Density Functional Theory and Optimal Transportation with Coulomb Cost

We present here novel insight into exchange‐correlation functionals in density functional theory, based on the viewpoint of optimal transport. We show that in the case of two electrons and in the semiclassical limit, the exact exchange‐correlation functional reduces to a very interesting functional that depends on an optimal transport map T associated with a given density ρ. The limit problem has been suggested, on grounds of formal arguments, in the physics literature, but it appears that it has not hitherto been interpreted as an optimal transport problem. Since the above limit is strongly correlated, the limit functional yields insight into electron correlations. We prove the existence and uniqueness of such an optimal map for any number of electrons and each ρ and determine the map explicitly in the case when ρ is radially symmetric. © 2012 Wiley Periodicals, Inc.     

Weber Correspondence Analysis: The One-Dimensional Case

When designing programs or software for the implementation of Monte Carlo (MC) hypothesis tests, we can save computation time by using sequential stopping boundaries. Such boundaries imply stopping resampling after relatively few replications if the early replications indicate a very large or a very small p value. We study a truncated sequential probability ratio test (SPRT) boundary and provide a tractable algorithm to implement it. We review two properties desired of any MC p value, the validity of the p value and a small resampling risk, where resampling risk is the probability that the accept/reject decision will be different than the decision from complete enumeration. We show how the algorithm can be used to calculate a valid p value and confidence intervals for any truncated SPRT boundary. We show that a class of SPRT boundaries is minimax with respect to resampling risk and recommend a truncated version of boundaries in that class by comparing their resampling risk (RR) to the RR of fixed boundaries with the same maximum resample size. We study the lack of validity of some simple estimators of p values and offer a new, simple valid p value for the recommended truncated SPRT boundary. We explore the use of these methods in a practical example and provide the MChtest R package to perform the methods.     

On an Optimal Filtration Problem for One-Dimensional Diffusion Processes

We find a method that reduces the solution of a problem of nonlinear filtration of one-dimensional diffusion processes to the solution of a linear parabolic equation with constant diffusion coefficients whose remaining coefficients are random and depend on the trajectory of the observable process. The method consists in reducing the initial filtration problem to a simpler problem with identity diffusion matrix and subsequently reducing the solution of the parabolic Itô equation for the filtered density to solving the above-mentioned parabolic equation. In addition, the filtered densities of both problems are connected by a sufficiently simple formula.     

EOQ Systems: The Case of an Increase in Purchase Cost

In an EOQ system, an anticipated price increase may require determination of an exceptiom lot. A criterion of minimizing the maximum error in terms of cumulative costs is used to determine this lot. This avoids the pitfalls raised on earlier works that compare out-of-phase inventory policies. The exception lot determined happens to be equal to the one obtained by an average cost model.     

最佳费用流

建立赋模糊数为费用权的容量－－费用网络中,据模糊决策来求解最佳费用流的网络模型,并给出这一模型的相应算法。     

Optimal Stopping of One-Dimensional Diffusions

In this paper we consider an optimal stopping problem for a time-homogeneous, onedimensional, regular diffusion. An essential tool in our approach is the MARTIN boundary theory. It is possible to determine explicitly the representing measure of a given β-excessive function. It is seen that this correspondence may be used to construct optimal stopping rules. In some specific cases, as demonstrated in the paper, the solution is reached directly and with ease. The so called condition of “smooth pasting” is seen to be a simple consequence of our results.     

Mass Transportation with LQ Cost Functions

We study the optimal transport problem in the Euclidean space where the cost function is given by the value function associated with a Linear Quadratic minimization problem. Under appropriate assumptions, we generalize Brenier’s Theorem proving existence and uniqueness of an optimal transport map. In the controllable case, we show that the optimal transport map has to be the gradient of a convex function up to a linear change of coordinates. We give regularity results and also investigate the non-controllable case.     

存在空箱调运的非对称运输市场定价策略研究

考虑空箱调运成本,本文对垄断和双寡头市场分别研究运输企业在两条相向路径上的定价问题。对于垄断市场,建立了运输企业最优定价策略,并刻画出无空箱调运的潜在需求不平衡区间。对于双寡头市场,考虑同一路径上不同企业潜在运输需求不等的现实情境,求解了非对称企业的伯川德纳什均衡,给出最优定价策略。研究发现,无空箱调运并不意味着较高利润,运输企业没有必要刻意消除空箱调运现象。另外,增加单位载货运输成本和竞争强度会降低企业利润,而提升单位空箱重置成本、价格敏感度和市场不对称程度都会增大企业利润。     

A Nonsmooth Global Optimization Technique Using Slopes: The One-Dimensional Case

In this paper we introduce a pruning technique based on slopes in the context of interval branch-and-bound methods for nonsmooth global optimization. We develop the theory for a slope pruning step which can be utilized as an accelerating device similar to the monotonicity test frequently used in interval methods for smooth problems. This pruning step offers the possibility to cut away a large part of the box currently investigated by the optimization algorithm. We underline the new technique's efficiency by comparing two variants of a global optimization model algorithm: one equipped with the monotonicity test and one equipped with the pruning step. For this reason, we compared the required CPU time, the number of function and derivative or slope evaluations, and the necessary storage space when solving several smooth global optimization problems with the two variants. The paper concludes on the test results for several nonsmooth examples.     

Optimal Control of One-Dimensional Partial Differential Algebraic Equations with Applications

We present an approach to compute optimal control functions in dynamic models based on one-dimensional partial differential algebraic equations (PDAE). By using the method of lines, the PDAE is transformed into a large system of usually stiff ordinary differential algebraic equations and integrated by standard methods. The resulting nonlinear programming problem is solved by the sequential quadratic programming code NLPQL. Optimal control functions are approximated by piecewise constant, piecewise linear or bang-bang functions. Three different types of cost functions can be formulated. The underlying model structure is quite flexible. We allow break points for model changes, disjoint integration areas with respect to spatial variable, arbitrary boundary and transition conditions, coupled ordinary and algebraic differential equations, algebraic equations in time and space variables, and dynamic constraints for control and state variables. The PDAE is discretized by difference formulae, polynomial approximations with arbitrary degrees, and by special update formulae in case of hyperbolic equations. Two application problems are outlined in detail. We present a model for optimal control of transdermal diffusion of drugs, where the diffusion speed is controlled by an electric field, and a model for the optimal control of the input feed of an acetylene reactor given in form of a distributed parameter system.     

一维扩散过程的最优停止问题的研究

本文研究了一维扩散过程的最优停止问题,论证了W iener过程和几何布朗运动是F e ller过程,同时给出了一般扩散过程的处理方法.     

Optimal Two-commodity Flows with Non-linear Cost Functions

We consider networks in which two different commodities have to be transported across undirected arcs, subject to a shared capacity on the arcs. For each arc and commodity there is an associated non-linear cost that depends on the amount of the commodity transported across the arc. The aim is to minimize the sum of the costs over all arcs and commodities. Efficient algorithms for solving this problem for two types of objective functions will be presented: in the first the cost depends on the absolute value of the flow and in the second the cost is a quadratic function of the flow. Previous work on multi-commodity flow has concentrated on linear cost problems or tackled non-linear cost problems with Lagrangian relaxation methods and other more general techniques. The algorithms in this paper, on the other hand, provide a very efficient way of dealing with two types of non-linear two-commodity optimal flow problems.     

Optimal Transportation for Generalized Lagrangian

This paper deals with the optimal transportation for generalized Lagrangian L = L(x, u, t), and considers the following cost function: c(x, y) = inf x(0)=x x(1)=y u∈U∫_0~1 L(x(s), u(x(s), s), s)ds, where U is a control set, and x satisfies the ordinary equation x(s) = f(x(s), u(x(s), s)).It is proved that under the condition that the initial measure μ0 is absolutely continuous w.r.t. the Lebesgue measure, the Monge problem has a solution, and the optimal transport map just walks along the characteristic curves of the corresponding Hamilton-Jacobi equation:V_t(t, x) + sup u∈UV_x(t, x), f(x, u(x(t), t), t)-L(x(t), u(x(t), t), t) = 0,V(0, x) = Φ0(x).     

Case Study: The Cost Minimization of Refuse Collection Operations

This paper describes an approach developed to minimize the cost of refuse collection operations in a city of 150,000 population. The method is general and applicable to a city of any size.A model is derived for collection which is a function of varied housing layouts, collection times, team sizes and the distance of the collection rounds from the disposal point. Then a computational method is developed which allocates collection operatives to vehicles and designs rounds which together minimize the total collection cost.The advantages of the method are:(a) It reduces collection costs. The “one-off” cost of applying the method to 10 per cent of the local authorities in the U.K. would be [pound]150,000 and yield a reduction in costs of [pound]500,000 p.a.(b) It enables management to replan rounds quickly as circumstances change.(c) Other applications, which are being investigated, are also discussed.     

Stabilization of a One-Dimensional Dam-River System: Nondissipative and Noncollocated Case

In this paper, we consider a one-dimensional dam-river system, described by a diffusive-wave equation and often used in hydraulic engineering to model the dynamic behavior of the unsteady flow in a river for shallow water when the flow variations are not important. We propose an integral boundary control which leads to a nondissipative closed-loop system with noncollocated actuators and sensors; hence, two main difficulties arise: first, how to show the C ₀-semigroup generation and second, how to achieve the stability of the system. To overcome this situation, the Riesz basis methodology is adopted to show that the closed-loop system generates an analytic semigroup. Concerning the stability, the shooting method is applied to assign the spectrum of the system in the open left-half plane and ensure its exponential stability as well as the output regulation. Numerical simulations are presented for a family of system parameters. The authors express their sincere thanks to Boumenir Amin for valuable comments and suggestions. The first author acknowledges the support of Sultan Qaboos University. The second author was supported by the National Natural Science Foundation of China.     

含有一阶导数的一维p-Laplace方程的正解

通过利用积分方程全连续算子的不动点指数对含有一阶导数的一维p-L ap lace方程建立了一个存在定理.这个定理表明此p-L ap lace方程必有一个正解,只要非线性项在某个有界集合上的“最大高度”是适当的.     

Spectrum of One-Dimensional p-Laplacian with an Indefinite Integrable Weight

Motivated by extremal problems of weighted Dirichlet or Neumann eigenvalues, we will establish two fundamental results on the dependence of weighted eigenvalues of the one-dimensional p-Laplacian on indefinite integrable weights. One is the continuous differentiability of eigenvalues in weights in the Lebesgue spaces L ^γ with the usual norms. Another is the continuity of eigenvalues in weights with respect to the weak topologies in L ^γ spaces. Here 1 ≤ γ ≤ ∞. In doing so, we will give a simpler explanation to the corresponding spectrum problems, with the help of several typical techniques in nonlinear analysis such as the Fréchet derivative and weak* convergence.     

Fractional Goal Programming for Fuzzy Solid Transportation Problem with Interval Cost

In this paper, we study a solid transportation problem with interval cost using fractional goal programming approach (FGP). In real life applications of the FGP problem with multiple objectives, it is difficult for the decision-maker(s) to determine the goal value of each objective precisely as the goal values are imprecise, vague, or uncertain. Therefore, a fuzzy goal programming model is developed for this purpose. The proposed model presents an application of fuzzy goal programming to the solid transportation problem. Also, we use a special type of non-linear (hyperbolic) membership functions to solve multi-objective transportation problem. It gives an optimal compromise solution. The proposed model is illustrated by using an example.     

Quantum Optimal Control Problems with a Sparsity Cost Functional

In this article, the investigation of a class of quantum optimal control problems with L¹ sparsity cost functionals is presented. The focus is on quantum systems modeled by Schrödinger-type equations with a bilinear control structure as it appears in many applications in nuclear magnetic resonance spectroscopy, quantum imaging, quantum computing, and in chemical and photochemical processes. In these problems, the choice of L¹ control spaces promotes sparse optimal control functions that are conveniently produced by laboratory pulse shapers. The characterization of L¹ quantum optimal controls and an efficient numerical semi-smooth Newton solution procedure are discussed.