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  • 1
    Electronic Resource
    Electronic Resource
    An interior point algorithm of $$O(\sqrt m \left| {\ln \varepsilon } \right|)$$ iterations forC 1-convex programming (1992)
    Springer
    Mathematical programming 57 (1992), S. 239-257 
    Details
    ISSN: 1436-4646
    Keywords: Complexity of algorithms ; convergence analysis ; interior point algorithms ; Newton's methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We present a theoretical result on a path-following algorithm for convex programs. The algorithm employs a nonsmooth Newton subroutine. It starts from a near center of a restricted constraint set, performs a partial nonsmooth Newton step in each iteration, and converges to a point whose cost is withinε accuracy of the optimal cost in $$O(\sqrt m \left| {\ln \varepsilon } \right|)$$ iterations, wherem is the number of constraints in the problem. Unlike other interior point methods, the analyzed algorithm only requires a first-order Lipschitzian condition and a generalized Hessian similarity condition on the objective and constraint functions. Therefore, our result indicates the theoretical feasibility of applying interior point methods to certainC 1-optimization problems instead ofC 2-problems. Since the complexity bound is unchanged compared with similar algorithms forC 2-convex programming, the result shows that the smoothness of functions may not be a factor affecting the complexity of interior point methods.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01581083
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    A trust region algorithm for minimization of locally Lipschitzian functions (1994)
    Springer
    Mathematical programming 66 (1994), S. 25-43 
    Details
    ISSN: 1436-4646
    Keywords: Trust region methods ; Locally Lipschitzian functions ; Global convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The classical trust region algorithm for smooth nonlinear programs is extended to the nonsmooth case where the objective function is only locally Lipschitzian. At each iteration, an objective function that carries both first and second order information is minimized over a trust region. The term that carries the first order information is an iteration function that may not explicitly depend on subgradients or directional derivatives. We prove that the algorithm is globally convergent. This convergence result extends the result of Powell for minimization of smooth functions, the result of Yuan for minimization of composite convex functions, and the result of Dennis, Li and Tapia for minimization of regular functions. In addition, compared with the recent model of Pang, Han and Rangaraj for minimization of locally Lipschitzian functions using a line search, this algorithm has the same convergence property without assuming positive definiteness and uniform boundedness of the second order term. Applications of the algorithm to various nonsmooth optimization problems are discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01581136
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Crystal structure of [Mn(II)(phen)(ClCH2COO)2]n (phen = 1,10−phenanthroline) (1999)
    Springer
    Journal of chemical crystallography 29 (1999), S. 199-201 
    Details
    ISSN: 1572-8854
    Keywords: Manganese (II) ; phenanthroline ; polymeric complex ; crystal structure
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Physics
    Notes: Abstract A new polymeric manganese(II) complex, [Mn(II)(phen)(ClCH2COO)2]n, was obtained from the reaction of Mn(ClCH2COO)2 with phen and its structure was determined by x-ray crystallography. The complex crystallizes in the monoclinic system, space group C2/c with a = 19.706(4), b = 11.381(3), c = 7.482(3) Å, β = 94.01(3)°, V = 1674.0(8) Å3, and Z = 4. The structure consists of an infinite chain. The manganese atom is located on a twofold axis and presents a distorted octahedral coordination sphere, which consists of the two N atoms of a phen ligand (Mn—N = 2.304(2) Å) and four carboxylato ligands. The Mn···Mn distance within the chain is 4.53 Å, and the carboxylato bridges present a syn-anti conformation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1009574128099
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  • 4
    Electronic Resource
    Electronic Resource
    A nonsmooth version of Newton's method (1993)
    Springer
    Mathematical programming 58 (1993), S. 353-367 
    Details
    ISSN: 1436-4646
    Keywords: Newton's methods ; generalized Jacobian ; semismoothness
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Newton's method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized Jacobian instead of the derivative. This extension includes the B-derivative version of Newton's method as a special case. Convergence theorems are proved under the condition of semismoothness. It is shown that the gradient function of the augmented Lagrangian forC 2-nonlinear programming is semismooth. Thus, the extended Newton's method can be used in the augmented Lagrangian method for solving nonlinear programs.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01581275
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    Paper (German National Licenses)
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