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  • Nonlinear programming
  • dynamic programming
  • 1975-1979  (18)
  • 1945-1949
  • 1978  (5)
  • 1977  (13)
  • 1
    Electronic Resource
    Electronic Resource
    The use of augmented lagrangian functions for sensitivity analysis in nonlinear programming (1977)
    Springer
    Mathematical programming 12 (1977), S. 281-284 
    Details
    ISSN: 1436-4646
    Keywords: Nonlinear programming ; Augmented Lagrangian functions ; Sensitivity analysis
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01593794
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  • 2
    Electronic Resource
    Electronic Resource
    New algorithms for constrained minimax optimization (1977)
    Springer
    Mathematical programming 13 (1977), S. 140-155 
    Details
    ISSN: 1436-4646
    Keywords: Minimax optimization ; Nonlinear programming ; Computer-aided network design
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract A constrained minimax problem is converted to minimization of a sequence of unconstrained and continuously differentiable functions in a manner similar to Morrison's method for constrained optimization. One can thus apply any efficient gradient minimization technique to do the unconstrained minimization at each step of the sequence. Based on this approach, two algorithms are proposed, where the first one is simpler to program, and the second one is faster in general. To show the efficiency of the algorithms even for unconstrained problems, examples are taken to compare the two algorithms with recent methods in the literature. It is found that the second algorithm converges faster with respect to the other methods. Several constrained examples are also tried and the results are presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01584333
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  • 3
    Electronic Resource
    Electronic Resource
    Simplicial decomposition in nonlinear programming algorithms (1977)
    Springer
    Mathematical programming 13 (1977), S. 49-68 
    Details
    ISSN: 1436-4646
    Keywords: Algorithm ; APL-code ; Barycentric representation ; Decomposition ; Nonlinear programming ; Pseudo-concave objective ; Simplex
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Simplicial decomposition is a special version of the Dantzig—Wolfe decomposition principle, based on Carathéodory's theorem. The associated class of algorithms has the following features and advantages: The master and the subprogram are constructed without dual variables; the methods remain therefore well-defined for non-concave objective functions, and pseudo-concavity suffices for convergence to global maxima. The subprogram produces affinely independent sets of feasible generator points defining simplices, which the master program keeps minimal by dropping redundant generator points and finding maximizers in the relative interiors of the resulting subsimplices. The use of parallel subspaces allows the direct application of any unrestricted optimization method in the master program; thus the best unconstrained procedure for any type of objective function can be used to find constrained maximizers for it. The paper presents the theory for this class of algorithms, the APL-code of a “demonstration” method and some computational experience with Colville's test problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01584323
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  • 4
    Electronic Resource
    Electronic Resource
    Generalized Lagrange multiplier technique for nonlinear programming (1977)
    Springer
    Journal of optimization theory and applications 21 (1977), S. 121-135 
    Details
    ISSN: 1573-2878
    Keywords: Nonlinear programming ; max-min problems ; Lagrange multiplier technique ; Newton's method
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Our aim here is to present numerical methods for solving a general nonlinear programming problem. These methods are based on transformation of a given constrained minimization problem into an unconstrained maximin problem. This transformation is done by using a generalized Lagrange multiplier technique. Such an approach permits us to use Newton's and gradient methods for nonlinear programming. Convergence proofs are provided, and some numerical results are given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00932516
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  • 5
    Electronic Resource
    Electronic Resource
    Square-root variable-metric methods for minimization (1977)
    Springer
    Journal of optimization theory and applications 21 (1977), S. 251-259 
    Details
    ISSN: 1573-2878
    Keywords: Nonlinear programming ; variable-metric methods ; parameter optimization ; function minimization ; mathematical programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Variable-metric methods are presented which do not need an accurate one-dimensional search and eliminate roundoff error problems which can occur in updating the metric for large-dimension systems. The methods are based on updating the square root of the metric, so that a positive-definite metric always results. The disadvantage of intentionally relaxing the accuracy of the one-dimensional search is that the number of iterations (and hence, gradient evaluations) increases. For problems involving a large number of variables, the square-root method is presented in a triangular form to reduce the amount of computation. Also, for usual optimization problems, the square-root procedure can be carried out entirely in terms of the metric, eliminating storage and computer time associated with computations of the square root of the metric.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00933529
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  • 6
    Electronic Resource
    Electronic Resource
    Optimal regulation policies for a multipurpose reservoir with seasonal input and return function (1977)
    Springer
    Journal of optimization theory and applications 21 (1977), S. 319-328 
    Details
    ISSN: 1573-2878
    Keywords: Storage theory ; dynamic programming ; water resources
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Optimal regulation policies for a multipurpose reservoir are characterized analytically for the case where the input and return functions are subject to seasonal variations and the return function is piecewise linear, concave, and separable in reservoir volume and discharge. Efficient upper and lower bounds on optimal policies are derived for two important special cases. Necessary and sufficient conditions are obtained for a finite-time horizon optimal policy to be also optimal for the infinite-time horizon.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00933533
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  • 7
    Electronic Resource
    Electronic Resource
    An economical method of calculating Lagrange multiplier estimates for nonlinear programming problems when many of the constraints are bounds on the variables (1977)
    Springer
    Journal of optimization theory and applications 21 (1977), S. 235-239 
    Details
    ISSN: 1573-2878
    Keywords: Nonlinear programming ; computing methods ; Lagrange multiplier estimates ; bounded variables ; large-scale problems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Some recent methods for solving nonlinear programming problems make use of estimates of the Lagrange multipliers. These estimates are usually calculated by solving a system oft linear equations, wheret is the number of active constraints. It is shown that, when a large proportion of the active constraints consists of simple upper or lower bounds on the variables, then computational effort can be saved by means of a reorganization of this linear system.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00932523
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  • 8
    Electronic Resource
    Electronic Resource
    Optimal control of diffusion processes with reflection (1977)
    Springer
    Journal of optimization theory and applications 22 (1977), S. 103-116 
    Details
    ISSN: 1573-2878
    Keywords: Optimal stochastic control ; partial differential equations ; existence theorems ; dynamic programming ; policy iteration
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, we consider the problem of optimally controlling a diffusion process on a closed bounded region ofR n with reflection at the boundary. Employing methods similar to Fleming (Ref. 1), we present a constructive proof that there exists an optimal Markov control that is measurable or lower semicontinuous. We prove further that the expected cost function corresponding to the optimal control is the unique solution of the quasilinear parabolic differential equation of dynamic programming with Neumann boundary conditions and that there exists a diffusion process (in the sense of Stroock and Varadhan) corresponding to the optimal control.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00936724
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  • 9
    Electronic Resource
    Electronic Resource
    A comparison of computational strategies for geometric programs (1978)
    Springer
    Journal of optimization theory and applications 26 (1978), S. 185-203 
    Details
    ISSN: 1573-2878
    Keywords: Nonlinear programming ; geometric programming ; computation ; comparisons
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Numerous algorithms for the solution of geometric programs have been reported in the literature. Nearly all are based on the use of conventional programming techniques specialized to exploit the characteristic structure of either the primal or the dual or a transformed primal problem. This paper attempts to elucidate, via computational comparisons, whether a primal, a dual, or a transformed primal solution approach is to be preferred.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00933403
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  • 10
    Electronic Resource
    Electronic Resource
    Iterative techniques for Riccati game equations (1977)
    Springer
    Journal of optimization theory and applications 22 (1977), S. 51-61 
    Details
    ISSN: 1573-2878
    Keywords: Differential games ; dynamic programming ; Riccati equation ; numerical methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We present a monotone iterative technique for the computation of a solution of a Riccati-type equation relevant to the theory of differential games. For this purpose, we show that the Kleinman algorithm for Riccati equation computations converges under extremely general conditions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00936720
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  • 11
    Electronic Resource
    Electronic Resource
    A new algorithm for solving certain variational problems (1977)
    Springer
    Journal of optimization theory and applications 22 (1977), S. 389-397 
    Details
    ISSN: 1573-2878
    Keywords: Calculus of variations ; dynamic programming ; reduction of dimensionality ; two-stage algorithm
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A technique for finding the solution of discrete, multistate dynamic programming problems is applied to solve certain variational problems. The algorithm is a method of successive approximations using a general two-stage solution. The advantage of the method is that it provides a means of reducing Bellman's “curse of dimensionality.” An example on the Plateau problem or the minimal surface area problem is considered, and the algorithm is found to be computationally efficient.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00932862
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  • 12
    Electronic Resource
    Electronic Resource
    A globally convergent method for nonlinear programming (1977)
    Springer
    Journal of optimization theory and applications 22 (1977), S. 297-309 
    Details
    ISSN: 1573-2878
    Keywords: Nonlinear programming ; global convergence ; exact penalty function
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Recently developed Newton and quasi-Newton methods for nonlinear programming possess only local convergence properties. Adopting the concept of the damped Newton method in unconstrained optimization, we propose a stepsize procedure to maintain the monotone decrease of an exact penalty function. In so doing, the convergence of the method is globalized.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00932858
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  • 13
    Electronic Resource
    Electronic Resource
    Standardization and optimum structural design by dynamic programming (1977)
    Springer
    Journal of optimization theory and applications 23 (1977), S. 183-191 
    Details
    ISSN: 1573-2878
    Keywords: Discrete systems ; structural optimization ; standardization ; minimum cost design ; dynamic programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The paper is concerned with tools for optimum structural design that can taken into account the industrial production of structural elements. Standardization is considered here as the result of minimizing the cost as a function of both element sizes and numbers of identical elements. The optimum value of the design variables obtained by traditional minimum weight (volume) design is taken as the starting point for the present procedure. Dynamic programming is used for standardization decisions. The method seems to be particularly effective when the cost function for each element size can be regarded as the sum of a constant term (initial investment) and a term that is proportional to the number of identical elements (cost of production).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00932306
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  • 14
    Electronic Resource
    Electronic Resource
    On the solution of the unidimensional local minimization problem (1977)
    Springer
    Journal of optimization theory and applications 23 (1977), S. 471-471 
    Details
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00933454
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  • 15
    Electronic Resource
    Electronic Resource
    Optimal economic growth with foreign trade and a learning industry (1978)
    Springer
    Journal of optimization theory and applications 24 (1978), S. 607-616 
    Details
    ISSN: 1573-2878
    Keywords: Economic growth ; dynamic programming ; singular arcs
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Ryder (Ref. 1) has developed a simple two-sector macroeconomic model incorporatinglearning by doing as a determinant of the growth of productivity-enhancing experience and its effect on foreign trade. In this paper, optimal foreign trade and capital allocation policies are determined, in the context of the above model, for ranges of the international trade price not considered by Ryder. An extension of Ryder's model to include a dual trade price structure is briefly discussed. A specific numerical example is used to ascertain the configuration of the various features occurring in the extremal fields at different price levels.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00935302
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  • 16
    Electronic Resource
    Electronic Resource
    Discounted Markov games: Generalized policy iteration method (1978)
    Springer
    Journal of optimization theory and applications 25 (1978), S. 125-138 
    Details
    ISSN: 1573-2878
    Keywords: Discounted two-person zero-sum stochastic games ; counterexample ; dynamic programming ; algorithms ; nearly-optimal strategies
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, we consider two-person zero-sum discounted Markov games with finite state and action spaces. We show that the Newton-Raphson or policy iteration method as presented by Pollats-chek and Avi-Itzhak does not necessarily converge, contradicting a proof of Rao, Chandrasekaran, and Nair. Moreover, a set of successive approximation algorithms is presented of which Shapley's method and a total-expected-rewards version of Hoffman and Karp's method are the extreme elements.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00933260
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  • 17
    Electronic Resource
    Electronic Resource
    Feasible directions in economic policy (1978)
    Springer
    Journal of optimization theory and applications 25 (1978), S. 383-406 
    Details
    ISSN: 1573-2878
    Keywords: Nonlinear programming ; mathematical programming ; computational methods ; applications to economics ; objective functions ; inequality constraints ; choice of endpoints
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Zoutendijk's method of feasible directions is used in this paper to derive numerical control strategies for the United Kingdom economy. The way in which the algorithm permits an examination of the sensitivity of the optimum short-term economic policy to changes in various assumptions demonstrates the versatility of the algorithm. Examined are the implications of different forms for the social welfare function; altering the length of the planning horizon, varying the magnitude of the terminal capital constraint, reducing the maximum permitted level of unemployment, changing the initial endowment of foreign currency reserves, fixing the interest rate for the whole planning period, and imposing a minimum growth rate for public expenditure.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00932901
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  • 18
    Electronic Resource
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    A finite steepest-ascent algorithm for maximizing piecewise-linear concave functions (1978)
    Springer
    Journal of optimization theory and applications 25 (1978), S. 437-442 
    Details
    ISSN: 1573-2878
    Keywords: Nonlinear programming ; nondifferentiable optimization ; subgradient search ; steepest-ascent algorithm
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this note, we describe a finitely convergent steepest-ascent scheme for maximizing piecewise-linear concave functions. Given any point, the algorithm moves along the direction of steepest ascent, that is, along the shortest subgradient, until a new ridge is reached. The overall process is then repeated by moving along the new steepest-ascent direction.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00932904
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