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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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  • 2
    Electronic Resource
    Electronic Resource
    On the computational utility of posynomial geometric programming solution methods (1982)
    Springer
    Mathematical programming 22 (1982), S. 163-201 
    Details
    ISSN: 1436-4646
    Keywords: Geometric Programming ; Code Comparisons ; Numerical Testing ; Nonlinear Programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Ten codes or code variants were used to solve the five equivalent posynomial GP problem formulations. Four of these codes were general NLP codes; six were specialized GP codes. A total of forty-two test problems was solved with up to twenty randomly generated starting points per problem. The convex primal formulation is shown to be intrinsically easiest to solve. The general purpose GRG code called OPT appears to be the most efficient code for GP problem solution. The reputed superiority of the specialized GP codes GGP and GPKTC appears to be largely due to the fact that these codes solve the convex primal formulation. The dual approaches are only likely to be competitive for small degree of difficulty, tightly constrained problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01581036
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  • 3
    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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  • 4
    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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  • 5
    Electronic Resource
    Electronic Resource
    A globally convergent constrained quasi-Newton method with an augmented lagrangian type penalty function (1982)
    Springer
    Mathematical programming 23 (1982), S. 75-86 
    Details
    ISSN: 1436-4646
    Keywords: Constrained Optimization ; Global Convergence ; Nonlinear Programming ; Penalty Function ; Quasi-Newton Method
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The recently proposed quasi-Newton method for constrained optimization has very attractive local convergence properties. To force global convergnce of the method, a descent method which uses Zangwill's penalty function and an exact line search has been proposed by Han. In this paper a new method which adopts a differentiable penalty function and an approximate line is presented. The proposed penalty function has the form of the augmented Lagrangian function. An algorithm for updating parameters which appear in the penalty function is described. Global convergence of the given method is proved.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01583780
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  • 6
    Electronic Resource
    Electronic Resource
    Simple bounds for solutions of monotone complementarity problems and convex programs (1985)
    Springer
    Mathematical programming 32 (1985), S. 32-40 
    Details
    ISSN: 1436-4646
    Keywords: Nonlinear Programming ; Complementarity ; Monotonicity ; Convexity ; Bounds
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract For a solvable monotone complementarity problem we show that each feasible point which is not a solution of the problem provides simple numerical bounds for some or all components of all solution vectors. Consequently for a solvable differentiable convex program each primal-dual feasible point which is not optimal provides simple bounds for some or all components of all primal-dual solution vectors. We also give an existence result and simple bounds for solutions of monotone compementarity problems satisfying a new, distributed constraint qualification. This result carries over to a simple existence and boundedness result for differentiable convex programs satisfying a similar constraint qualification.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01585657
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  • 7
    Electronic Resource
    Electronic Resource
    On uniqueness of Kuhn-Tucker multipliers in nonlinear programming (1985)
    Springer
    Mathematical programming 32 (1985), S. 242-246 
    Details
    ISSN: 1436-4646
    Keywords: Nonlinear Programming ; Kuhn—Tucker Multipliers ; Constraint Qualifications ; Second Order Conditions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Recently Fujiwara, Han and Mangasarian introduced a new constraint qualification which is a slight tightening of the well-known Mangasarian—Fromovitz constraint qualification. We show that this new qualification is a necessary and sufficient condition for the uniqueness of Kuhn—Tucker multipliers. We also show that it implies the satisfaction of second order necessary optimality conditions at a local minimum.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01586095
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  • 8
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    On optimality conditions for structured families of nonlinear programming problems (1982)
    Springer
    Mathematical programming 22 (1982), S. 82-92 
    Details
    ISSN: 1436-4646
    Keywords: Generic Programming ; Optimality Conditions ; Nonlinear Programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Optimality conditions for families of nonlinear programming problems inR n are studied from a generic point of view. The objective function and some of the constraints are assumed to depend on a parameter, while others are held fixed. Techniques of differential topology are used to show that under suitable conditions, certain strong second-order conditions are necessary for optimality except possibly for parameter values lying in a negligible set.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01581027
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  • 9
    Electronic Resource
    Electronic Resource
    Nonlinear programming via an exact penalty function: Global analysis (1982)
    Springer
    Mathematical programming 24 (1982), S. 137-161 
    Details
    ISSN: 1436-4646
    Keywords: Nonlinear Programming ; Exact Penalty Methods ; Successive Quadratic Programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper we motivate and describe an algorithm to solve the nonlinear programming problem. The method is based on an exact penalty function and possesses both global and superlinear convergence properties. We establish the global qualities here (the superlinear nature is proven in [7]). The numerical implementation techniques are briefly discussed and preliminary numerical results are given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01585101
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  • 10
    Electronic Resource
    Electronic Resource
    Fast Givens rotations for orthogonal similarity transformations (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 47-56 
    Details
    ISSN: 0945-3245
    Keywords: AMS (MOS): 65F30 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Fast Givens rotations with half as many multiplications are proposed for orthogonal similarity transformations and a matrix notation is introduced to describe them more easily. Applications are proposed and numerical results are examined for the Jacobi method, the reduction to Hessenberg form and the QR-algorithm for Hessenberg matrices. It can be seen that in general fast Givens rotations are competitive with Householder reflexions and offer distinct advantages for sparse matrices.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01459074
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  • 11
    Electronic Resource
    Electronic Resource
    Exact solution of linear equations usingP-adic expansions (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 137-141 
    Details
    ISSN: 0945-3245
    Keywords: (MR 1980) AMS(MOS) ; 65F05, 15A06, 10M10, 10A30 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A method is described for computing the exact rational solution to a regular systemAx=b of linear equations with integer coefficients. The method involves: (i) computing the inverse (modp) ofA for some primep; (ii) using successive refinements to compute an integer vector $$\bar x$$ such that $$A\bar x \equiv b$$ (modp m ) for a suitably large integerm; and (iii) deducing the rational solutionx from thep-adic approximation $$\bar x$$ . For matricesA andb with entries of bounded size and dimensionsn×n andn×1, this method can be implemented in timeO(n 3(logn)2) which is better than methods previously used.
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    URL: http://dx.doi.org/10.1007/BF01459082
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  • 12
    Electronic Resource
    Electronic Resource
    Error bounds for computed eigenvalues and eigenvectors. II (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 201-206 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F05, 65F15, 65H10 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper, motivated by Symm-Wilkinson's paper [5], we describe a method which finds the rigorous error bounds for a computed eigenvalue λ(0) and a computed eigenvectorx (0) of any matrix A. The assumption in a previous paper [6] that λ(0),x (0) andA are real is not necessary in this paper. In connection with this method, Symm-Wilkinson's procedure is discussed, too.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01400539
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  • 13
    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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  • 14
    Electronic Resource
    Electronic Resource
    Nonlinear programming via an exact penalty function: Asymptotic analysis (1982)
    Springer
    Mathematical programming 24 (1982), S. 123-136 
    Details
    ISSN: 1436-4646
    Keywords: Nonlinear Programming ; Exact Penalty Methods ; Successive Quadratic Programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper we consider the final stage of a ‘global’ method to solve the nonlinear programming problem. We prove 2-step superlinear convergence. In the process of analyzing this asymptotic behavior, we compare our method (theoretically) to the popular successive quadratic programming approach.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01585100
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  • 15
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    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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  • 16
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    Second order sensitivity analysis and asymptotic theory of parametrized nonlinear programs (1985)
    Springer
    Mathematical programming 33 (1985), S. 280-299 
    Details
    ISSN: 1436-4646
    Keywords: Marginal Function ; Optimal Value ; Nonlinear Programming ; Sensitivity Analysis ; Parametric Programming ; Asymptotic Distribution
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper we study second-order differential properties of an optimal-value functionϕ(x). It is shown that under certain conditionsϕ(x) possesses second-order directional derivatives, which can be calculated by solving corresponding quadratic programs. Also upper and lower bounds on these derivatives are introduced under weaker assumptions. In particular we show that the second-order directional derivative is infinite if the corresponding quadratic program is unbounded. Finally sensitivity results are applied to investigate asymptotics of estimators in parametrized nonlinear programs.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01584378
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  • 17
    Electronic Resource
    Electronic Resource
    LU decompositions of generalized diagonally dominant matrices (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 57-69 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F05 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Using the simple vehicle ofM-matrices, the existence and stability ofLU decompositions of matricesA which can be scaled to diagonally dominant (possibly singular) matrices are investigated. Bounds on the growth factor for Gaussian elimination onA are derived. Motivation for this study is provided in part by applications to solving homogeneous systems of linear equationsAx=0, arising in Markov queuing networks, input-output models in economics and compartmental systems, whereA or −A is an irreducible, singularM-matrix. This paper extends earlier work by Funderlic and Plemmons and by Varga and Cai.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01459075
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  • 18
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    Computing theCS decomposition of a partitioned orthonormal matrix (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 297-306 
    Details
    ISSN: 0945-3245
    Keywords: AMS (MOS): 65F25 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper describes an algorithm for simultaneously diagonalizing by orthogonal transformations the blocks of a partitioned matrix having orthonormal columns.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396447
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  • 19
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    On fixed point linear equations (1982)
    Springer
    Numerische Mathematik 38 (1982), S. 53-59 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F10 ; 47B55 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary By means of successive partial substitutions it is possible to obtain new fixed point linear equations from old ones and it is interesting to determine how the spectral radius of the corresponding matrices varies. We prove that, when the original matrix is nonnegative, this variation is decreasing or increasing, depending on whether the original matrix has its spectral radius smaller or greater than 1. We answer in this way a question posed by F. Robert in [5].
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01395808
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  • 20
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    On theLU factorization ofM-matrices (1981)
    Springer
    Numerische Mathematik 38 (1981), S. 179-192 
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    ISSN: 0945-3245
    Keywords: AMS(MOS) 15 A 23 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper, we give in Theorem 1 a characterization, based on graph theory, of when anM-matrixA admits anLU factorization intoM-matrices, whereL is a nonsingular lower triangularM-matrix andU is an upper triangularM-matrix. This result generalizes earlier factorization results of Fiedler and Pták (1962) and Kuo (1977). As a consequence of Theorem 1, we show in Theorem 3 that the conditionx T A≧0 T for somex〉0, for anM-matrixA, is both necessary and sufficient forPAP T to admit such anLU factorization for everyn×n permutation matrixP. This latter result extends recent work of Funderlic and Plemmons (1981). Finally, Theorem 1 is extended in Theorem 5 to give a characterization, similarly based on graph theory, of when anM-matrixA admits anLU factorization intoM-matrices.
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    URL: http://dx.doi.org/10.1007/BF01397088
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  • 21
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    Ein linear konvergentes zyklisches jacobiähnliches Verfahren für beliebige reelle Matrizen (1981)
    Springer
    Numerische Mathematik 38 (1981), S. 245-253 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65 F 15 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A modification of Eberleins jacobi-like algorithm for solving the eigenproblem of arbitrary real matrices is described. The algorithm is always convergent. The convergence is linear for diagonalizable matrices.
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    URL: http://dx.doi.org/10.1007/BF01397093
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  • 22
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    A modified bisection algorithm for the determination of the eigenvalues of a symmetric tridiagonal matrix (1982)
    Springer
    Numerische Mathematik 38 (1982), S. 417-419 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F10 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A modification to the well known bisection algorithm [1] when used to determine the eigenvalues of a real symmetric matrix is presented. In the new strategy the terms in the Sturm sequence are computed only as long as relevant information on the required eigenvalues is obtained. The resulting algorithm usingincomplete Sturm sequences can be shown to minimise the computational work required especially when only a few eigenvalues are required. The technique is also applicable to other computational methods which use the bisection process.
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    URL: http://dx.doi.org/10.1007/BF01396441
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  • 23
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    A contribution to the theory of condition (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 85-96 
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    ISSN: 0945-3245
    Keywords: AMS65F05 ; 65F 15 ; 65F 20 ; 65F 35 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper the use of the condition number of a problem, as defined by Rice in 1966, is discussed. For the eigenvalue, eigenvector, and linear least squares problems either condition numbers according to various norms are determined or lower and upper bounds for them are derived.
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    URL: http://dx.doi.org/10.1007/BF01399313
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  • 24
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    A Newton iteration process for inverse eigenvalue problems (1981)
    Springer
    Numerische Mathematik 37 (1981), S. 349-354 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F15 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We suppose an inverse eigenvalue problem which includes the classical additive and multiplicative inverse eigenvalue problems as special cases. For the numerical solution of this problem we propose a Newton iteration process and compare it with a known method. Finally we apply it to a numerical example.
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    URL: http://dx.doi.org/10.1007/BF01400314
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  • 25
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    Inversion of tridiagonal matrices (1982)
    Springer
    Numerische Mathematik 38 (1982), S. 333-345 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 15 A 09, 15-04 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper presents a simple algorithm for inverting nonsymmetric tridiagonal matrices that leads immediately to closed forms when they exist. Ukita's theorem is extended to characterize the class of matrices that have tridiagonal inverses.
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    URL: http://dx.doi.org/10.1007/BF01396436
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  • 26
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    On the convergence of the symmetric SOR method for matrices with red-black ordering (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 113-117 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F10 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We prove that if the matrixA has the structure which results from the so-called “red-black” ordering and ifA is anH-matrix then the symmetric SOR method (called the SSOR method) is convergent for 0〈ω〈2. In the special case thatA is even anM-matrix we show that the symmetric single-step method cannot be accelerated by the SSOR method. Symmetry of the matrixA is not assumed.
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    URL: http://dx.doi.org/10.1007/BF01399315
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  • 27
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    Zur Konvergenz eines Verfahrens von D.J. Evans zur iterativen Verbesserung einer Näherung für die inverse Matrix (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 163-173 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F10 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Recently D.J. Evans introduced an implicit matrix inversion process showing asymptotic behaviour which is superior to that of the well known Schulz-method. In this paper we give sufficient conditions for convergence, prove some error bounds and show that under certain conditions the iterates are converging monotonously.
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    URL: http://dx.doi.org/10.1007/BF01408691
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