Error estimation in nonlinear optimization

Methods are developed and analyzed for estimating the distance to a local minimizer of a nonlinear programming problem. One estimate, based on the solution of a constrained convex quadratic program, can be used when strict complementary slackness and the second-order sufficient optimality conditions hold. A second estimate, based on the solution of an unconstrained nonconvex, nonsmooth optimization problem, is valid even when strict complementary slackness is violated. Both estimates are valid in a neighborhood of a local minimizer. An active set algorithm is developed for computing a stationary point of the nonsmooth error estimator. Each iteration of the algorithm requires the solution of a symmetric, positive semidefinite linear system, followed by a line search. Convergence is achieved in a finite number of iterations. The error bounds are based on stability properties for nonlinear programs. The theory is illustrated by some numerical examples.     

Stability in the presence of degeneracy and error estimation

Received February 10, 1997 / Revised version received June 6, 1998 Published online October 9, 1998     

Non-Glycosidic Azides of D-Altro- and D-Gulopyranose

ABSTRACT

Starting with methyl 2-O-cyclohexylcarbamoyl-3,4-O-(2,2,2-trichloroethylidene)-α-D-altropyranoside (1), methyl 4-O-cyclohexylcarbamoyl-2,3-O-(2,2,2-trichloroethylidene)-β-D-gulopyranoside (12), and methyl 6-O-cyclohexylcarbamoyl-2,3-O-(2,2,2-trichloroethylidene)-β-D-gulopyranoside (21), the 6-azido-6-deoxyaltroses 4, 6, 11, the 6-azido-6-deoxy-D-gulose 14, the 4-azido-4,6-dideoxy-D-gulose 20, and the 4-azido-4-deoxy-D-gulose 26 were synthesised via iodinated or tosylated precursors. Additionally, two gluco-configured azides, the 3-azido-3,6-dideoxy-D-glucose (19) and the 3-azido-3-deoxy-D-glucose (25), were obtained besides the desired 4-azido-4-deoxy-D-gulosides 20 and 26, when methyl 6-deoxy-4-O-tosyl-β-D-gulopyranoside (18) and methyl 6-O-cyclohexylcarbamoyl-4-O-tosyl-β-D-gulopyranoside, respectively, were reacted with sodium azide. An X-ray analysis is presented for methyl 2,4-di-O-acetyl-3-azido-3,6-dideoxy-zl-D-glucose (19).     

Active constraints,indefinite quadratic test problems,and complexity

The observation that at leasts constraints are active when the Hessian of the Lagrangian hass negative eigenvalues at a local minimizer is used to obtain two results: (i) a class of nearly concave quadratic minimization problem can be solved in polynomial time; (ii) a class of indefinite quadratic test problems can be constructed with a specified number of positive and negative eigenvalues and with a known global minimizer.The authors thank the reviewers for their constructive comments. The first author was supported by the National Science Foundation Grant DMS-85-20926 and by the Air Force Office of Scientific Research Grant AFOSR-ISSA-86-0091.     

Mass measurements of neutron-deficient nuclides close to A = 80 with a Penning trap

The masses of ^{80, 81, 82, 83}Y, ^{83, 84, 85, 86, 88}Zr and ^{85, 86, 87, 88}Nb have been measured with a typical precision of 7keV by using the Penning trap setup at IGISOL. The mass of ⁸⁴Zr has been measured for the first time. These precise mass measurements have improved S_p and Q_EC values for astrophysically important nuclides.     

Aetherische Oele

Ohne Zusammenfassung     

Reply to the Comment by S. J. Kowalski,TIPM 40, 113, 2000

Transport in Porous Media -