An Inexact Newton Method Derived from Efficiency Analysis

We consider solving an unconstrained optimization problem by Newton-PCG like methods in which the preconditioned conjugate gradient method is applied to solve the Newton equations. The main question to be investigated is how efficient Newton-PCG like methods can be from theoretical point of view. An algorithmic model with several parameters is established. Furthermore, a lower bound of the efficiency measure of the algorithmic model is derived as a function of the parameters. By maximizing this lower bound function, the parameters are specified and therefore an implementable algorithm is obtained. The efficiency of the implementable algorithm is compared with Newtons method by theoretical analysis and numerical experiments. The results show that this algorithm is competitive.Mathematics Subject Classification: 90C30, 65K05.This work was supported by the National Science Foundation of China Grant No. 10371131, and Hong Kong Competitive Earmarked Research Grant CityU 1066/00P from Hong Kong University Grant Council     

Objective Function Features Providing Barriers to Rapid Global Optimization

The purposes of this discussion paper are twofold. First, features of an objective function landscape which provide barriers to rapid finding of the global optimum are described. Second, stochastic algorithms are discussed and their performance examined, both theoretically and computationally, as the features change. The paper lays a foundation for the later findings paper.     

Entropic Penalties in Finite Games

The main objects here are finite-strategy games in which entropic terms are subtracted from the payoffs. After such subtraction each Nash equilibrium solves an explicit, unconstrained, nonlinear system of smooth equations. That system, while characteristic of perturbed best responses, is amenable in computation. It also facilitates analysis of fictitious play, learning by reinforcement, and evolutionary dynamics.     

Robust$${\cal {H}}_{\infty}$$ Stabilization of Stochastic Hybrid Systems with Wiener Process

This paper deals with the class of uncertain continuous-time linear stochastic hybrid systems with Wiener process. The uncertainties that we are considering are of the norm-bounded type. The robust stochastic stabilization problem is treated. LMIs based sufficient conditions are developed to design the state feedback controller that robustly and stochastically stabilizes the studied class of systems and at the same time rejects a disturbance of desired level. The minimum disturbance rejection is also determined. A numerical example is provided to show the validity of the proposed results.     

Weighted BMO and discrete time hedging within the Black-Scholes model

The paper combines two objects rather different at first glance: spaces of stochastic processes having weighted bounded mean oscillation (weighted BMO) and the approximation of certain stochastic integrals, driven by the geometric Brownian motion, by integrals over piece-wise constant integrands. The consideration of the approximation error with respect to weighted BMO implies L_p and uniform distributional estimates for the approximation error by a John-Nirenberg type theorem. The general results about weighted BMO are given in the first part of the paper and applied to our approximation problem in the second one.     

Stochastic nonlinear beam equations

An extensible beam equation with a stochastic force of a white noise type is studied, Lyapunov functions techniques being used to prove existence of global mild solutions and asymptotic stability of the zero solution.This research was supported in part by the GA R Grants no. 201/98/1454, 201/01/1197 and by a Royal Society grant     

On the relationship between differentiability conditions and existence of a strong gradient

It is proved that for any n 2 there exists continuous function f : ⁿ which is differentiable almost everywhere, but has no strong gradient almost everywhere.Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 93–98.Original Russian Text Copyright © 2005 by G. G. Oniani.This revised version was published online in April 2005 with a corrected issue number.