Pseudospectral methods for solving infinite-horizon optimal control problems

An important aspect of numerically approximating the solution of an infinite-horizon optimal control problem is the manner in which the horizon is treated. Generally, an infinite-horizon optimal control problem is approximated with a finite-horizon problem. In such cases, regardless of the finite duration of the approximation, the final time lies an infinite duration from the actual horizon at t=+∞. In this paper we describe two new direct pseudospectral methods using Legendre–Gauss (LG) and Legendre–Gauss–Radau (LGR) collocation for solving infinite-horizon optimal control problems numerically. A smooth, strictly monotonic transformation is used to map the infinite time domain t∈[0,∞) onto a half-open interval τ∈[−1,1). The resulting problem on the finite interval is transcribed to a nonlinear programming problem using collocation. The proposed methods yield approximations to the state and the costate on the entire horizon, including approximations at t=+∞. These pseudospectral methods can be written equivalently in either a differential or an implicit integral form. In numerical experiments, the discrete solution exhibits exponential convergence as a function of the number of collocation points. It is shown that the map ?:[−1,+1)→[0,+∞) can be tuned to improve the quality of the discrete approximation.     

Optimization of projection methods for solving ill-posed problems

In this paper we propose a modification of the projection scheme for solving ill-posed problems. We show that this modification allows to obtain the best possible order to accuracy of Tikhonov Regularization using an amount of information which is far less than for the standard projection technique.     

Direct analytical methods for solving Poisson equations in computervision problems

Direct analytical methods are discussed for solving Poisson equations of the general form Δu=f on a rectangular domain. Some embedding techniques that may be useful when boundary conditions (obtained from stereo and occluding boundary) are defined on arbitrary contours are described. The suggested algorithms are computationally efficient owing to the use of fast orthogonal transforms. Applications to shape from shading, lightness and optical flow problems are also discussed. A proof for the existence and convergence of the flow estimates is given. Experiments using synthetic images indicate that results comparable to those using multigrid can be obtained in a very small number of iterations     

Spline methods for solving system of second-order boundary-value problems

In this paper, we use cubic polynomial splines to derive some consistency relations which are then used to develop a numerical method for computing smooth approximations to the solution and its derivatives for a system of second order boundary value problems associated with obstacle, unilateral and contact problems. We show that the present method gives approximations which are better than that produced by other collocation, finite difference and spline methods. Numerical example is presented to illustrate the applicability of the new method.     

Differential dynamic programming methods for solving bang-bang control problems

Differential dynamic programming is a technique, based on dynamic programming rather than the calculus of variations, for determining the optimal control function of a nonlinear system. Unlike conventional dynamic programming where the optimal cost function is considered globally, differential dynamic programming applies the principle of optimality in the neighborhood of a nominal, possibly nonoptimal, trajectory. This allows the coefficients of a linear or quadratic expansion of the cost function to be computed in reverse time along the trajectory: these coefficients may then be used to yield a new improved trajectory (i.e., the algorithms are of the "successive sweep" type). A class of nonlinear control problems, linear in the control variables, is studied using differential dynamic programming. It is shown that for the free-end-point problem, the first partial derivatives of the optimal cost function are continuous throughout the state space, and the second partial derivatives experience jumps at switch points of the control function. A control problem that has an aualytic solution is used to illustrate these points. The fixed-end-point problem is converted into an equivalent free-end-point problem by adjoining the end-point constraints to the cost functional using Lagrange multipliers: a useful interpretation for Pontryagin's adjoint variables for this type of problem emerges from this treatment. The above results are used to devise new second- and first-order algorithms for determining the optimal bang-bang control by successively improving a nominal guessed control function. The usefulness of the proposed algorithms is illustrated by the computation of a number of control problem examples.     

Fast corrected Uzawa methods for solving symmetric saddle point problems

Abstract We consider the solution of linear systems of saddle point problems by two nonlinear iterative methods, which are similar to Uzawa-type methods and called corrected Uzawa methods. Their convergence rates are analyzed. The results of numerical experiments are presented when we apply them to solve the Stokes equations discretized by mixed finite elements. Keywords: Saddle point problem, Uzawa-type algorithm, Schur complement, Stokes equation     

Tseng type methods for solving inclusion problems and its applications

In this paper, we introduce two modifications of the forward–backward splitting method with a new step size rule for inclusion problems in real Hilbert spaces. The modifications are based on Mann and viscosity-ideas. Under standard assumptions, such as Lipschitz continuity and monotonicity (also maximal monotonicity), we establish strong convergence of the proposed algorithms. We present two numerical examples, the first in infinite dimensional spaces, which illustrates mainly the strong convergence property of the algorithm. For the second example, we illustrate the performances of our scheme, compared with the classical forward–backward splitting method for the problem of recovering a sparse noisy signal. Our result extend some related works in the literature and the primary experiments might also suggest their potential applicability.     

非单调信赖域方法求解无约束非光滑优化问题

提出了非单调信赖域算法求解无约束非光滑优化问题,并和经典的信赖域方法作比较分析。同时,设定了一些条件,在这些假设条件下证明了该算法是整体收敛的。数值实验结果表明,非单调策略对无约束非光滑优化问题的求解是行之有效的,拓展了非单调信赖域算法的应用领域。     

Network dual steepest-edge methods for solving capacitated multicommodity network problems

This paper presents a two-phased network dual steepest-edge method for solving capacitated multicommodity network problems. In the first phase, an advanced starting solution in concert with a dual steepest-edge method is applied to solve each capacitated single-commodity network problem. At each iteration, either the primal infeasibility is improved or the dual objective value is inceased. In the second phase, the steepest-edge selection criterion is used to determine the leaving infeasible coupling constraint. By maintaining dual feasibility while improving the dual objective value, the number of infeasible coupling constraints is monotonically reduced to zero. The finite convergency property of this algorithm is shown. Finally, this algorithm is coded using Pascal language and tested in several problems. Results show this algorithm is promising.     

多期证券投资组合问题的区间多目标规划求解方法

投资组合问题主要研究如何将有限的资金合理地分配到不同的金融资产中,以实现收益最大化与风险最小化之间的均衡.然而,证券市场往往具有很强的不确定性,投资者对于证券的期望收益率和风险损失率难以用精确数值描述,区间规划则是处理这类不确定性问题的有力工具.鉴于此,首先基于区间多目标规划建立一个以预期收益率、风险损失率和流动性为目标函数的多期投资组合选择模型;然后通过设计一个定向变异算子,改进基于偏好多面体的交互式遗传算法,并将上述算法的运算机制与所建模型的多期特性相结合以求解模型;最后在不确定交互进化优化系统上进行实证分析.实验结果表明,所提出算法能够根据投资者的不同需要得到相应最满意的多期资产组合.     

计算机安全问题与保密技术的解决策略

随着当前中国经济的不断发展,计算机网络现已在企业中得到广泛的应用。针对信息的安全性一直存在着严重的弊端,但是随着科技信息技术的不断改革,对计算机网络系统中的配件设置进行优化升级,其中包括对硬件设备的改造,操作软件系统的升级等方面都进行了全方位的优化。保证计算机设备系统的安全性,使其确保计算机工作运行有一个良好的运作环境。使得外界传输系统中不明代码难以攻克防防火墙火墙系统,保证个人信息的有效性。     

Coupling of continuous and discontinuous Galerkin methods for transport problems

In this paper, we formulate a coupled discontinuous/continuous Galerkin method for the numerical solution of convection–diffusion (transport) equations, where convection may be dominant. One motivation for this approach is to use a discontinuous method where the solution is rough, e.g., in regions of high gradients, and use a continuous method where the solution is smooth. In this approach, the domain is decomposed into two regions, and appropriate transmission conditions are applied at the interface between regions. In one region, a local discontinuous Galerkin method is applied, and in the other region a standard continuous Galerkin method is employed. Stability and a priori error estimates for the coupled method are derived, and numerical results in one space dimension are given for smooth problems and problems with sharp fronts.     

On the convergence of spectral multigrid methods for solving periodic problems

The spectral multigrid method combines the efficiencies of the spectral method and the multigrid method. In this paper, we show that various spectral multigrid methods have constant convergence rates (independent of the number of unknowns in the linear system, to be solved) in their multilevel iterations for solving periodic problems.