A posteriori joint detection of reference fragments in a quasi-periodic sequence

The problem of joint detection of quasi-periodic reference fragments (of given size) in a numerical sequence and its partition into segments containing series of recurring reference fragments is solved in the framework of the a posteriori approach. It is assumed that (i) the number of desired fragments is not known, (ii) an ordered reference tuple of sequences to be detected is given, (iii) the index of the sequence member corresponding to the beginning of a fragment is a deterministic (not random) value, and (iv) a sequence distorted by an additive uncorrelated Gaussian noise is available for observation. It is established that the problem consists of testing a set of hypotheses about the mean of a random Gaussian vector. The cardinality of the set grows exponentially as the vector dimension (i.e., the sequence length) increases. It is shown that the search for a maximum-likelihood hypothesis is equivalent to the search for arguments that minimize an auxiliary objective function. It is proved that the minimization problem for this function can be solved in polynomial time. An exact algorithm for its solution is substantiated. Based on the solution to an auxiliary extremum problem, an efficient a posteriori algorithm producing an optimal (maximum-likelihood) solution to the partition and detection problem is proposed. The results of numerical simulation demonstrate the noise stability of the algorithm.     

Extremum conditions for a nonsmooth function in terms of exhausters and coexhausters

The notions of upper and lower exhausters and coexhausters are discussed and necessary conditions for an unconstrained extremum of a nonsmooth function are derived. The necessary conditions for a minimum are formulated in terms of an upper exhauster (coexhauster) and the necessary conditions for a maximum are formulated in terms of a lower exhauster (coexhauster). This involves the problem of transforming an upper exhauster (coexhauster) into a lower exhauster (coexhauster) and vice versa. The transformation is carried out by means of a conversion operation (converter). Second-order approximations obtained with the help of second-order (upper and lower) coexhausters are considered. It is shown how a secondorder upper coexhauster can be converted into a lower coexhauster and vice versa. This problem is reduced to using a first-order conversion operator but in a space of a higher dimension. The obtained result allows one to construct second-order methods for the optimization of nonsmooth functions (Newton-type methods).     

Adjoint Coexhausters in Nonsmooth Analysis and Extremality Conditions

In the classical (“smooth”) mathematical analysis, a differentiable function is studied by means of the derivative (gradient in the multidimensional space). In the case of nondifferentiable functions, the tools of nonsmooth analysis are to be employed. In convex analysis and minimax theory, the corresponding classes of functions are investigated by means of the subdifferential (it is a convex set in the dual space), quasidifferentiable functions are treated via the notion of quasidifferential (which is a pair of sets). To study an arbitrary directionally differentiable function, the notions of upper and lower exhausters (each of them being a family of convex sets) are used. It turns out that conditions for a minimum are described by an upper exhauster, while conditions for a maximum are stated in terms of a lower exhauster. This is why an upper exhauster is called a proper one for the minimization problem (and an adjoint exhauster for the maximization problem) while a lower exhauster will be referred to as a proper one for the maximization problem (and an adjoint exhauster for the minimization problem). The directional derivatives (and hence, exhausters) provide first-order approximations of the increment of the function under study. These approximations are positively homogeneous as functions of direction. They allow one to formulate optimality conditions, to find steepest ascent and descent directions, to construct numerical methods. However, if, for example, the maximizer of the function is to be found, but one has an upper exhauster (which is not proper for the maximization problem), it is required to use a lower exhauster. Instead, one can try to express conditions for a maximum in terms of upper exhauster (which is an adjoint one for the maximization problem). The first to get such conditions was Roshchina. New optimality conditions in terms of adjoint exhausters were recently obtained by Abbasov. The exhauster mappings are, in general, discontinuous in the Hausdorff metric, therefore, computational problems arise. To overcome these difficulties, the notions of upper and lower coexhausters are used. They provide first-order approximations of the increment of the function which are not positively homogeneous any more. These approximations also allow one to formulate optimality conditions, to find ascent and descent directions (but not the steepest ones), to construct numerical methods possessing good convergence properties. Conditions for a minimum are described in terms of an upper coexhauster (which is, therefore, called a proper coexhauster for the minimization problem) while conditions for a maximum are described in terms of a lower coexhauster (which is called a proper one for the maximization problem). In the present paper, we derive optimality conditions in terms of adjoint coexhausters.     

Intertemporal incentive allocation in simple hierarchies

The problem of optimally allocating intertemporal incentive payments in a hierarchy to coordinate the observable actions of subordinate members is posed and solved for a principal, desiring to maximize the present value of his share of organizational rent. The analysis includes: (1) an endogenous determination of the equilibrium state of the hierarchy, (2) a derivation of an optimal sequence of intertemporal allocation patterns for the principal, (3) a characterization of the most-rapid approach to the equilibrium state, and (4) an analysis of the distribution of organizational rent between participants at each moment.     

On a periodic motion of a rigid body carrying a material point in the presence of impacts with a horizontal plane

A material system consisting of an outer rigid body (a shell) and an inner body (a material point) is considered. The system moves in a uniform field of gravity over a fixed absolutely smooth horizontal plane. The central ellipsoid of inertia of the shell is an ellipsoid of rotation. The material point moves according to the harmonic law along a straight-line segment rigidly attached to the shell and lying on its axis of dynamical symmetry. During its motion, the shell may collide with the plane. The coefficient of restitution for an impact is supposed to be arbitrary. The periodic motion of the shell is found when its symmetry axis is situated along a fixed vertical, and the shell rotates around this vertical with an arbitrary constant angular velocity. The conditions for existence of this periodic motion are obtained, and its linear stability is studied.     

On the numerical solution of the quadratic eigenvalue complementarity problem

The Quadratic Eigenvalue Complementarity Problem (QEiCP) is an extension of the Eigenvalue Complementarity Problem (EiCP) that has been introduced recently. Similar to the EiCP, the QEiCP always has a solution under reasonable hypotheses on the matrices included in its definition. This has been established in a previous paper by reducing a QEiCP of dimension n to a special 2n-order EiCP. In this paper we propose an enumerative algorithm for solving the QEiCP by exploiting this equivalence with an EiCP. The algorithm seeks a global minimum of a special Nonlinear Programming Problem (NLP) with a known global optimal value. The algorithm is shown to perform very well in practice but in some cases terminates with only an approximate optimal solution to NLP. Hence, we propose a hybrid method that combines the enumerative method with a fast and local semi-smooth method to overcome the latter drawback. This algorithm is also shown to be useful for computing a positive eigenvalue for an EiCP under similar assumptions. Computational experience is reported to demonstrate the efficacy and efficiency of the hybrid enumerative method for solving the QEiCP.     

A boundary integral formulation for elastically deformable particles in a viscous fluid

This paper reports a formulation and implementation of a mixed (both direct and indirect) boundary element method using the double layer and its adjoint in a form suitable for solving Stokes flow problems involving elastically deformable particles. The formulation is essentially the Completed Double Layer Boundary Element Method for solving an exterior traction problem for the surrounding fluid or solid phase, followed by an interior displacement, and a mobility problem (if required) for the elastic particles. At the heart of the method is a deflation procedure that allows iterative solution strategies to be adopted, effectively opens the way for large-scale simulations of suspensions of deformable particles to be performed. Several problems are considered, to illustrate and benchmark the method. In particular, an analytical solution for an elastic sphere in an elongational flow is derived. The stresslet calculations for an elastic sphere in shear and elongational flows indicate that elasticity of the inclusions can potentially lead to positive second normal stress difference in shear flow, and an increase in the tensile resistance in elongational flow.This work is supported by a grant from the Australian Research Grant Council. X-J F wishes to acknowledge the support of the National Natural Science Foundation of China.     

An efficient computational method for a stochastic dynamic lot-sizing problem under service-level constraints

We provide an efficient computational approach to solve the mixed integer programming (MIP) model developed by Tarim and Kingsman [8] for solving a stochastic lot-sizing problem with service level constraints under the static-dynamic uncertainty strategy. The effectiveness of the proposed method hinges on three novelties: (i) the proposed relaxation is computationally efficient and provides an optimal solution most of the time, (ii) if the relaxation produces an infeasible solution, then this solution yields a tight lower bound for the optimal cost, and (iii) it can be modified easily to obtain a feasible solution, which yields an upper bound. In case of infeasibility, the relaxation approach is implemented at each node of the search tree in a branch-and-bound procedure to efficiently search for an optimal solution. Extensive numerical tests show that our method dominates the MIP solution approach and can handle real-life size problems in trivial time.     

The inverse solution of the atomic mixing equations by an operator-splitting method

The quantification problem of recovering the original material distribution from secondary ion mass spectrometry (SIMS) data is considered in this paper. It is an inverse problem, is ill-posed and hence it requires a special technique for its solution. The quantification problem is essentially an inverse diffusion or (classically) a backward heat conduction problem. In this paper an operator-splitting method (that is proposed in a previous paper by the first author for the solution of inverse diffusion problems) is developed for the solution of the problem of recovering the original structure from the SIMS data. A detailed development of the quantification method is given and it is applied to typical data to demonstrate its effectiveness.     

FCM聚类算法中模糊加权指数m的优选方法

模糊c-均值(FCM)聚类算法是一种通过目标函数的极小化来获得数据集模糊划分的方法。其中,模糊加权指数m对FCM算法的分类性能有着重要的影响,而调用FCM算法进行模糊聚类分析时又必须给m赋值。因此,模糊加权指数m的优选研究就变得很有意义。基于模糊决策的方法本文给出了一种对m的优选方法,实验结果表明该方法是有效的。     

The splittance of a graph

The splittance of an arbitrary graph is the minimum number of edges to be added or removed in order to produce a split graph (i.e. a graph whose vertex set can be partitioned into a clique and an independent set). The splittance is seen to depend only on the degree sequence of the graph, and an explicit formula for it is derived. This result allows to give a simple characterization of the degree sequences of split graphs. Worst cases for the splittance are determined for some classes of graphs (the class of all graphs, of all trees and of all planar graphs).     

A review of Hopfield neural networks for solving mathematical programming problems

The Hopfield neural network (HNN) is one major neural network (NN) for solving optimization or mathematical programming (MP) problems. The major advantage of HNN is in its structure can be realized on an electronic circuit, possibly on a VLSI (very large-scale integration) circuit, for an on-line solver with a parallel-distributed process. The structure of HNN utilizes three common methods, penalty functions, Lagrange multipliers, and primal and dual methods to construct an energy function. When the function reaches a steady state, an approximate solution of the problem is obtained. Under the classes of these methods, we further organize HNNs by three types of MP problems: linear, non-linear, and mixed-integer. The essentials of each method are also discussed in details. Some remarks for utilizing HNN and difficulties are then addressed for the benefit of successive investigations. Finally, conclusions are drawn and directions for future study are provided.     

On the organization of large shared Model bases

Central to the Model Management (MM) function is the creation and maintenance of a knowledge-based model repository. The Model Knowledge Base (MKB) provides the basis by which information about models can be shared to facilitate consistent and controlled utilization of existing models for decision making, as well as the development of new models. Various schemes for representing individual models have been proposed in the literature. This paper focuses on how best to structure, control, and administer a large MKB to support organization-wide modeling activities. Guided by a recently proposed systems framework for MM, we describe a number of concepts which are useful for capturing the semantics and structural relationships of models in an MKB. These concepts, and the nature of the MMS functions to be supported, are then used to derive specific information management requirements for model bases. Four major requirements are identified: (1) management of composite model configurations; (2) management of model version histories; (3) support for the model consultation and selection functions of an MMS; and (4) support for multiple logical MKBs (private, group, and public). We argue that traditional record-based approaches to data management appear to fall short of capturing the rich semantics present in an MM environment. The paper proposes an architecture for an MMS, focusing on its major component — the MKB Management Subsystem. An implementation of this architecture is briefly described.     

Distributed Solution of Constrained Optimal Control Problems Using the Alternating Direction Method of Multipliers

Many control problems in science and engineering can be formulated as optimal control problems (OCP) such as load changes in process control or point-to-point motion of industrial robots in a time-optimal or energy-optimal way while accounting for physical or security constraints. Hence, an efficient way to handle constrained OCPs is an important topic of research. A promising approach to address this issue is a transformation technique allowing to reformulate an inequality constrained OCP into an equality constrained counterpart. The reformulated OCP reveals to possess a particular structure that is favorable for a decomposition and the application of distributed optimization methods. Motivated by the Augmented Lagrangian approach, the structure of the reformulated OCP can be exploited to derive a decomposition method for splitting up the entire OCP into smaller subproblems. In addition, an algorithm is presented that follows the ideas of the Alternating Direction Method of Multipliers (ADMM) and solves the resulting subproblems in a distributed manner. The approach is applied to a mechatronic example system to demonstrate the performance of the presented method. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Erlang loss bounds for OT–ICU systems

In hospitals, patients can be rejected at both the operating theater (OT) and the intensive care unit (ICU) due to limited ICU capacity. The corresponding ICU rejection probability is an important service factor for hospitals. Rejection of an ICU request may lead to health deterioration for patients, and for hospitals to costly actions and a loss of precious capacity when an operation is canceled. There is no simple expression available for this ICU rejection probability that takes the interaction with the OT into account. With c the ICU capacity (number of ICU beds), this paper proves and numerically illustrates a lower bound by an M|G|c|c system and an upper bound by an M|G|c-1|c-1 system, hence by simple Erlang loss expressions. The result is based on a product form modification for a special OT–ICU tandem formulation and proved by a technically complicated Markov reward comparison approach. The upper bound result is of particular practical interest for dimensioning an ICU to secure a prespecified service quality. The numerical results include a case study.     

Using OR for diagnosis and facilitation in change programmes: a university application

In this paper, a methodology for the ongoing diagnosis and facilitation (when required) of organizational change management programmes in an Australian University is described. The application of this methodology on an annual basis requires the assessment of research attitudes and behaviour, areas that have assumed considerable importance within universities around the world in the last decade. This increasing importance associated with research stems in part (a significant part) from the linkage of research quantum (output) of universities to their funding from government. The methodology developed in this paper embraces a wide range of OR-type techniques as well as a range of change management tools from human resources management. These two sets of techniques and tools work together as tools in their own right as well as providing the infrastructure to achieve the objectives. The developed methodology is an interesting blend of hard techniques and ‘soft’ approaches implemented through a soft heuristic, indeed this application is an example of mixed-mode modelling. The OR techniques (conventional in both their nature and application) are comprised of social judgement theory (for bench-marking research attitudes) and integer linear programming (for setting research targets). The results of the application of the developed methodology are discussed in terms of the effect on research quantum over a 5-year period. A similar methodology could be developed for a change process covering aspects of an organization other than research output in a number of countries.     

A small-gain approach to networked control systems with asynchronous communication

The aim of this work is to give a procedure to compute an ISS Lyapunov function (ISS - input-to-state stability) for a large-scale networked control system (NCS) in which sensors, controllers and actuators are connected via multiple communication networks that operate asynchronously, [1]. The main idea is that the NCS is modeled as an interconnection of controlled subsystems and (hybrid) error systems, for which local ISS Lyapunov functions can be derived from the underlying medium access control (MAC) protocol systematically. The application of a small-gain result leads to the construction of an ISS Lyapunov function for the whole NCS. In particularly, maximally allowable transfer intervals (MATIs) and maximally allowable delays (MADs) for each communication network can be computed that are directly related to the ISS gains and convergence rates. In this respect, tradeoffs between the quality-of-control (QoC) of the whole NCS and the quality-of-service (QoS) of the communication networks become explicit. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

SOME THEOREMS ON CONVEX HYPERSURFACES IN AN AFFINE SPACE

Let M and M~* be hypersurfaces in an affine space A~(n+1) of dimension n+1. The main results of this paper are the following:(1) Two types of integral formulas for M and M~*.(2) Some conditions for an affine convex hypersurface to be an affine sphere.(3) Some conditions for M and M~* to be different only by an affine transformation or a translation.     

Dynamics of hepatitis C under optimal therapy and sampling based analysis

We examine two models for hepatitis C viral (HCV) dynamics, one for monotherapy with interferon (IFN) and the other for combination therapy with IFN and ribavirin. Optimal therapy for both the models is determined using the steepest gradient method, by defining an objective functional which minimizes infected hepatocyte levels, virion population and side-effects of the drug(s). The optimal therapies for both the models show an initial period of high efficacy, followed by a gradual decline. The period of high efficacy coincides with a significant decrease in the viral load, whereas the efficacy drops after hepatocyte levels are restored.We use the Latin hypercube sampling technique to randomly generate a large number of patient scenarios and study the dynamics of each set under the optimal therapy already determined. Results show an increase in the percentage of responders (indicated by drop in viral load below detection levels) in case of combination therapy (72%) as compared to monotherapy (57%). Statistical tests performed to study correlations between sample parameters and time required for the viral load to fall below detection level, show a strong monotonic correlation with the death rate of infected hepatocytes, identifying it to be an important factor in deciding individual drug regimens.     

Construction of Lyapunov functions for nonlinear planar systems by linear programming

Recently the authors proved the existence of piecewise affine Lyapunov functions for dynamical systems with an exponentially stable equilibrium in two dimensions (Giesl and Hafstein, 2010 [7]). Here, we extend these results by designing an algorithm to explicitly construct such a Lyapunov function. We do this by modifying and extending an algorithm to construct Lyapunov functions first presented in Marinosson (2002) [17] and further improved in Hafstein (2007) [10]. The algorithm constructs a linear programming problem for the system at hand, and any feasible solution to this problem parameterizes a Lyapunov function for the system. We prove that the algorithm always succeeds in constructing a Lyapunov function if the system possesses an exponentially stable equilibrium. The size of the region of the Lyapunov function is only limited by the region of attraction of the equilibrium and it includes the equilibrium.