Fractional programming

The paper reviews some of the major developments in fractional programming. The emphasis is on applications and algorithms. Theoretical results are only outlined to the extent as they are needed to describe the algorithms.The survey is subjective in nature. It tries to critically evaluate developments in fractional programming after more than 20 years of research with over 500 articles. The authors hope that this somewhat critical report will serve as a guide for meaningful future work in this growing area of nonlinear programming.     

Fractional programming revisited

A new format is proposed for fractional programming problems. This format gives full expression to the fact that the parametric approach to fractional programming problems is rooted in a first-order necessary and sufficient optimality condition. It is thus shown that although traditionally it has not been construed as such, the parametric approach is in fact classical par excellence.     

Fractional programming without differentiability

The notion of quasi-differentiability is examined and related to fractional programming. Necessary and sufficient conditions are given and various other properties of quasi-differentiable functions are discussed. Differentiability is not assumed.This research was partially supported by N.R.C. Grants A7751 and A7675.     

Fractional programming by lower subdifferentiability techniques

The notion of lower subdifferentiability is applied to the analysis of convex fractional programming problems. In particular, duality results and optimality conditions are presented, and the applicability of a cutting-plane algorithm using lower subgradients is discussed. These methods are useful also in generalized fractional programming, where, in the linear case, the performance of the cutting-plane algorithm is compared with that of the most efficient version of the Dinkelbach method, which is based on the solution of a parametric linear programming problem.The authors wish to thank Mr. Jaume Timoneda for his help in the implementation of the numerical methods on the computer and the referees for valuable comments and suggestions; the present improved statement and proof of Proposition 2.1 is due to one of them. Financial support from the Dirección General de Investigación Científica y Técnica (DGICYT), under project PS89-0058, is gratefully acknowledged.     

Fractional programming approach to fuzzy weighted average

This paper proposes a fractional programming approach to construct the membership function for fuzzy weighted average. Based on the -cut representation of fuzzy sets and the extension principle, a pair of fractional programs is formulated to find the -cut of fuzzy weighted average. Owing to the special structure of the fractional programs, in most cases, the optimal solution can be found analytically. Consequently, the exact form of the membership function can be derived by taking the inverse function of the -cut. For other cases, a discrete but exact solution to fuzzy weighted average is provided via an efficient solution method. Examples are given for illustration.     

Fractional programming approach to two stochastic inventory problems

This paper considers two popular inventory models: the continuous review and periodic review reorder-point, order-quantity, control systems. Specifically we present two procedures which determine optimal values for the two control parameters (i.e., reorder-point and order-quantity) when the holding-and-shortage costs are non-quasi-convex. This cost structure may arise when non-linear cost rate is considered, for instance when the shortage cost is the shadow cost of a service-level constraint. The algorithms based on a fractional programming method are intuitive and efficient, and as the holding-and-shortage cost functions become quasi-convex, they are compatible to existing algorithms.     

Fractional programming and characterization of some vertices of the feasible region

This paper considers a problem of nonlinear programming in which the objective function is the ratio of two linear functions and the constraints define a bounded and connected feasible region. Using a coordinate transformation, this problem is transformed into a simpler one, whose geometric interpretation is of particular significance. The transformation leads to a characterization of some special vertices of the feasible region from both the theoretical and operational points of view.     

分数因子和分数哈密顿图

本文介绍了图的分数方面,将图中基于整数的定义和变量转化为分数形式．介绍了分数图论的一些新结果,特别是关于分数因子和分数哈密顿图的新结果,其中包括了作者最近得到的一些关于分数(g,f)-因子的若干结果．进而,提出了还没有解决的几个新问题．     

Stochastic Fractional Anderson Models with Fractional Noises

The authors are concerned with a class of one-dimensional stochastic Anderson models with double-parameter fractional noises, whose differential operators are fractional. A unique solution for the model in some appropriate Hilbert space is constructed. Moreover, the Lyapunov exponent of the solution is estimated, and its HSlder continuity is studied. On the other hand, the absolute continuity of the solution is also discussed.     

Fractional eigenvalues

We study the non-local eigenvalue problem $$\begin{aligned} 2\, \int \limits _{\mathbb{R }^n}\frac{|u(y)-u(x)|^{p-2}\bigl (u(y)-u(x)\bigr )}{|y-x|^{\alpha p}}\,dy +\lambda |u(x)|^{p-2}u(x)=0 \end{aligned}$$ for large values of $p$ and derive the limit equation as $p\rightarrow \infty $ . Its viscosity solutions have many interesting properties and the eigenvalues exhibit a strange behaviour.     

Fractional Opial Type Inequalities and Fractional Differential Equations

A set of very general Opial type inequalities ([7]) is established involving fractional derivatives of different orders. These are based on Taylor’s formula for fractional derivatives. These results are applied in proving uniqueness to the solutions of very general fractional initial value problems of fractional ordinary differential equations.     

Fractional Planks

Abstract. In 1950 Bang proposed a conjecture which became known as ``the plank conjecture': Suppose that a convex set S contained in the unit cube of R ⁿ and touching all its sides is covered by planks. (A plank is a set of the form {(x ₁ , ..., x _n ): x _j ∈ I} for some j ∈ {1, ...,n} and a measurable subset I of [0, 1]. Its width is defined as |I| .) Then the sum of the widths of the planks is at least 1 . We consider a version of the conjecture in which the planks are fractional. Namely, we look at n -tuples f ₁ , ..., f _n of nonnegative-valued measurable functions on [0,1] which cover the set S in the sense that ∑ f _j (x _j ) ≥ 1 for all (x ₁ , ..., x _n )∈ S . The width of a function f _j is defined as ∈t ₀ ¹ f _j (x) dx . In particular, we are interested in conditions on a convex subset of the unit cube in R ⁿ which ensure that it cannot be covered by fractional planks (functions) whose sum of widths (integrals) is less than 1 . We prove that this (and, a fortiori, the plank conjecture) is true for sets which touch all edges incident with two antipodal points in the cube. For general convex bodies inscribed in the unit cube in R ⁿ we prove that the sum of widths must be at least 1/n (the true bound is conjectured to be 2/n ).     

A Note on the Fractional Integrated Fractional Brownian Motion

We characterize the lower classes of the fractional integrated fractional Brownian motion by an integral test.     

关于分数稳定过程分数参数的注

注意到分数稳定过程的样本性质和分布特征都密切关联于其分数参数,本文进一步刻画分数稳定过程关于分数参数的正则性及其对过程协差的影响.     

分数阶微分方程多点分数阶边值问题

研究分数阶微分方程多点分数阶边值问题解的存在性与唯一性,利用不动点定理,得到了边值问题存在唯一解和至少存在1个解的充分条件.     

Fractional Brownian Flows

We consider a stochastic flow on ? ⁿ driven by a fractional Brownian motion with Hurst parameter \(H\in(\frac{1}{2},1)\) and study a tangent flow and the growth of the Hausdorff measure of sub-manifolds of ? ⁿ as they evolve under the flow.The main result is a bound on the rate of (global) growth in terms of the (local) Hölder norm of the flow.     

Fractional Sobolev-type inequalities

Here we present univariate Sobolev-type fractional inequalities involving fractional derivatives of Canavati, Riemann–Liouville and Caputo types. The results are general L _p inequalities forward and converse on a closed interval. We give an application to a fractional ODE. We present also the mean Sobolev-type fractional inequalities.     

股票价格服从分式Brown运动的股票期权保险精算定价

期权定价的保险精算方法由M ogens B ladt和H ina Hv iid R ydberg于1998年首次提出,由于无任何市场假设,所以它不光对无套利、均衡、完备的市场有效,且对有套利、非均衡、不完备的市场也有效.本文利用保险精算方法讨论了股票价格服从分式B row n运动的欧式期权定价问题.