Convex Analysis and Duality over Discrete Domains

The aim of this paper is to establish a fundamental theory of convex analysis for the sets and functions over a discrete domain.By introducing conjugate/biconjugate functions and a discrete duality notion for the cones over discrete domains,we study duals of optimization problems whose decision parameters are integers.In particular,we construct duality theory for integer linear programming,provide a discrete version of Slater’s condition that implies the strong duality and discuss the relationship between integrality and discrete convexity.

ANALYSIS SOLUTION OF BULGING CONVEXITY OF THE HYDRAULIC ELASTIC BULGING ROLL

The hydraulic elastic bulging roll system is a new roll system being developed in recent years.It is the first time in home that dead load test and study on this system has been made.Based on the experiment,the analysis solution of the bulging convexity of the hydraulic elastic bulging roll was obtained by selecting appropriate displacement function and adopting inverse solution of the elasticity theory.In evaluating,a variable displacement function and its solving method was advanced.The analysis results are in accord with the experiment's.Some conclusions have laid the foundation of establishing close-loop controlled mathematical model of the hydraulic elastic bulging roll system.

CASH SUBADDITIVE RISK MEASURES FOR PORTFOLIO VECTORS

In this paper, from the viewpoint of the time value of money, we study the risk measures for portfolio vectors with discount factor. Cash subadditive risk measures for portfolio vectors are proposed. Representation results are given by two different methods which are convex analysis and enlarging space. Especially, the method of convex analysis make the line of reasoning and the representation result be simpler. Meanwhile, spot and forward risk measures for portfolio vectors are also introduced, and the relationships between them are investigated.

Alternative Approaches of Convolution within Network Calculus

Network Calculus is a powerful mathematical theory for the performance evaluation of communication systems;among others it allows to determine worst-case performance measures. This is why it is often used to appoint Quality of Service guarantees in packet-switched systems like the internet. The main mathematical operation within this deterministic queuing theory is the min- plus convolution of two functions. For example the convolution of the arrival and service curve of a system which reflects the data’s departure. Considering Quality of Service measures and performance evaluation, the convolution operation plays a considerable important role, similar to classical system theory. Up to the present day, in many cases it is not practical and simple to perform this operation. In this article we describe approaches to simplify the min-plus convolution and, accordingly, facilitate the corresponding calculations.

OPTIMAL INVESTMENT CONSUMPTION MODEL WITH A HIGHER INTEREST RATE FOR BORROWING

This paper considers a consumption and investment decision problem with a higher interest rate for borrowing as well as the dividend rate.Wealth is divided into a riskless asset and risky asset with logrithmic Brownian motion price fluctuations.The stochastic control problem of maximizating expected utility from terminal wealth and consumption is studied.Equivalent conditions for optimality are obtained.By using duality methods,the existence of optimal portfolio consumption is proved,and the explicit solutions leading to feedback formulae are derived for deteministic coefficients.

Reachable set modeling and engagement analysis of exoatmospheric interceptor

A novel reachable set（RS） model is developed within a framework of exoatmospheric interceptor engagement analysis. The boost phase steering scheme and trajectory distortion mechanism of the interceptor are firstly explored. A mathematical model of the distorted RS is then formulated through a dimension–reduction analysis. By treating the outer boundary of the RS on sphere surface as a spherical convex hull, two relevant theorems are proposed and the RS envelope is depicted by the computational geometry theory. Based on RS model, the algorithms of intercept window analysis and launch parameters determination are proposed, and numerical simulations are carried out for interceptors with different energy or launch points. Results show that the proposed method can avoid intensive on-line computation and provide an accurate and effective approach for interceptor engagement analysis. The suggested RS model also serves as a ready reference to other related problems such as interceptor effectiveness evaluation and platform disposition.

Representation of measures of noncompactness and its applications related to an initial value problem in Banach spaces

This paper is devoted to studying the representation of measures of non-generalized compactness,in particular,measures of noncompactness,of non-weak compactness and of non-super weak compactness,defined on Banach spaces and its applications.With the aid of a three-time order-preserving embedding theorem,we show that for every Banach space X,there exist a Banach function space C(K)for some compact Hausdorff space K and an order-preserving affine mapping T from the super space B of all the nonempty bounded subsets of X endowed with the Hausdorff metric to the positive cone C(K)^(+) of C(K),such that for every convex measure,in particular,the regular measure,the homogeneous measure and the sublinear measure of non-generalized compactnessμon X,there is a convex function F on the cone V=T(B)which is Lipschitzian on each bounded set of V such that F(T(B))=μ(B),■B∈B.As its applications,we show a class of basic integral inequalities related to an initial value problem in Banach spaces,and prove a solvability result of the initial value problem,which is an extension of some classical results due to Bana′s and Goebel(1980),Goebel and Rzymowski(1970)and Rzymowski(1971).