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  • Articles  (5)
  • Other Sources
  • Nonlinear Programming  (5)
  • 1985-1989
  • 1980-1984  (5)
  • 1981  (5)
  • Mathematics  (5)
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  • Articles  (5)
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  • 1985-1989
  • 1980-1984  (5)
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  • 1
    Electronic Resource
    Electronic Resource
    On solving a primal geometric program by partial dual optimization (1981)
    Springer
    Mathematical programming 21 (1981), S. 319-330 
    Details
    ISSN: 1436-4646
    Keywords: Nonlinear Programming ; Geometric Programming ; Duality Theory
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The several published methods for mapping a dual solution estimate to a primal solution estimate in posynomial geometric programming provide no criteria for deciding how much deviation from primal feasibility, or discrepancy between the primal and dual objective function values, should be permitted before the primal solution estimate is accepted by the designer. This paper presents a new and simple dual-to-primal conversion method that uses the cost coefficients to provide a sound economic criterion for determining when to accept a primal solution estimate. The primal solution estimate generated is the exact solution to a modified primal obtained from the given primal by modifying the cost coefficients, with the exponent matrix left unchanged. The method is shown to have desirable properties when coupled with a convergent dual algorithm.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01584252
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    An elementary survey of general duality theory in mathematical programming (1981)
    Springer
    Mathematical programming 21 (1981), S. 241-261 
    Details
    ISSN: 1436-4646
    Keywords: Dual problem ; Duality Theory ; Optimality Conditions ; Price Functions ; Nonlinear Programming ; Integer Programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We survey some recent developments in duality theory with the idea of explaining and unifying certain basic duality results in both nonlinear and integer programming. The idea of replacing dual variables (prices) by price functions, suggested by Everett and developed by Gould, is coupled with an appropriate dual problem with the consequence that many of the results resemble those used in linear programming. The dual problem adopted has a (traditional) economic interpretation and dual feasibility then provides a simple alternative to concepts such as conjugate functions or subdifferentials used in the study of optimality. In addition we attempt to make precise the relationship between primal, dual and saddlepoint results in both the traditional Lagrangean and the more general duality theories and to see the implications of passing from prices to price functions. Finally, and perhaps surprisingly, it appears that all the standard algorithms terminate by constructing primal and dual feasible solutions of equal value, i.e., by satisfying generalised optimality conditions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01584248
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    A method for globally minimizing concave functions over convex sets (1981)
    Springer
    Mathematical programming 20 (1981), S. 22-32 
    Details
    ISSN: 1436-4646
    Keywords: Concave Programming ; Extreme Point Solutions ; Global Optimization ; Nonconvex Programming ; Nonlinear Programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract A method is described for globally minimizing concave functions over convex sets whose defining constraints may be nonlinear. The algorithm generates linear programs whose solutions minimize the convex envelope of the original function over successively tighter polytopes enclosing the feasible region. The algorithm does not involve cuts of the feasible region, requires only simplex pivot operations and univariate search computations to be performed, allows the objective function to be lower semicontinuous and nonseparable, and is guaranteed to converge to the global solution. Computational aspects of the algorithm are discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01589330
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  • 4
    Electronic Resource
    Electronic Resource
    Elimination of bounds in optimization problems by transforming variables (1981)
    Springer
    Mathematical programming 20 (1981), S. 110-121 
    Details
    ISSN: 1436-4646
    Keywords: Bounded Variables ; Constrained Optimization ; Nonlinear Programming ; Nonnegativity Constraints ; Transformation of Variables
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The problem of minimizing a nonlinear objective function ofn variables, with continuous first and second partial derivatives, subject to nonnegativity constraints or upper and lower bounds on the variables is studied. The advisability of solving such a constrained optimization problem by making a suitable transformation of its variables in order to change the problem into one of unconstrained minimization is considered. A set of conditions which guarantees that every local minimum of the new unconstrained problem also satisfies the first-order necessary (Kuhn—Tucker) conditions for a local minimum of the original constrained problem is developed. It is shown that there are certain conditions under which the transformed objective function will maintain the convexity of the original objective function in a neighborhood of the solution. A modification of the method of transformations which moves away from extraneous stationary points is introduced and conditions under which the method generates a sequence of points which converges to the solution at a superlinear rate are given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01589336
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  • 5
    Electronic Resource
    Electronic Resource
    The application of optimal control methodology to nonlinear programming problems (1981)
    Springer
    Mathematical programming 21 (1981), S. 331-347 
    Details
    ISSN: 1436-4646
    Keywords: Optimization ; Nonlinear Programming ; Unconstrained Optimization ; Discrete Optimal Control ; Differential Dynamic Programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Dynamic programming techniques have proven to be more successful than alternative nonlinear programming algorithms for solving many discrete-time optimal control problems. The reason for this is that, because of the stagewise decomposition which characterizes dynamic programming, the computational burden grows approximately linearly with the numbern of decision times, whereas the burden for other methods tends to grow faster (e.g.,n 3 for Newton's method). The idea motivating the present study is that the advantages of dynamic programming can be brought to bear on classical nonlinear programming problems if only they can somehow be rephrased as optimal control problems. As shown herein, it is indeed the case that many prominent problems in the nonlinear programming literature can be viewed as optimal control problems, and for these problems, modern dynamic programming methodology is competitive with respect to processing time. The mechanism behind this success is that such methodology achieves quadratic convergence without requiring solution of large systems of linear equations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01584253
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