Solution of optimal control problem in class of piecewise-constant functions

A numerical method of solution of problems of optimal control of objects that may be described by systems of ordinary differential equations in the class of piecewise-constant controls is proposed in the article. At the same time, both the piecewise-constant values of the controls as well as the constancy intervals of these values are optimized. Analytic formulas of the gradient of the functional with respect to the optimized parameters are obtained. The formulas for the gradient of the functional obtained for an initial continuous optimal control problem and the corresponding discretized optimal control problem are compared and results of numerical experiments are presented.     

Explicit preconditioned domain decomposition schemes for solving nonlinear boundary value problems

A new class of inner-outer iterative procedures in conjunction with Picard-Newton methods based on explicit preconditioning iterative methods for solving nonlinear systems is presented. Explicit preconditioned iterative schemes, based on the explicit computation of a class of domain decomposition generalized approximate inverse matrix techniques are presented for the efficient solution of nonlinear boundary value problems on multiprocessor systems. Applications of the new composite scheme on characteristic nonlinear boundary value problems are discussed and numerical results are given.     

Two-step explicit methods for second-order IVPs with oscillatory solutions

In this paper, we present a class of non-conventional two-step explicit methods for solving the second-order initial value problems (IVPs) with periodic or oscillatory solutions. The methods have second algebraic order, a large interval of periodicity and high phase-lag order. For multi-dimensional problems, we give the vector form of the methods with the aid of a special vector operation. Some numerical results are reported to illustrate the efficiency of our methods.     

Monotonicity for Runge–Kutta Methods: Inner Product Norms

An important class of ordinary differential systems is that whose solutions satisfy a monotonicity property for a given norm. For these problems, a natural requirement for the numerical solution is the reflection of this monotonicity property, perhaps under certain stepsize restriction. For Runge–Kutta methods, when the applied norm is an arbitrary one, the stepsize restrictions depend on the radius of absolute monotonicity. However for many problems, monotonicity holds for inner product norms and therefore it makes sense to restrict the analysis to this class of norms to obtain, if possible, less restrictive results. In this paper, we consider monotonicity issues for Runge–Kutta methods when the applied norm is an inner product norm.     

Robust stability analysis of systems with real parametric uncertainty: A global optimization approach

This paper presents an optimization framework for the robustness analysis of linear and nonlinear systems with real parameter uncertainty. For linear systems, a nonlinear programming formulation for the exact calculation of the stability margin is presented. The potential of decomposition-based global optimization methods for the solution of this nonconvex problem is discussed. Next the concept of the stability margin is extended to a class of nonlinear systems. A nonlinear stability margin and a uniqueness margin are defined to address the effect of parametric uncertainty on the stability of a particular steady state, as well as on the number of steady states of the system. This analysis allows for the derivation of necessary and sufficient conditions for robust stability and robust uniqueness of the steady state of the system in the presence of parametric uncertainty.     

基于区间优化的神经网络全局优化方法

Hopfield神经网络被广泛应用于优化问题的求解中,而传统的Hopfield网络通常基于梯度下降法,此方法容易陷入局部极小而得到次最优解或收敛到问题的不可行解。另外,当用于训练网络样本的输入/输出数据无法精确给出,而只能以一定的范围的形式给出时,传统的神经网络学习方法就无能为力了。论文提出了一种基于区间优化的神经网络学习算法,可以很好地解决上面所提到的传统神经网络学习算法的缺点。     

Block methods for parabolic equations

Block methods for the finite difference solution of linear one dimensional parabolic partial differential equations are considered. These schemes use two linear multistep formulae which, when applied simultaneously, advance the numerical solution by two time steps. No special starting procedure is required for their implementation. By careful choice of the coefficients in these formulae, all of the block methods derived in this paper are unconditionally stable and have high order accuracy. In addition, some of these schemes are suitable for problems involving a discontinuity between the initial and boundary conditions. The results of numerical experiments on two test problems are presented.     

One-step explicit methods for the numerical integration of perturbed oscillators

In this paper, we present a class of one-step explicit zero-dissipative nonlinear methods for the numerical integration of perturbed oscillators, which have second algebraic order and high phase-lag order. For multi-dimensional problems, we give the vector form of the methods with the aid of a special vector operation. Some numerical results are reported to illustrate the efficiency of our methods.     

A.D.I. Methods for the solution of diffusion problems in r−θ geometry

In this paper the application of a particular class of iterative methods, the alternating direction implicit methods applied to the solution of time dependent and steady state diffusion problems in regions involving circular symmetry, is presented.A comparison between the convergence rates for commutative and non-commutative versions of the finite difference equations and a peripheral (azimuthal) block over-relaxation method for the steady-state model problem is given.     

A coupled gradient network approach for static and temporalmixed-integer optimization

Utilizes the ideas of artificial neural networks to propose new solution methods for a class of constrained mixed-integer optimization problems. These new solution methods are more suitable to parallel implementation than the usual sequential methods of mathematical programming. Another attractive feature of the proposed approach is that some global search mechanisms may be easily incorporated into the computation, producing results which are more globally optimal. To formulate the solution method proposed in this paper, a penalty function approach is used to define a coupled gradient-type network with an appropriate architecture, energy function and dynamics such that high-quality solutions may be obtained upon convergence of the dynamics. Finally, it is shown how the coupled gradient net may be extended to handle temporal mixed-integer optimization problems, and simulations are presented which demonstrate the effectiveness of the approach.     

Disjunctive optimization,max-separable problems and extremal algebras

The paper was motivated by solution methods suggested in the literature for solving linear optimization problems over (max,+)- or (max,min)-algebras and certain class of so called max-separable optimization problems. General features of these optimization problems, which play a crucial role in the optimization methods were used to formulate a general class of optimization problems with disjunctive constraints and a max-separable objective function and suggest a solution procedure for solving such problems. Linear problems over (max,+)-algebras and the max-separable problems are contained in this general class of optimization problems as special cases.     

On obtaining the ‘best’ multipliers for a lagrangean relaxation for integer programming

In a workshop on Integer Programming held in Germany in 1975, Geoffrion presented as an open important area for research, “What are the Cost Effective Ways to Determine the Best Multipliers for Generalized Lagrangean Relaxation?” In this paper, a method for computing the Lagrangean multipliers is presented. The method is applicable for a relatively wide class of Integer Programming problems. Four examples are presented in which, by applying this approach, it was possible to directly compute the multipliers. First, the general approach is outlined, and then the examples are presented. It is hoped that this approach and those specific examples will be of use in developing more general methods for computing the optimal Lagrangean multipliers and reduce the solution time of Integer Programming problems that fall within this class.Preliminary computational tests have been performed on a variety of interval-bounded Zero-one Knapsack problems, using a Branch and Bound algorithm and a Lagrangean Relaxation that applied a method similar to the one presented in this research to obtain the best multipliers. The computational tests were very favorable.     

一类切换系统的输入-状态稳定性分析与优化控制

利用驻留时间法和 Gronwall-Bellman不等式研究了一类切换系统的输入一状态稳定性分析与优化控制问题.在保证切换系统输入-状态稳定的前提下,将切换时刻和切换次数约束条件转化为线性约束,提出了一种新的切换系统优化问题目标函数的形式.与已有的方法相比,该方法无需引入新的状态变量,无需同时满足构造输入-状态稳定控制李亚普诺夫函数和所有子系统都是输入-状态稳定的条件,为控制器的优化设计提供了便利.最后,通过算例仿真证实了文中所提方法的可行性.     

Parallel and Systolic Solution of Normalized Explicit Approximate Inverse Preconditioning

A new class of normalized approximate inverse matrix techniques, based on the concept of sparse normalized approximate factorization procedures are introduced for solving sparse linear systems derived from the finite difference discretization of partial differential equations. Normalized explicit preconditioned conjugate gradient type methods in conjunction with normalized approximate inverse matrix techniques are presented for the efficient solution of sparse linear systems. Theoretical results on the rate of convergence of the normalized explicit preconditioned conjugate gradient scheme and estimates of the required computational work are presented. Application of the new proposed methods on two dimensional initial/boundary value problems is discussed and numerical results are given. The parallel and systolic implementation of the dominant computational part is also investigated.     

Lossless convexification of a class of optimal control problems with non-convex control constraints

We consider a class of finite time horizon optimal control problems for continuous time linear systems with a convex cost, convex state constraints and non-convex control constraints. We propose a convex relaxation of the non-convex control constraints, and prove that the optimal solution of the relaxed problem is also an optimal solution for the original problem, which is referred to as the lossless convexification of the optimal control problem. The lossless convexification enables the use of interior point methods of convex optimization to obtain globally optimal solutions of the original non-convex optimal control problem. The solution approach is demonstrated on a number of planetary soft landing optimal control problems.     

三种基于偏好的区间多目标进化算法及应用

区间参数多目标优化问题是普遍存在且非常重要的。目前直接求解该类问题的进化优化方法非常少,且已有方法的目的是找到收敛性好且分布均匀的Pareto最优解集。为得到符合决策者偏好的最满意解,本文综述3种基于偏好的区间多目标进化算法,并将其应用于特定环境下机器人路径规划问题,比较3种算法的性能。研究结果可丰富特定环境下机器人路径规划的求解方法,提高机器人路径优化效果。     

Domain decomposition preconditioners for the spectral collocation method

We propose and analyze several block iteration preconditioners for the solution of elliptic problems by spectral collocation methods in a region partitioned into several rectangles. It is shown that convergence is achieved with a rate that does not depend on the polynomial degree of the spectral solution. The iterative methods here presented can be effectively implemented on multiprocessor systems due to their high degree of parallelism.     

A family of high-order multistep methods with vanished phase-lag and its derivatives for the numerical solution of the Schrödinger equation

Many simulation algorithms (chemical reaction systems, differential systems arising from the modelling of transient behaviour in the process industries etc.) contain the numerical solution of systems of differential equations. For the efficient solution of the above mentioned problems, linear multistep methods or Runge-Kutta single-step methods are used. For the simulation of chemical procedures the radial Schrödinger equation is used frequently. In the present paper we will study a class of linear multistep methods. More specifically, the purpose of this paper is to develop an efficient algorithm for the approximate solution of the radial Schrödinger equation and related problems. This algorithm belongs in the category of the multistep methods. In order to produce an efficient multistep method the phase-lag property and its derivatives are used. Hence the main result of this paper is the development of an efficient multistep method for the numerical solution of systems of ordinary differential equations with oscillating or periodical solutions. The reason of their efficiency, as the analysis proved, is that the phase-lag and its derivatives are eliminated. Another reason of the efficiency of the new obtained methods is that they have high algebraic order     

跳变双线性随机系统饱和执行器的镇定

对于一类具有Markov跳变参数的双线性离散随机系统,研究其饱和执行器问题.分别采用一般二次型Lyapunov函数、饱和关联Lyapunov函数进行系统随机稳定性分析,以椭圆不变集构造随机稳定域,提出两种依赖于模态跳变率的饱和状态控制器设计方法,两种方法均以线性矩阵不等式的形式给出.