Delayed dynamics in heterogeneous competition with product differentiation

Heterogeneous duopolies with product differentiation and isoelastic price functions are examined, in which one firm is quantity setter and the other is price setter. The reaction functions and the Cournot–Bertrand (CB) equilibrium are first determined. It is shown that the best response dynamics with continuous time scales and without time delays, is always locally asymptotically stable. This stability can be, however, lost in the presence of time delays. Both fixed and continuously distributed time delays are examined, stability conditions derived and the stability regions determined and illustrated. The results are compared to Cournot–Cournot (CC) and Bertrand–Bertrand (BB) dynamics. It turns out that continuously distributed lags have a smaller destabilizing effect on the equilibria than fixed lags, and both homogeneous (CC and BB) competitions are more stable than the heterogeneous competitions.     

Power law fluid model for blood flow through a tapered artery with a stenosis

In the present paper, blood flow through a tapered artery with a stenosis is analyzed, assuming the flow is steady and blood is treated as non-Newtonian power law fluid model. Exact solution has been evaluated for velocity, resistance impedance, wall shear stress and shearing stress at the stenosis throat. The graphical results of different types of tapered arteries (i.e. converging tapering, diverging tapering, non-tapered artery) have been examined for different parameters of interest. Some special cases of the problem are also presented.     

New Brownian bridge construction in quasi-Monte Carlo methods for computational finance

Quasi-Monte Carlo (QMC) methods have been playing an important role for high-dimensional problems in computational finance. Several techniques, such as the Brownian bridge (BB) and the principal component analysis, are often used in QMC as possible ways to improve the performance of QMC. This paper proposes a new BB construction, which enjoys some interesting properties that appear useful in QMC methods. The basic idea is to choose the new step of a Brownian path in a certain criterion such that it maximizes the variance explained by the new variable while holding all previously chosen steps fixed. It turns out that using this new construction, the first few variables are more “important” (in the sense of explained variance) than those in the ordinary BB construction, while the cost of the generation is still linear in dimension. We present empirical studies of the proposed algorithm for pricing high-dimensional Asian options and American options, and demonstrate the usefulness of the new BB.     

A Barzilai–Borwein type method for stochastic linear complementarity problems

We consider the expected residual minimization (ERM) formulation of stochastic linear complementarity problem (SLCP). By employing the Barzilai–Borwein (BB) stepsize and active set strategy, we present a BB type method for solving the ERM problem. The global convergence of the proposed method is proved under mild conditions. Preliminary numerical results show that the method is promising.     

R-linear convergence of the Barzilai and Borwein gradient method

Combined with non-monotone line search, the Barzilai and Borwein(BB) gradient method has been successfully extended for solvingunconstrained optimization problems and is competitive withconjugate gradient methods. In this paper, we establish theR-linear convergence of the BB method for any-dimensional stronglyconvex quadratics. One corollary of this result is that theBB method is also locally R-linear convergent for general objectivefunctions, and hence the stepsize in the BB method will alwaysbe accepted by the non-monotone line search when the iterateis close to the solution.     

解大型无约束优化的新的非单调简单模型BB-TR方法（英文）

The trust region(TR) method for optimization is a class of effective methods.The conic model can be regarded as a generalized quadratic model and it possesses the good convergence properties of the quadratic model near the minimizer.The Barzilai and Borwein(BB) gradient method is also an effective method,it can be used for solving large scale optimization problems to avoid the expensive computation and storage of matrices.In addition,the BB stepsize is easy to determine without large computational efforts.In this paper,based on the conic trust region framework,we employ the generalized BB stepsize,and propose a new nonmonotone adaptive trust region method based on simple conic model for large scale unconstrained optimization.Unlike traditional conic model,the Hessian approximation is an scalar matrix based on the generalized BB stepsize,which resulting a simple conic model.By adding the nonmonotone technique and adaptive technique to the simple conic model,the new method needs less storage location and converges faster.The global convergence of the algorithm is established under certain conditions.Numerical results indicate that the new method is effective and attractive for large scale unconstrained optimization problems.     

Decomposition-based Inner- and Outer-Refinement Algorithms for Global Optimization

Traditional deterministic global optimization methods are often based on a Branch-and-Bound (BB) search tree, which may grow rapidly, preventing the method to find a good solution. Motivated by decomposition-based inner approximation (column generation) methods for solving transport scheduling problems with over 100 million variables, we present a new deterministic decomposition-based successive approximation method for general modular and/or sparse MINLPs. The new method, called Decomposition-based Inner- and Outer-Refinement, is based on a block-separable reformulation of the model into sub-models. It generates inner- and outer-approximations using column generation, which are successively refined by solving many easier MINLP and MIP subproblems in parallel (using BB), instead of searching over one (global) BB search tree. We present preliminary numerical results with Decogo (Decomposition-based Global Optimizer), a new parallel decomposition MINLP solver implemented in Python and Pyomo.     

Alternate step gradient method*

The Barzilai and Borwein (BB) gradient method does not guarantee a descent in the objective function at each iteration, but performs better than the classical steepest descent (SD) method in practice. So far, the BB method has found many successful applications and generalizations in linear systems, unconstrained optimization, convex-constrained optimization, stochastic optimization, etc. In this article, we propose a new gradient method that uses the SD and the BB steps alternately. Hence the name “alternate step (AS) gradient method.” Our theoretical and numerical analyses show that the AS method is a promising alternative to the BB method for linear systems. Unconstrained optimization algorithms related to the AS method are also discussed. Particularly, a more efficient gradient algorithm is provided by exploring the idea of the AS method in the GBB algorithm by Raydan (1997).

To establish a general R-linear convergence result for gradient methods, an important property of the stepsize is drawn in this article. Consequently, R-linear convergence result is established for a large collection of gradient methods, including the AS method. Some interesting insights into gradient methods and discussion about monotonicity and nonmonotonicity are also given.     

Performance modeling and analysis of blood flow in elastic arteries

The effect of magnetic field has been examined on rheological models of blood. One. of them is the Power Law model and the other is the generalized Maxwell model. It is noticed that the magnetic field has a significant effect on the flow phenomena. The investigation shows that the model considered here is capable of taking into account the rheological properties affecting the blood flow and hemodynamic features, which may be important for medical doctors to predict diseases for individuals on the basis of the pattern of flow for an elastic artery in the presence of a transverses magnetic field. The effects of a magnetic field have been used to control the flow, which may be useful in certain hypertension cases, etc.     

Robustness of connected balanced block designs

Summary It is well known that the connected incomplete block designs with the highest intrablock efficiency factors are balanced. In a connected balanced block (BB) design, every elementary contrast is estimated with the same variance. If a treatment is lost in a connected BB design, then the residual design is not balanced for the most part. In this paper, an upper bound and a lower bound for efficiency of a residual design are derived with some illustrations. Moreover, from these discussions it is conceivable that the loss of efficiency even in the unbalanced case is, in general, small.     

Mass transfer to blood flowing through arterial stenosis

The present investigation deals with a mathematical model representing the mass transfer to blood streaming through the arteries under stenotic condition. The mass transport refers to the movement of atherogenic molecules, that is, blood-borne components, such as oxygen and low-density lipoproteins from flowing blood into the arterial walls or vice versa. The blood flowing through the artery is treated to be Newtonian and the arterial wall is considered to be rigid having differently shaped stenoses in its lumen arising from various types of abnormal growth or plaque formation. The nonlinear unsteady pulsatile flow phenomenon unaffected by concentration-field of the macromolecules is governed by the Navier–Stokes equations together with the equation of continuity while that of mass transfer is controlled by the convection-diffusion equation. The governing equations of motion accompanied by appropriate choice of the boundary conditions are solved numerically by MAC(Marker and Cell) method and checked numerical stability with desired degree of accuracy. The quantitative analysis carried out finally includes the respective profiles of the flow-field and concentration along with their distributions over the entire arterial segment as well. The key factors like the wall shear stress and Sherwood number are also examined for further qualitative insight into the flow and mass transport phenomena through arterial stenosis. The present results show quite consistency with several existing results in the literature which substantiate sufficiently to validate the applicability of the model under consideration.      

Mass transfer to blood flowing through arterial stenosis

The present investigation deals with a mathematical model representing the mass transfer to blood streaming through the arteries under stenotic condition. The mass transport refers to the movement of atherogenic molecules, that is, blood-borne components, such as oxygen and low-density lipoproteins from flowing blood into the arterial walls or vice versa. The blood flowing through the artery is treated to be Newtonian and the arterial wall is considered to be rigid having differently shaped stenoses in its lumen arising from various types of abnormal growth or plaque formation. The nonlinear unsteady pulsatile flow phenomenon unaffected by concentration-field of the macromolecules is governed by the Navier–Stokes equations together with the equation of continuity while that of mass transfer is controlled by the convection-diffusion equation. The governing equations of motion accompanied by appropriate choice of the boundary conditions are solved numerically by MAC(Marker and Cell) method and checked numerical stability with desired degree of accuracy. The quantitative analysis carried out finally includes the respective profiles of the flow-field and concentration along with their distributions over the entire arterial segment as well. The key factors like the wall shear stress and Sherwood number are also examined for further qualitative insight into the flow and mass transport phenomena through arterial stenosis. The present results show quite consistency with several existing results in the literature which substantiate sufficiently to validate the applicability of the model under consideration.     

A new gradient method via quasi-Cauchy relation which guarantees descent

We propose a new monotone algorithm for unconstrained optimization in the frame of Barzilai and Borwein (BB) method and analyze the convergence properties of this new descent method. Motivated by the fact that BB method does not guarantee descent in the objective function at each iteration, but performs better than the steepest descent method, we therefore attempt to find stepsize formula which enables us to approximate the Hessian based on the Quasi-Cauchy equation and possess monotone property in each iteration. Practical insights on the effectiveness of the proposed techniques are given by a numerical comparison with the BB method.     

无约束最优化的信赖域BB法

梯度法是求解无约束最优化的一类重要方法.步长选取的好坏与梯度法的数值表现息息相关.注意到BB步长隐含了目标函数的二阶信息,本文将BB法与信赖域方法相结合,利用BB步长的倒数去近似目标函数的Hesse矩阵,同时利用信赖域子问题更加灵活地选取梯度法的步长,给出求解无约束最优化问题的单调和非单调信赖域BB法.在适当的假设条件下,证明了算法的全局收敛性.数值试验表明,与已有的求解无约束优化问题的BB类型的方法相比,非单调信赖域BB法中e_k=‖x_k-x~*‖的下降呈现更明显的阶梯状和单调性,因此收敛速度更快.     

Gradient Methods with Adaptive Step-Sizes

Motivated by the superlinear behavior of the Barzilai-Borwein (BB) method for two-dimensional quadratics, we propose two gradient methods which adaptively choose a small step-size or a large step-size at each iteration. The small step-size is primarily used to induce a favorable descent direction for the next iteration, while the large step-size is primarily used to produce a sufficient reduction. Although the new algorithms are still linearly convergent in the quadratic case, numerical experiments on some typical test problems indicate that they compare favorably with the BB method and some other efficient gradient methods.     

A Branch-and-Bound Algorithm for Concave Network Flow Problems

In this paper a Branch-and-Bound (BB) algorithm is developed to obtain an optimal solution to the single source uncapacitated minimum cost Network Flow Problem (NFP) with general concave costs. Concave NFPs are NP-Hard, even for the simplest version therefore, there is a scarcity of exact methods to address them in their full generality. The BB algorithm presented here can be used to solve optimally single source uncapacitated minimum cost NFPs with any kind of concave arc costs. The bounding is based on the computation of lower bounds derived from state space relaxations of a dynamic programming formulation. The relaxations, which are the subject of the paper (Fontes et al., 2005b) and also briefly discussed here, involve the use of non-injective mapping functions, which guarantee a reduction on the cardinality of the state space. Branching is performed by either fixing an arc as part of the final solution or by removing it from the final solution. Computational results are reported and compared to available alternative methods for addressing the same type of problems. It could be concluded that our BB algorithm has better performance and the results have also shown evidence that it has a sub-exponential time growth.     

Prediction-correction method with BB step sizes

In the prediction-correction method for variational inequality (VI) problems, the step size selection plays an important role for its performance. In this paper, we employ the Barzilai-Borwein (BB) strategy in the prediction step, which is efficient for many optimization problems from a computational point of view. To guarantee the convergence, we adopt the line search technique, and relax the conditions to accept the BB step sizes as large as possible. In the correction step, we utilize a longer step length to calculate the next iteration point. Finally, we present some preliminary numerical results to show the efficiency of the algorithms.     

Some projection methods with the BB step sizes for variational inequalities

Since the appearance of the Barzilai-Borwein (BB) step sizes strategy for unconstrained optimization problems, it received more and more attention of the researchers. It was applied in various fields of the nonlinear optimization problems and recently was also extended to optimization problems with bound constraints. In this paper, we further extend the BB step sizes to more general variational inequality (VI) problems, i.e., we adopt them in projection methods. Under the condition that the underlying mapping of the VI problem is strongly monotone and Lipschitz continuous and the modulus of strong monotonicity and the Lipschitz constant satisfy some further conditions, we establish the global convergence of the projection methods with BB step sizes. A series of numerical examples are presented, which demonstrate that the proposed methods are convergent under mild conditions, and are more efficient than some classical projection-like methods.     

An evaluation of inventory and transportation policies of a regional blood distribution system

Blood is a valuable but perishable community resource. Regional blood centers coordinate the blood drawing and inventory policies of the hospital blood banks in a region for more efficient use of the resource. This study used a simulation model to analyze the costs and effects of several different operational policies for a regional blood center. Simulated experiments were carried out in four areas: (1) increasing the amount of blood available, (2) changing the number of delivery vehicles, (3) comparing two types of inventory consignment policies (shipping blood to hospitals with a recall privilege and frequent redistribution of blood amongst regional hospitals) to a direct-sale, no-redistribution policy, and (4) examining the effect of sending fresher blood supplies as inventories to hospitals with lower probabilities of transfusion. The results of the third experiment are presented in detail. It was found that periodic redistribution of the regional inventory yielded lower expiration rates and lower shortage rates. The results of the other three experiments are presented briefly.     

A new adaptive Barzilai and Borwein method for unconstrained optimization

In this paper we view the Barzilai and Borwein (BB) method from a new angle, and present a new adaptive Barzilai and Borwein (NABB) method with a nonmonotone line search for general unconstrained optimization. In the proposed method, the scalar approximation to the Hessian matrix is updated by the Broyden class formula to generate an adaptive stepsize. It is remarkable that the new stepsize is chosen adaptively in the interval which contains the two well-known BB stepsizes. Moreover, for the negative curvature direction, a strategy for the choice of the stepsize is designed to accelerate the convergence rate of the NABB method. Furthermore, we apply the NABB method without any line search to strictly convex quadratic minimization. The numerical experiments show the NABB method is very promising.