A UNIVERSAL APPROACH FOR CONTINUOUS OR DISCRETE NONLINEAR PROGRAMMINGS WITH MULTIPLE VARIABLES AND CONSTRAINTS

A universal numerical approach for nonlinear mathematic programming problems is presented with an application of ratios of first-order differentials/differences of objective functions to constraint functions with respect to design variables. This approach can be efficiently used to solve continuous and, in particular, discrete programmings with arbitrary design variables and constraints. As a search method, this approach requires only computations of the functions and their partial derivatives or differences with respect to design variables, rather than any solution of mathematic equations. The present approach has been applied on many numerical examples as well as on some classical operational problems such as one-dimensional and two-dimensional knap-sack problems, one-dimensional and two-dimensional resource-distribution problems, problems of working reliability of composite systems and loading problems of machine, and more efficient and reliable solutions are obtained than traditional methods. The present approach can be used without limitation of modeling scales of the problem. Optimum solutions can be guaranteed as long as the objective function, constraint functions and their First-order derivatives/differences exist in the feasible domain or feasible set. There are no failures of convergence and instability when this approach is adopted.

Global Convergence of a New Restarting Conjugate Gradient Method for Nonlinear Optimizations

Conjugate gradient optimization algorithms depend on the search directions with different choices for the parameters in the search directions. In this note, by combining the nice numerical performance of PR and HS methods with the global convergence property of the class of conjugate gradient methods presented by HU and STOREY(1991), a class of new restarting conjugate gradient methods is presented. Global convergences of the new method with two kinds of common line searches, are proved. Firstly, it is shown that, using reverse modulus of continuity function and forcing function, the new method for solving unconstrained optimization can work for a continously dif ferentiable function with Curry-Altman's step size rule and a bounded level set. Secondly, by using comparing technique, some general convergence properties of the new method with other kind of step size rule are established. Numerical experiments show that the new method is efficient by comparing with FR conjugate gradient method.

PARALLEL ALGORITHMS OF ONE-LEG METHOD AND ITERATED DEFECT CORRECTION METHODS

The parallel algorithms of iterated defect correction methods (PIDeCM’s) are constructed, which are of efficiency and high order B-convergence for general nonlinear stiff systems in ODE’S. As the basis of constructing and discussing PIDeCM’s. a class of parallel one-leg methods is also investigated, which are of particular efficiency for linear systems.

THE EXISTENCE AND BLOW-UP OF THE RADIAL SOLUTIONS OF A(k_(1),k_(2))-HESSIAN SYSTEM INVOLVING A NONLINEAR OPERATOR AND GRADIENT

In this paper,we are concerned with the existence of the positive bounded and blow-up radial solutions of the(k1,k2)-Hessian system■where G is a nonlinear operator,Ki=Ski(λ(D^(2) z_(i)))+ψ_(i)(|x|)|▽_(zi)|^(ki),i=1,2.Under the appropriate conditions on gi and g2,our main results are obtained by using the monotone iterative method and the Arzela-Ascoli theorem.Furthermore,our main results also extend the previous existence results for both the single equation and systems.

A GENERAL TECHNIQUE FOR DEALING WITH DEGENERACY IN REDUCED GRADIENT METHODS FOR LINEARLY CONSTRAINED NONLINEAR PROGRAMMING

In this paper we discuss the degeneracy in nonlinear programming with linear constraints, and give a technique for dealing with degeneracy in a general model of reduced gradient algorithms. Under the assumption that the objective function is continuously differentiable, we prove that either the iterative sequence {xk} generated by the method terminates at a Kuhn-Tucker point after a finite number of iterations, or any cluster point of the sequence {xk} is a KuhnTucker point.

Global Convergence of a New Restarting Three Terms Conjugate Gradient Method for Non-linear Optimizations

In this note,by combining the nice numerical performance of PR and HS methods with the global convergence property of FR method,a class of new restarting three terms conjugate gradient methods is presented.Global convergence properties of the new method with two kinds of common line searches are proved.

Solving the Optimal Control Problems of Nonlinear Duffing Oscillators By Using an Iterative Shape Functions Method

In the optimal control problem of nonlinear dynamical system,the Hamiltonian formulation is useful and powerful to solve an optimal control force.However,the resulting Euler-Lagrange equations are not easy to solve,when the performance index is complicated,because one may encounter a two-point boundary value problem of nonlinear differential algebraic equations.To be a numerical method,it is hard to exactly preserve all the specified conditions,which might deteriorate the accuracy of numerical solution.With this in mind,we develop a novel algorithm to find the solution of the optimal control problem of nonlinear Duffing oscillator,which can exactly satisfy all the required conditions for the minimality of the performance index.A new idea of shape functions method(SFM)is introduced,from which we can transform the optimal control problems to the initial value problems for the new variables,whose initial values are given arbitrarily,and meanwhile the terminal values are determined iteratively.Numerical examples confirm the high-performance of the iterative algorithms based on the SFM,which are convergence fast,and also provide very accurate solutions.The new algorithm is robust,even large noise is imposed on the input data.

Seismic fluid identification using a nonlinear elastic impedance inversion method based on a fast Markov chain Monte Carlo method

Elastic impedance inversion with high efficiency and high stability has become one of the main directions of seismic pre-stack inversion. The nonlinear elastic impedance inversion method based on a fast Markov chain Monte Carlo （MCMC） method is proposed in this paper, combining conventional MCMC method based on global optimization with a preconditioned conjugate gradient （PCG） algorithm based on local optimization, so this method does not depend strongly on the initial model. It converges to the global optimum quickly and efficiently on the condition that effi- ciency and stability of inversion are both taken into consid- eration at the same time. The test data verify the feasibility and robustness of the method, and based on this method, we extract the effective pore-fluid bulk modulus, which is applied to reservoir fluid identification and detection, and consequently, a better result has been achieved.

Separating iterative solution model of generalized nonlinear dynamic least squares for data processing in building of digital earth

Data coming from different sources have different types and temporal states. Relations between one type of data and another ones, or between data and unknown parameters are almost nonlinear. It is not accurate and reliable to process the data in building the digital earth with the classical least squares method or the method of the common nonlinear least squares. So a generalized nonlinear dynamic least squares method was put forward to process data in building the digital earth. A separating solution model and the iterative calculation method were used to solve the generalized nonlinear dynamic least squares problem. In fact, a complex problem can be separated and then solved by converting to two sub problems, each of which has a single variable. Therefore the dimension of unknown parameters can be reduced to its half, which simplifies the original high dimensional equations.

Strong-form framework for solving boundary value problems with geometric nonlinearity

In this paper, we present a strong-form framework for solving the boundary value problems with geometric nonlinearity, in which an incremental theory is developed for the problem based on the Newton-Raphson scheme. Conventionally, the finite ele- ment methods （FEMs） or weak-form based meshfree methods have often been adopted to solve geometric nonlinear problems. However, issues, such as the mesh dependency, the numerical integration, and the boundary imposition, make these approaches com- putationally inefficient. Recently, strong-form collocation methods have been called on to solve the boundary value problems. The feasibility of the collocation method with the nodal discretization such as the radial basis collocation method （RBCM） motivates the present study. Due to the limited application to the nonlinear analysis in a strong form, we formulate the equation of equilibrium, along with the boundary conditions, in an incremental-iterative sense using the RBCM. The efficacy of the proposed framework is numerically demonstrated with the solution of two benchmark problems involving the geometric nonlinearity. Compared with the conventional weak-form formulation, the pro- posed framework is advantageous as no quadrature rule is needed in constructing the governing equation, and no mesh limitation exists with the deformed geometry in the increment al-it erative process.

NONLINEAR FILTERING ALGORITHM FOR IN S INITIAL ALIGNMENT

The initial alignment error equation of an INS (Inertial Navigation System) with large initial azimuth error has been derived and nonlinear characteristics are included. When azimuth error is fairly small, the nonlinear equation can be reduced to a linear one. Extended Kalman filter, iterated filter and second order filter formulas are derived for the nonlinear state equation with linear measurement equation. Simulations results show that the accuracy of azimuth error estimation using extended Kalman filter is better than that of using standard Kalman filter while the iterated filter and second order filter can give even better estimation accuracy.

Fast First-Order Methods for Minimizing Convex Composite Functions

Two new versions of accelerated first-order methods for minimizing convex composite functions are proposed. In this paper, we first present an accelerated first-order method which chooses the step size 1/ Lk to be 1/ L0 at the beginning of each iteration and preserves the computational simplicity of the fast iterative shrinkage-thresholding algorithm. The first proposed algorithm is a non-monotone algorithm. To avoid this behavior, we present another accelerated monotone first-order method. The proposed two accelerated first-order methods are proved to have a better convergence rate for minimizing convex composite functions. Numerical results demonstrate the efficiency of the proposed two accelerated first-order methods.

Intelligent Iterated Local Search Methods for Solving Vehicle Routing Problem with Different Fleets

To solve vehicle routing problem with different fleets, two methodologies are developed. The first methodology adopts twophase strategy. In the first phase, the improved savings method is used to assign customers to appropriate vehicles. In the second phase, the iterated dynasearch algorithm is adopted to route each selected vehicle with the assigned customers. The iterated dynasearch algorithm combines dynasearch algorithm with iterated local search algorithm based on random kicks. The second methodplogy adopts the idea of cyclic transfer which is performed by using dynamic programming algorithm, and the iterated dynasearch algorithm is also embedded in it. The test results show that both methodologies generate better solutions than the traditional method, and the second methodology is superior to the first one.

Computational Methods in the Theory of Synthesis of Radio and Acoustic Radiating Systems

A brief review of the works of the author and his co-authors on the application of nonlinear analysis, numerical and analytical methods for solving the nonlinear inverse problems (synthesis problems) for optimizing the different types of radiating systems, is presented in the paper. The synthesis problems are formulated in variational statements and further they are reduced to research and numerical solution of nonlinear integral equations of Hammerstein type. The existence theorems are proof, the investigation methods of nonuniqueness problem of solutions and numerical algorithms of finding the optimal solutions are proved.

Solving method of generalized nonlinear dynamic least squares for data processing in building of digital mine

Data are very important to build the digital mine. Data come from many sources, have different types and temporal states. Relations between one class of data and the other one, or between data and unknown parameters are more nonlinear. The unknown parameters are non random or random, among which the random parameters often dynamically vary with time. Therefore it is not accurate and reliable to process the data in building the digital mine with the classical least squares method or the method of the common nonlinear least squares. So a generalized nonlinear dynamic least squares method to process data in building the digital mine is put forward. In the meantime, the corresponding mathematical model is also given. The generalized nonlinear least squares problem is more complex than the common nonlinear least squares problem and its solution is more difficultly obtained because the dimensions of data and parameters in the former are bigger. So a new solution model and the method are put forward to solve the generalized nonlinear dynamic least squares problem. In fact, the problem can be converted to two sub problems, each of which has a single variable. That is to say, a complex problem can be separated and then solved. So the dimension of unknown parameters can be reduced to its half, which simplifies the original high dimensional equations. The method lessens the calculating load and opens up a new way to process the data in building the digital mine, which have more sources, different types and more temporal states.

基于遗传算法-序列二次规划的磁共振被动匀场优化方法

为了解决磁共振成像(MRI)系统中固有的主磁场(B0)不均匀的问题,提出遗传算法-序列二次规划(GASQP)算法,以提高7 T磁共振的主磁场均匀性.从被动匀场数学模型的角度出发,该混合算法利用GA算法获得稳定的初始解,实现主磁场的第1次优化,再通过SQP算法的快速求解,在较少的时间内实现主磁场的第2次优化,同时提高磁共振主磁场的均匀性.采用正则化方法减少磁场均匀所需的铁片质量,并且获得稀疏的铁片分布.在仿真建模的案例研究中,7 T磁共振裸磁场均匀度可以从462×10-6优化到4.5×10-6,并且在匀场空间上仅消耗0.8 kg的铁片.相比于传统的GA优化方法,新方案的磁场均匀性提高了96.7%,总铁片消耗质量减少了85.7%.实验结果表明,GA-SQP算法比其他优化算法具有更强的鲁棒性和竞争力.

基于二分迭代法的铁路线路纵断面优化算法研究

为提高铁路线路纵断面优化自动化水平,加强变坡点的寻找精度,减少人工调节工作量,基于二分迭代法设计了1种自动化程度和精度更高的纵断面优化算法,该算法采用最大矢距值二分查找方式,精确定位变坡点位置,并对查找过程进行有效的阈值控制。为实现该优化算法开发了1套高程自动优化软件,进一步采用试验数据对比分析验证该算法的实用性,与传统算法相比,提高了自动化程度,且计算结果精度满足精捣作业需求。

一类非线性两层规划问题的递阶优化解法

提出一种求解一类非线性两层规划问题的新方法.通过引入解耦向量将非线性两层规划问题分解为独立且易于求解的子问题,利用两级递阶结构第1级求解若干优化的子问题,而在第2级利用第1级求解的结果调整解耦向量.所提出的方法借助于分解-协调原理并按迭代方式最终求得问题的最优解.对于含整数的规划问题,通过连续化处理后也可按该方法方便地求解.算例表明所提出的算法是简便而有效的.

求解离散无功优化的非线性原—对偶内点算法

针对无功优化计算中离散变量和连续变量共存问题 ,提出用直接非线性原—对偶内点法内嵌罚函数的新算法。通过对几个不同规模试验系统计算分析 ,并与 Tabu搜索法求得的结果比较 ,证明了该方法是有效的 ,而且在计算速度、收敛性和优化精度上都优于 Tabu搜索法。

基于逻辑程序的安全协议验证

安全协议本质上是分布式并发程序 ,可以自然地描述为多个子进程的并发合成系统 .将安全协议对应的并发合成系统抽象为逻辑程序进行消解 ,能够对安全协议无穷多个会话的交叠运行进行验证 .该文提出了安全协议逻辑程序中逻辑规则的一个分类方法 ,基于该分类方法提出了安全协议逻辑程序不动点的迭代计算方法 .逻辑规则的分类优化了安全协议逻辑程序不动点的迭代计算和安全性质验证过程中的计算 .由于安全协议逻辑程序不动点迭代计算过程不一定终止 ,文中提出了每进行k 1步安全协议逻辑程序不动点迭代计算验证一次安全性质的验证策略 .