; dynamic programming ; minimum weight design ; minimum of maximum deviation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The optimal design of elastic trusses is discussed from a dynamic programming point of view. Emphasis is placed on minimum volume design of statically determinate trusses with displacement and stress constraints in the discrete case, i.e., when the cross-sectional areas of the bars are available from a discrete set of values. This, a design constraint usually very difficult to handle with standard nonlinear programming algorithms, is naturally incorporated in the present formulation. In addition, the functional equation approach is shown to furnish a direct solution to the problem of determining a design, among all possible ones satisfying certain volume and displacement constraints, for which the maximum stress is a minimum. A successive approximation approach is briefly indicated as an extension of the method to solve statically indeterminate trusses. Finally, several numerical examples are presented and the main features of the methods are briefly exposed.
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  • 9
    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.
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  • 10
    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.
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