Nondifferentiable mathematical programming and convex-concave functions

For a convex-concave functionL(x, y), we define the functionf(x) which is obtained by maximizingL with respect toy over a specified set. The minimization problem with objective functionf is considered. We derive necessary conditions of optimality for this problem. Based upon these necessary conditions, we define its dual problem. Furthermore, a duality theorem and a converse duality theorem are obtained. It is made clear that these results are extensions of those derived in studies on a class of nondifferentiable mathematical programming problems.This work was supported by the Japan Society for the Promotion of Sciences.     

Properties of updating methods for the multipliers in augmented Lagrangians

The convergence properties of different updating methods for the multipliers in augmented Lagrangians are considered. It is assumed that the updating of the multipliers takes place after each line search of a quasi-Newton method. Two of the updating methods are shown to be linearly convergent locally, while a third method has superlinear convergence locally. Modifications of the algorithms to ensure global convergence are considered. The results of a computational comparison with other methods are presented.This work was supported by the Swedish Institute of Applied Mathematics.     

Capital budgeting with benders decomposition

Capital budgeting problems with different interest rates for borrowing and lending and with possible limits on borrowing are applied to dual and primal decomposition. While the former fails if a dual gap exists, the latter becomes attractive. The paper elaborates dual and primal decomposition to capital budgeting models and discusses variants of the Benders scheme. A computer implementation is described and results of extensive computer runs with different strategies are reported which give proof of the efficiency of the implemented decomposition procedure.     

Infrared Variables for the SU(3) Yang–Mills Field

We generalize the self-dual parameterization of the SU(2) Yang–Mills field proposed by Niemi and Faddeev for describing the infrared limit of the theory to the case of the gauge group SU(3). We demonstrate that the duality property intrinsic to the SU(2) gauge field cannot be transferred automatically to the higher-rank group case. We interpret the algebraic structures appearing in the Lagrangian for the new compact variables in terms of the group products SU(2)³.     

Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps

Let be a nonempty closed convex subset of a real Banach space and be a Lipschitz pseudocontractive self-map of with . An iterative sequence is constructed for which as . If, in addition, is assumed to be bounded, this conclusion still holds without the requirement that Moreover, if, in addition, has a uniformly Gâteaux differentiable norm and is such that every closed bounded convex subset of has the fixed point property for nonexpansive self-mappings, then the sequence converges strongly to a fixed point of . Our iteration method is of independent interest.

     

A note on a paper by G. Mastroianni and G. Monegato

Recently, Mastroianni and Monegato derived error estimates for a numerical approach to evaluate the integral

where or or and is a smooth function (see G. Mastroianni and G. Monegato, Error estimates in the numerical evaluation of some BEM singular integrals, Math. Comp. 70 2001, 251-267). The error bounds for the quadrature rule approximating the inner integral given in Theorems 3, 4 of that paper are not correct according to the proof. However, following a different approach, we are able to improve the pointwise error estimates given in that paper.

     

Almost flat locally finite coverings of the sphere

For any subset A of the unit sphere of a Banach space X and for [0,2) the notion of -flatness is introduced as a measure of non-flatness of A. For any positive , construction of locally finite tilings of the unit sphere by -flat sets is carried out under suitable -renormings of X in a quite general context; moreover, a characterization of spaces having separable dual is provided in terms of the existence of such tilings. Finally, relationships between the possibility of getting such tilings of the unit sphere in the given norm and smoothness properties of the norm are discussed.     

Poincaré duality in P.A. Smith theory

Let , or more generally be a finite -group, where is an odd prime. If acts on a space whose cohomology ring fulfills Poincaré duality (with appropriate coefficients ), we prove a mod congruence between the total Betti number of and a number which depends only on the -module structure of . This improves the well known mod congruences that hold for actions on general spaces.

     

Maximal monotonicity, conjugation and the duality product

Recently, the authors studied the connection between each maximal monotone operator and a family of convex functions. Each member of this family characterizes the operator and satisfies two particular inequalities.

The aim of this paper is to establish the converse of the latter fact. Namely, that every convex function satisfying those two particular inequalities is associated to a unique maximal monotone operator.