Convergence rate analysis of iteractive algorithms for solving variational inequality problems

 We present a unified convergence rate analysis of iterative methods for solving the variational inequality problem. Our results are based on certain error bounds; they subsume and extend the linear and sublinear rates of convergence established in several previous studies. We also derive a new error bound for $\gamma$-strictly monotone variational inequalities. The class of algorithms covered by our analysis in fairly broad. It includes some classical methods for variational inequalities, e.g., the extragradient, matrix splitting, and proximal point methods. For these methods, our analysis gives estimates not only for linear convergence (which had been studied extensively), but also sublinear, depending on the properties of the solution. In addition, our framework includes a number of algorithms to which previous studies are not applicable, such as the infeasible projection methods, a separation-projection method, (inexact) hybrid proximal point methods, and some splitting techniques. Finally, our analysis covers certain feasible descent methods of optimization, for which similar convergence rate estimates have been recently obtained by Luo [14]. Received: April 17, 2001 / Accepted: December 10, 2002 Published online: April 10, 2003 RID="⋆" ID="⋆" Research of the author is partially supported by CNPq Grant 200734/95–6, by PRONEX-Optimization, and by FAPERJ. Key Words. Variational inequality – error bound – rate of convergence Mathematics Subject Classification (2000): 90C30, 90C33, 65K05     

Precise measurement of the positive muon anomalous magnetic moment

A precise measurement of the anomalous g value, a(mu) = (g-2)/2, for the positive muon has been made at the Brookhaven Alternating Gradient Synchrotron. The result a(mu+) = 11 659 202(14) (6) x 10(-10) (1.3 ppm) is in good agreement with previous measurements and has an error one third that of the combined previous data. The current theoretical value from the standard model is a(mu)(SM) = 11 659 159.6(6.7) x 10(-10) (0.57 ppm) and a(mu)(exp) - a(mu)(SM) = 43(16) x 10(-10) in which a(mu)(exp) is the world average experimental value.     

High-resolution Q-branches spectrum of C2H2 at 9366 and 9407 cm−1

The acetylene absorption spectrum in the Nd-laser range 9240–9520 cm^?1 has been recorded earlier (1), using a highly sensitive intracavity laser spectrometer. Two weak absorption perpendicular bands 2100⁰1¹ ← 0000⁰0⁰ (ν₀ = 9366.6 cm^?1) and 1200°3¹ ← 0000°0° (ν₀ = 9407.7 cm^?1) were studied and spectroscopic constants were obtained. However, the structure of the Q branches of these bands with J < 7 was unresolved because the resolution of the spectrometer was not very high.     

Incremental Gradient Algorithms with Stepsizes Bounded Away from Zero

We consider the class of incremental gradient methods for minimizing a sum of continuously differentiable functions. An important novel feature of our analysis is that the stepsizes are kept bounded away from zero. We derive the first convergence results of any kind for this computationally important case. In particular, we show that a certain -approximate solution can be obtained and establish the linear dependence of on the stepsize limit. Incremental gradient methods are particularly well-suited for large neural network training problems where obtaining an approximate solution is typically sufficient and is often preferable to computing an exact solution. Thus, in the context of neural networks, the approach presented here is related to the principle of tolerant training. Our results justify numerous stepsize rules that were derived on the basis of extensive numerical experimentation but for which no theoretical analysis was previously available. In addition, convergence to (exact) stationary points is established when the gradient satisfies a certain growth property.     

FTIR spectroscopy of 2-0 band of carbon monoxide confined in silica aerogels with different pore sizes

The absorption spectra of carbon monoxide confined in three aerogel samples with different pore sizes have been recorded within the 4100–4400?cm^-1 spectral region at room temperature. The measurements were made using a Bruker IFS 125HR Fourier-transform infrared spectrometer. Lineshift and half-width values for CO were obtained. The influence of pore sizes on dependence of CO line half-width values on rotational quantum numbers was studied.     

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.     

Error bounds for 2-regular mappings with Lipschitzian derivatives and their applications

We obtain local estimates of the distance to a set defined by equality constraints under assumptions which are weaker than those previously used in the literature. Specifically, we assume that the constraints mapping has a Lipschitzian derivative, and satisfies a certain 2-regularity condition at the point under consideration. This setting directly subsumes the classical regular case and the twice differentiable 2-regular case, for which error bounds are known, but it is significantly richer than either of these two cases. When applied to a certain equation-based reformulation of the nonlinear complementarity problem, our results yield an error bound under an assumption more general than b-regularity. The latter appears to be the weakest assumption under which a local error bound for complementarity problems was previously available. We also discuss an application of our results to the convergence rate analysis of the exterior penalty method for solving irregular problems. Received: February 2000 / Accepted: November 2000?Published online January 17, 2001     

A regularized smoothing method for fully parameterized convex problems with applications to convex and nonconvex two-stage stochastic programming

Mathematical Programming - We present an approach to regularize and approximate solution mappings of parametric convex optimization problems that combines interior penalty (log-barrier) solutions...