一类线性奇异摄动最优控制问题的渐近解

研究了一类线性奇异摄动最优控制问题的空间对照结构,讨论了初始点固定,终端自由的情形.首先根据变分法得到了一阶最优性条件,其次运用退化最优控制问题的解证明了异宿轨道的存在性,从而结合奇异摄动理论证明了原问题空间对照结构解的存在性.进一步根据解的结构,利用边界层函数法构造了奇异摄动最优控制问题一致有效的形式渐近解.最后,通...     

Step-like contrast structure for a nonlinear system of singularly perturbed differential equations in the critical case

For an n-dimensional singularly perturbed system of differential equations, we construct the asymptotics of a solution with a step-like contrast structure in the critical case. We prove the existence of a solution and obtain an estimate for the remainder terms of the asymptotic representation of this solution.     

Stability of a supersonic flow past a wedge with adjoint weak neutrally stable shock wave

We study the classical problem of a supersonic stationary flow of a nonviscous nonheat-conducting gas in local thermodynamic equilibrium past an infinite plane wedge. Under the Lopatinski? condition on the shock wave (neutral stability), we prove the well-posedness of the linearized mixed problem (the main solution is a weak shock wave), obtain a representation of the classical solution, where, in this case (in contrast to the case of the uniform Lopatinski? condition—an absolutely stable shock wave), plane waves additionally appear in the representation. If the initial data have compact support, the solution reaches the given regime in infinite time.     

Characterization of Approximate Solutions of Vector Optimization Problems with a Variable Order Structure

In this paper, we deal with approximate solutions in vector-optimization problems with respect to a variable order structure. In the case of exact solutions of a vector optimization problem, especially in the variable order case, authors use a cone or a pointed convex cone-valued map in order to describe the solution concepts but in this paper, we use a set-valued map and this map is not a (pointed convex) cone-valued map necessarily. We characterize these solution concepts by a general scalarization method by means of nonlinear functionals. In the last section, an extension of Ekeland’s variational principle for a vector optimization problem with a variable order structure is given.     

冲裁件有约束最优剪切方式的设计

本文讨论冲裁件有约束最优剪切方式的设计问题 .阐明最优剪切排样方式的规范结构 ;采用分支定界法求解冲裁件无约束排样问题 ;将有约束排样问题转换为求解一系列的无约束排样问题 ,并通过对解的性质分析提高算法效率 .实验计算结果说明本文算法十分有效 .最后给出一例题的最优排样方式 .     

An FPTAS for optimizing a class of low-rank functions over a polytope

We present a fully polynomial time approximation scheme (FPTAS) for optimizing a very general class of non-linear functions of low rank over a polytope. Our approximation scheme relies on constructing an approximate Pareto-optimal front of the linear functions which constitute the given low-rank function. In contrast to existing results in the literature, our approximation scheme does not require the assumption of quasi-concavity on the objective function. For the special case of quasi-concave function minimization, we give an alternative FPTAS, which always returns a solution which is an extreme point of the polytope. Our technique can also be used to obtain an FPTAS for combinatorial optimization problems with non-linear objective functions, for example when the objective is a product of a fixed number of linear functions. We also show that it is not possible to approximate the minimum of a general concave function over the unit hypercube to within any factor, unless P = NP. We prove this by showing a similar hardness of approximation result for supermodular function minimization, a result that may be of independent interest.     

Optimal capital allocation in a hierarchical corporate structure

We consider capital allocation in a hierarchical corporate structure where stakeholders at two organizational levels (e.g., board members vs line managers) may have conflicting objectives, preferences, and beliefs about risk. Capital allocation is considered as the solution to an optimization problem whereby a quadratic deviation measure between individual losses (at both levels) and allocated capital amounts is minimized. Thus, this paper generalizes the framework of Dhaene et al. (2012), by allowing potentially diverging risk preferences in a hierarchical structure. An explicit unique solution to this optimization problem is given. In several examples, it is shown how the optimal capital allocation achieves a compromise between conflicting views of risk within the organization.     

Algorithms for projecting a point onto a level surface of a continuous function on a compact set

Given an equation f(x) = 0, the problem of finding its solution nearest to a given point is considered. In contrast to the authors’ previous works dealing with this problem, exact algorithms are proposed assuming that the function f is continuous on a compact set. The convergence of the algorithms is proved, and their performance is illustrated with test examples.     

The simplified partial digest problem: Approximation and a graph-theoretic model

The goal of the simplified partial digest problem (SPDP) is motivated by the reconstruction of the linear structure of a DNA chain with respect to a given nucleotide pattern, based on the multiset of distances between the adjacent patterns (interpoint distances) and the multiset of distances between each pattern and the two unlabeled endpoints of the DNA chain (end distances). We consider optimization versions of the problem, called SPDP-Min and SPDP-Max. The aim of SPDP-Min (SPDP-Max) is to find a DNA linear structure with the same multiset of end distances and the minimum (maximum) number of incorrect (correct) interpoint distances. Results are presented on the worst-case efficiency of approximation algorithms for these problems. We suggest a graph-theoretic model for SPDP-Min and SPDP-Max, which can be used to reduce the search space for an optimal solution in either of these problems. We also present heuristic polynomial time algorithms based on this model. In computational experiments with randomly generated and real-life input data, our best algorithm delivered an optimal solution in 100% of the instances for a number of restriction sites not greater than 50.     

Superconvergence of collocation methods for a class of weakly singular Volterra integral equations

We discuss the application of spline collocation methods to a certain class of weakly singular Volterra integral equations. It will be shown that, by a special choice of the collocation parameters, superconvergence properties can be obtained if the exact solution satisfies certain conditions. This is in contrast with the theory of collocation methods for Abel type equations. Several numerical examples are given which illustrate the theoretical results.     

Inverse problem for a quasilinear hyperbolic equation with a nonlocal boundary condition containing a delay argument

We consider a problem for a quasilinear hyperbolic equation with a nonlocal condition that contains a retarded argument. By reducing this problem to a nonlinear integrofunctional equation, we prove the existence and uniqueness theorem for its solution. We pose an inverse problem of finding a solution-dependent coefficient of the equation on the basis of additional information on the solution; the information is given at a fixed point in space and is a function of time. We prove the uniqueness theorem for the solution of the inverse problem. The proof is based on the derivation and analysis of an integro-functional equation for the difference of two solutions of the inverse problem.     

具有边值条件的高维吉洪诺夫系统的阶梯状空间对照结构

借助于首次积分构造高维的空间异宿轨道,利用指数二分法的一些性质和Fredholm交换引理,在求解高阶边界函数的同时确定了转移点t~*.利用边界函数法构造形式渐近解,用k+σ交换引理证明了高维吉洪诺夫系统阶梯状空间对照结构解的存在性和形式渐近解的一致有效性.最后举例验证本文的结果.     

Improving the Structure of a Distribution System

An account is given of how an existing distribution structure was examined. Two questions were answered. First, what is the minimum cost solution given that one was starting up a new system? Secondly, what changes are needed to the present structure to obtain the maximum benefit. The steps taken in building simple models to describe the system; and the transition from the answer given to the first question to a programme for implementation are outlined. The proposals were implemented and the savings actually achieved were comparable with those anticipated.     

Differential equations for the analytic singular value decomposition of a matrix

Summary This paper is concerned with finding a smooth singular value decomposition for a matrix which is smoothly dependent on a parameter. A previous approach to this problem was based on minimisation techniques, here, in contrast, a system of ordinary differential equations is derived for the decomposition. It is shown that the numerical solution of an initial value problem associated with these differential equations provides a feasible approach to the solution of this problem. Particular consideration is given to the situation which arises with equal modulus singular values which lead to indeterminacies in the evaluations needed for the numerical solution. Examples which illustrate the behaviour of the method are included.     

Solving a bi-criterion cutting stock problem with open-ended demand: a case study

We consider a real-life cutting stock problem with two types of orders. All orders have to be cut from a given number of raws (also known as stock unit, master reel or jumbo). For each order the width of the final (also known as reels or units) and the number of finals is given. An order is called an exact order when the given number of finals must be produced exactly. An order is called an open order when at least the given number of finals must be produced. There is a given maximum on the number of finals that can be produced from a single raw which is determined by the number of knives on the machine. A pattern specifies the number of finals of a given width that will be produced from one raw. A solution consists of specifying a pattern for each raw such that in total the number of finals of exact orders is produced exactly and at least the number of finals of open orders is produced. There are two criteria defined for a solution. One criterion is the cutting loss: the total width of the raws minus the total width of the produced finals. The second criterion is the number of different patterns used in the solution. We describe a branch-and-bound algorithm that produces all Pareto-optimal solutions.     

On non-transferable utility games with coalition structure

We introduce a solution function for Non-transferable Utility (NTU) games when prior coalition structure is given. This solution function generalizes both the Harsanyi solution function forNTU games and the Owen solution forTU games with coalition structure.I would like to thank Sergiu Hart, Bezalel Peleg and Shmuel Zamir for some conversations and constructive remarks on an earlier version of this paper. Part of this research was supported by the Sonderforschungsbereich 303 in the university of Bonn.     

Scheduling for a processor sharing system with linear slowdown

We consider the problem of scheduling arrivals to a congestion system with a finite number of users having identical deterministic demand sizes. The congestion is of the processor sharing type in the sense that all users in the system at any given time are served simultaneously. However, in contrast to classical processor sharing congestion models, the processing slowdown is proportional to the number of users in the system at any time. That is, the rate of service experienced by all users is linearly decreasing with the number of users. For each user there is an ideal departure time (due date). A centralized scheduling goal is then to select arrival times so as to minimize the total penalty due to deviations from ideal times weighted with sojourn times. Each deviation penalty is assumed quadratic, or more generally convex. But due to the dynamics of the system, the scheduling objective function is non-convex. Specifically, the system objective function is a non-smooth piecewise convex function. Nevertheless, we are able to leverage the structure of the problem to derive an algorithm that finds the global optimum in a (large but) finite number of steps, each involving the solution of a constrained convex program. Further, we put forward several heuristics. The first is the traversal of neighbouring constrained convex programming problems, that is guaranteed to reach a local minimum of the centralized problem. This is a form of a “local search”, where we use the problem structure in a novel manner. The second is a one-coordinate “global search”, used in coordinate pivot iteration. We then merge these two heuristics into a unified “local–global” heuristic, and numerically illustrate the effectiveness of this heuristic.     

Asymptotics of a Solution of a Three-Dimensional Nonlinear Wave Equation near a Butterfly Catastrophe Point

We consider an optimal distributed control problem in a planar convex domain with smooth boundary and a small parameter at the highest derivatives of an elliptic operator. The zero Dirichlet condition is given on the boundary of the domain, and the control is included additively in the inhomogeneity. The set of admissible controls is the unit ball in the corresponding space of square integrable functions. Solutions of the obtained boundary value problems are considered in the generalized sense as elements of a Hilbert space. The optimality criterion is the sum of the squared norm of the deviation of the state from a given state and the squared norm of the control with a coefficient. This structure of the optimality criterion makes it possible to strengthen, if necessary, the role of either the first or the second term of the criterion. In the first case, it is more important to achieve the desired state, while, in the second case, it is preferable to minimize the resource consumption. We study in detail the asymptotics of the problem generated by the sum of the Laplace operator with a small coefficient and a first-order differential operator. A feature of the problem is the presence of the characteristics of the limit operator which touch the boundary of the domain. We obtain a complete asymptotic expansion of the solution of the problem in powers of the small parameter in the case where the optimal control is an interior point of the set of admissible controls.     

Global existence of a restoring force realizing a prescribed half-period

We consider an inverse problem to determine a nonlinearity of an autonomous differential equation so that each solution of the equation has a half-period that is a prescribed function of a half-amplitude of the solution. A global existence of the nonlinearity is proved under the assumption that the prescribed function is Lipschitz continuous. A reconstructive way of the nonlinearity is also given.     

A stochastic optimal control approach to a class of production and inventory problems

A comprehensible and unified system control approach is presented to solve a class of production/inventory smoothing problems. A nonstationary, non-Gaussian, finite-time linear optimal solution with an attractive computation scheme is obtained for a general quadratic and linear cost structure. A complete solution to a classical production/inventory control problem is given as an example. A general solution to the discrete-time optimal regulator with arbitrary but known disturbance is provided and discussed in detail. A computationally attractive closed-loop suboptimal scheme is presented for problems with constraints or nonquadratic costs. Implementation and interpretation of the results are discussed.