On Approximations with Finite Precision in Bundle Methods for Nonsmooth Optimization

Journal of Optimization Theory and Applications - We consider the proximal form of a bundle algorithm for minimizing a nonsmooth convex function, assuming that the function and subgradient values...     

Stationary Points of Bound Constrained Minimization Reformulations of Complementarity Problems

We consider two merit functions which can be used for solving the nonlinear complementarity problem via nonnegatively constrained minimization. One of the functions is the restricted implicit Lagrangian (Refs. 1–3), and the other appears to be new. We study the conditions under which a stationary point of the minimization problem is guaranteed to be a solution of the underlying complementarity problem. It appears that, for both formulations, the same regularity condition is needed. This condition is closely related to the one used in Ref. 4 for unrestricted implicit Lagrangian. Some new sufficient conditions are also given.     

On the limits of generalization of the Kolmogorov integral

The generalization of the Kolmogorov integral to functions with values in a Banach space is considered. It is proved that the resulting integral turns out to be essentially more general than the Bochner integral and is exactly equivalent to an integral of McShane type, whose definition requires that the scaling function be measurable.Translated from Matematicheskie Zametki, vol. 77, no. 2, 2005, pp. 258–272.Original Russian Text Copyright © 2005 by A. P. Solodov.This revised version was published online in April 2005 with a corrected issue number.     

Henstock and McShane integrals for Banach-valued functions

We solve the problem of the equivalence of theH L-integral and the Henstock integral in Banach spaces. Namely, we prove that the Saks-Henstock lemma is valid in a Banach space if and only if it is finite-dimensional. Translated fromMatematicheskie Zametki, Vol. 65, No. 6, pp. 860–870, June, 1999.     

Nonlinear complementarity as unconstrained and constrained minimization

The nonlinear complementarity problem is cast as an unconstrained minimization problem that is obtained from an augmented Lagrangian formulation. The dimensionality of the unconstrained problem is the same as that of the original problem, and the penalty parameter need only be greater than one. Another feature of the unconstrained problem is that it has global minima of zero at precisely all the solution points of the complementarity problem without any monotonicity assumption. If the mapping of the complementarity problem is differentiable, then so is the objective of the unconstrained problem, and its gradient vanishes at all solution points of the complementarity problem. Under assumptions of nondegeneracy and linear independence of gradients of active constraints at a complementarity problem solution, the corresponding global unconstrained minimum point is locally unique. A Wolfe dual to a standard constrained optimization problem associated with the nonlinear complementarity problem is also formulated under a monotonicity and differentiability assumption. Most of the standard duality results are established even though the underlying constrained optimization problem may be nonconvex. Preliminary numerical tests on two small nonmonotone problems from the published literature converged to degenerate or nondegenerate solutions from all attempted starting points in 7 to 28 steps of a BFGS quasi-Newton method for unconstrained optimization.Dedicated to Phil Wolfe on his 65th birthday, in appreciation of his major contributions to mathematical programming.This material is based on research supported by Air Force Office of Scientific Research Grant AFOSR-89-0410 and National Science Foundation Grant CCR-9101801.     

Water vapor line strengths in the 1-μm region

The line strenghts of H₂O bands ν₁ + ν₂ + ν₃ and ν₂ + 2ν₃ have been measured using a high-resolution Nd-glass single-mode laser spectrophotometer. The parameters of 33 lines were found by fitting a Voigt contour to experimental line profile. The vibration-rotation interaction affecting the line strength was taken into account by using the F factor in the three-parametric form, and its constants for ν₁ + ν₂ + ν₃ and ν₂ + 2ν₃ bands were found. There is a good agreement between line intensities calculated using these constants and the experimental values.     

Study of the e+e- to hadrons via ISR at BABAR

Experimental data from the PEP-Ⅱ B-factory at 10.6 GeV center-of-mass (c.m.) energy, obtained via initial-state radiation (ISR) with the BABAR detector, are presented. The cross sections for many hadronic processes have been measured from the production threshold to 4-5 GeV of the e+e- c.m. Energy. The obtained data allow to study a number of intermediate states and determine the parameters of known resonances and their branching fractions. The exclusive cross section for some number of hadronic sub-processes are presented.     

IR spectroscopy of the structural states of ethylene in aerogel nanopores

We have investigated for the first time the combination vibrational absorption bands ν₅ + ν₉ and ν₁ + ν₁₁ of ethylene adsorbed by nanopores of a silicon dioxide aerogel in the 5800–6300 cm^–1 region. Our measurements show significant differences in the absorption spectra of gas-phase ethylene and ethylene found in the nanoporous structure of the aerogel, including a change in the shape of the absorption band contours, their frequency shift, and an increase in the absorption intensity.     

Search for the ϕ → π+π−γγ and ϕ → π+π−η decays with the CMD-2 detector

The ? → π⁺π^?γγ and ? → π⁺π^?η decays are sought in the experimental data obtained with the CMD-2 detector in the region of the ?-meson resonance. Upper limits ?(? → ππγγ)< 1.2 × 10^?4 and ?(? → ππη) < 6.1 × 10^?5 (both at a C.L. of 90%) are determined for the branching ratios of these decays.