A LQP BASED INTERIOR PREDICTION-CORRECTION METHOD FOR NONLINEAR COMPLEMENTARITY PROBLEMS

To solve nonlinear complementarity problems （NCP）, at each iteration, the classical proximal point algorithm solves a well-conditioned sub-NCP while the Logarithmic-Quadratic Proximal （LQP） method solves a system of nonlinear equations （LQP system）. This paper presents a practical LQP method-based prediction-correction method for NCP. The predictor is obtained via solving the LQP system approximately under significantly relaxed restriction, and the new iterate （the corrector） is computed directly by an explicit formula derived from the original LQP method. The implementations are very easy to be carried out. Global convergence of the method is proved under the same mild assumptions as the original LQP method. Finally, numerical results for traffic equilibrium problems are provided to verify that the method is effective for some practical problems.     

A new criterion for the inexact logarithmic-quadratic proximal method and its derived hybrid methods

To solve nonlinear complementarity problems, the inexact logarithmic-quadratic proximal (LQP) method solves a system of nonlinear equations (LQP system) approximately at each iteration. Therefore, the efficiencies of inexact-type LQP methods depend greatly on the involved inexact criteria used to solve the LQP systems. This paper relaxes inexact criteria of existing inexact-type LQP methods and thus makes it easier to solve the LQP system approximately. Based on the approximate solutions of the LQP systems, a descent method, and a prediction–correction method are presented. Convergence of the new methods are proved under mild assumptions. Numerical experiments for solving traffic equilibrium problems demonstrate that the new methods are more efficient than some existing methods and thus verify that the new inexact criterion is attractive in practice.     

一种求解单调变分不等式的部分并行分裂LQP交替方向法

LQP交替方向法是求解可分离结构型单调变分不等式问题的一种非常有效的方法.它不仅可以充分地利用目标函数的可分结构,将原问题分解为多个更易求解的子问题,还更适合求解大规模问题.对于带有三个可分离算子的单调变分不等式问题,结合增广拉格朗日算法和LQP交替方向法提出了一种部分并行分裂LQP交替方向法,构造了新算法的两个下降方向,结合这两个下降方向得到了一个新的下降方向,沿着这个新的下降方向给出了最优步长.并在较弱的假设条件下,证明了新算法的全局收敛性.     

关于一类变分不等式的LQP算法

本文研究一类ξ-单调的变分不等式问题.利用KKT条件将原问题转换为非线性互补问题(nonlinear complementarity problem,NCP)的方法,获得了基于logarithmic-quadratic proximal(LQP)的算法及其改进形式,推广了LQP算法的适用范围.     

A new logarithmic-quadratic proximal method for nonlinear complementarity problems

In this paper, we propose a new modified logarithmic-quadratic proximal (LQP) method for solving nonlinear complementarity problems (NCP). We suggest using a prediction-correction method to solve NCP. The predictor is obtained via solving the LQP system approximately under significantly relaxed accuracy criterion and the new iterate is computed by using a new step size α_k. Under suitable conditions, we prove that the new method is globally convergent. We report preliminary computational results to illustrate the efficiency of the proposed method. This new method can be considered as a significant refinement of the previously known methods for solving nonlinear complementarity problems.     

An inexact logarithmic-quadratic proximal augmented Lagrangian method for a class of constrained variational inequalities

The augmented Lagrangian method is attractive in constraint optimizations. When it is applied to a class of constrained variational inequalities, the sub-problem in each iteration is a nonlinear complementarity problem (NCP). By introducing a logarithmic-quadratic proximal term, the sub-NCP becomes a system of nonlinear equations, which we call the LQP system. Solving a system of nonlinear equations is easier than the related NCP, because the solution of the NCP has combinatorial properties. In this paper, we present an inexact logarithmic-quadratic proximal augmented Lagrangian method for a class of constrained variational inequalities, in which the LQP system is solved approximately under a rather relaxed inexactness criterion. The generated sequence is Fejér monotone and the global convergence is proved. Finally, some numerical test results for traffic equilibrium problems are presented to demonstrate the efficiency of the method.      

A self-adaptive descent LQP alternating direction method for the structured variational inequalities

Numerical Algorithms - In this paper, by combining the logarithmic-quadratic proximal (LQP) method and alternating direction method, we proposed an LQP alternating direction method for solving...     

On the O(1/t) convergence rate of the LQP prediction–correction method

The logarithmic-quadratic proximal (abbreviated by LQP) prediction–correction method is attractive for structured monotone variational inequalities, and it ensures the global convergence under some suitable conditions. In this paper, we are interested in investigating the convergence rate of the LQP prediction–correction method. Motivated by the research work about the convergence rate or iteration complexity for various first-order algorithms in the literature, we provide a simple proof to show the $O(1/t)$ convergence rate for the LQP prediction–correction method.     

非线性互补问题的两种数值解法

本文研究了非线性互补问题的两类数值求解方法.在经典LQP算法及LevenbergMarquardt算法的基础上,构造了两种新算法,并证明了这两种新算法的收敛性.数值实验表明,新算法对测试问题优于已有算法.     

An improved LQP-based method for solving nonlinear complementarity problems

The well-known logarithmic-quadratic proximal (LQP)method has motivated a number of efficient numerical algorithms for solving nonlinear complementarity problems (NCPs). In this paper,we aim at improving one of them, i.e., the LQP-based interior prediction-correction method proposed in [He, Liao and Yuan, J. Comp. Math., 2006, 24(1): 33–44], via identifying more appropriate step-sizes in the correction steps. Preliminary numerical results for solving some NCPs arising in traffic equilibrium problems are reported to verify the theoretical assertions.     

A Logarithmic-Quadratic Proximal Prediction-Correction Method for Structured Monotone Variational Inequalities

Inspired by the Logarithmic-Quadratic Proximal (LQP) method for variational inequalities, we present a prediction-correction method for structured monotone variational inequalities. Each iteration of the new method consists of a prediction and a correction. Both the predictor and the corrector are obtained easily with tiny computational load. In particular, the LQP system that appears in the prediction is approximately solved under significantly relaxed inexactness restriction. Global convergence of the new method is proved under mild assumptions. In addition, we present a self-adaptive version of the new method that leads to easier implementations. Preliminary numerical experiments for traffic equilibrium problems indicate that the new method is effectively applicable in practice. Presented at the 6th International conference on Optimization: Techniques and Applications, Ballarat Australia, December 9–11, 2004. This author was supported by NSFC Grant 10571083, the MOEC grant 20020284027 and Jiangsu NSF grant BK2002075     

On descent alternating direction method with LQP regularization for the structured variational inequalities

Optimization Letters - By reviewing the Logarithmic–quadratic proximal (LQP) method, in this paper we suggest and analyze a new LQP alternating direction scheme for the separable constrained...     

The relationship among the solutions of two auxiliary ordinary differential equations

In a recent article [Phys. Lett. A 356 (2006) 124], Sirendaoreji extended their auxiliary equation method by introducing a new auxiliary ordinary differential equation (NAODE) and its 14 solutions. Then the author studied some nonlinear evolution equations (NLEEs) and got more exact travelling wave solutions. In this paper, we will show that the 14 solutions of the NAODE are actually the same as the solutions obtained by original auxiliary equation method, and they are only different in the form.     

Symbolic computation of some new nonlinear partial differential equations of nanobiosciences using modified extended tanh-function method

By means of computerized symbolic computation and a modified extended tanh-function method the multiple travelling wave solutions of nonlinear partial differential equations is presented and implemented in a computer algebraic system. Applying this method, we consider some of nonlinear partial differential equations of special interest in nanobiosciences and biophysics namely, the transmission line models of microtubules for nano-ionic currents. The nonlinear equations elaborated here are quite original and first proposed in the context of important nanosciences problems related with cell signaling. It could be even of basic importance for explanation of cognitive processes in neurons. As results, we can successfully recover the previously known solitary wave solutions that had been found by other sophisticated methods. The method is straightforward and concise, and it can also be applied to other nonlinear equations in physics.     

扩充偏微分方程(组)守恒律和对称的辅助方程方法及微分形式吴方法的应用

首先,我们给出了引入伴随方程(组)扩充原方程(组)的策略使给定偏微分方程(组)的扩充方程组具有对应泛瓯即,成为Lagrange系统的方法,以此为基础提出了作为偏微分方程(组)传统守恒律和对称概念的一种推广-偏微分方程(组)扩充守恒律和扩充对称的概念;其次,以得到的Lagrange系统为基础给定了确定原方程(组)扩充守恒律和扩充对称的方法,从而达到扩充给定偏微分方程(组)的首恒律和对称的目的;第三,提出了适用于一般形式微分方程(组)的计算固有守恒律的方法;第四,实现以上算法过程中,我们先把计算(扩充)守恒律和对称问题均归结为求解超定线性齐次偏微分方程组(确定方程组)的问题.然后,对此关键问题我们提出了用微分形式吴方法处理的有效算法;最后,作为方法的应用我们计算确定了非线性电报方程组在内的五个发展方程(组)的新守恒律和对称,同时也说明了方法的有效性.     

Controlling based method for modelling chaotic dynamical systems from time series

A simple method is introduced for modelling chaotic dynamical systems from the time series, based on the concept of controlling of chaos by constant bias. In this method, a modified system is constructed by including some constants (controlling constants) into the given (original) system. The system parameters and the controlling constants are determined by solving a set of implicit nonlinear simultaneous algebraic equations which is obtained from the relation connecting original and modified systems. The method is also extended to find the form of the evolution equation of the system itself. The important advantage of the method is that it needs only a minimal number of time series data and is applicable to dynamical systems of any dimension. It also works extremely well even in the presence of noise in the time series. The method is illustrated in some specific systems of both discrete and continuous cases.     

An LQP Method for Pseudomonotone Variational Inequalities

In this paper, we proposed a modified Logarithmic-Quadratic Proximal (LQP) method [Auslender et al.: Comput. Optim. Appl. 12, 31–40 (1999)] for solving variational inequalities problems. We solved the problem approximately, with constructive accuracy criterion. We show that the method is globally convergence under that the operator is pseudomonotone which is weaker than the monotonicity and the solution set is nonempty. Some preliminary computational results are given.The author was supported by the NSFC grants Nos: 70571033 and 10571083.     

Saddle-Point Criteria in an η-Approximation Method for Nonlinear Mathematical Programming Problems Involving Invex Functions

In this paper, the η-approximation method introduced by Antczak (Ref. 1) for solving a nonlinear constrained mathematical programming problem involving invex functions with respect to the same function η is extended. In this method, a so-called η-approximated optimization problem associated with the original mathematical programming problems is constructed; moreover, an η-saddle point and an η-Lagrange function are defined. By the help of the constructed η-approximated optimization problem, saddle-point criteria are obtained for the original mathematical programming problem. The equivalence between an η-saddle point of the η-Lagrangian of the associated η-approximated optimization problem and an optimal solution in the original mathematical programming problem is established.     

Approximation theorem for a nonlinear control system with sliding modes

We consider the question of validity of the extension of a nonlinear control system by introducing the so-called sliding modes (i.e., by convexifying the set of admissible velocities) in the presence of constraints imposed on the endpoints of trajectories. We prove that a trajectory of the extended system can be approximated by trajectories of the original system if the equality constraints of the extended system are nondegenerate in the first order. The proof is based on a nonlocal estimate for the distance to the zero set of the nonlinear operator corresponding to the extended system, and involves a specific iteration process of corrections. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2007, Vol. 256, pp. 102–114.     

Master partial differential equations for a Type II hidden symmetry

An approach for determining a class of master partial differential equations from which Type II hidden point symmetries are inherited is presented. As an example a model nonlinear partial differential equation (PDE) reduced to a target PDE by a Lie symmetry gains a Lie point symmetry that is not inherited (hidden) from the original PDE. On the other hand this Type II hidden symmetry is inherited from one or more of the class of master PDEs. The class of master PDEs is determined by the hidden symmetry reverse method. The reverse method is extended to determine symmetries of the master PDEs that are not inherited. We indicate why such methods are necessary to determine the genesis of Type II symmetries of PDEs as opposed to those that arise in ordinary differential equations (ODEs).