A non-interior implicit smoothing approach to complementarity problems for frictionless contacts

This paper presents a non-interior point method for solving frictionless contact problems in large deformations, where we solve the problems in an incremental path-following method from warm start. We propose a novel reformulation of the nonlinear complementarity problem, which is based on the smoothed Fischer–Burmeister function but is distinguished from the conventional formulations in the following two particular aspects: (i) the smoothing parameter is considered as an independent variable; (ii) an equality constraint is added so that the smoothing parameter serves as a measure of the residual of the complementarity conditions. The reduced system of nonlinear equations is solved with a conventional Newton method for nonlinear equations from the initial point which is defined by using the solution of the preceding loading stage. Throughout numerical examples it is shown that in many cases the solution can be found within four Newton iterations.     

A non-monotone inexact regularized smoothing Newton method for solving nonlinear complementarity problems

In the paper [S.P. Rui and C.X. Xu, A smoothing inexact Newton method for nonlinear complementarity problems, J. Comput. Appl. Math. 233 (2010), pp. 2332–2338], the authors proposed an inexact smoothing Newton method for nonlinear complementarity problems (NCP) with the assumption that F is a uniform P function. In this paper, we present a non-monotone inexact regularized smoothing Newton method for solving the NCP which is based on Fischer–Burmeister smoothing function. We show that the proposed algorithm is globally convergent and has a locally superlinear convergence rate under the weaker condition that F is a P ₀ function and the solution of NCP is non-empty and bounded. Numerical results are also reported for the test problems, which show the effectiveness of the proposed algorithm.     

应用非单调线搜索求解一类互补问题

考虑一类含非Lipschtizian连续函数的非线性互补问题。引入plus函数的一类广义光滑函数,讨论其性质。应用所引入函数将互补问题重构为一系列光滑方程组,提出一个具有非单调线搜索的Newton算法求解重构的方程组以得到原问题的解。在很弱的条件下,该算法具有全局收敛性和局部二次收敛性。利用该算法求解一自由边界问题,其数值结果显示该算法是有效的。     

一类非光滑优化及其在控制系统稳定化中的应用

研究一类来自控制系统稳定化中的非光滑优化问题．考虑Lyapunov函数是非光滑的，特别是有限个光滑函数的极大值函数．建立了相应的非光滑优化模型，进一步导出了这类非光滑优化的KKT系统，然后基于非线性互补函数将KKT系统转化成一个非光滑方程组，最后分别用广义牛顿法和光滑化牛顿法求解此非光滑方程组。使得此类稳定化设计可以具体实现．     

A smooth Newton method with 3-1 piecewise NCP function for generalized nonlinear complementarity problem

In this paper, based on the 3-1 piecewise nonlinear complementarity problem (NCP) function, we proposed a smoothing Newton-type method for the generalized nonlinear complementarity problem (GNCP) with a modified non-monotone line search. The algorithm for a GNCP is more difficult than that for an NCP, because two functions must be considered in the problem. We reformulate the (GNCP) to a smoothing system of equations by two independent variables, and then develop a smoothing Newton-type method for solving it. Under reasonable conditions, we obtain the global convergent properties. Also, the numerical experiments are reported in this paper.     

非线性互补问题的光滑化拟牛顿算法

为求解非线性互补问题,给出了一种新的基于光滑对称扰动Fischer-Burmeister函数的光滑化拟牛顿算法。该算法利用了无导数线搜索。数值实验表明,算法是有效的。     

A predictor-corrector smoothing Newton method for solving the mixed complementarity problem with a P 0-function

The mixed complementarity problem (denoted by MCP(F)) can be reformulated as the solution of a nonsmooth system of equations. In the paper, based on a perturbed mid function, we contract a new smoothing function. The existence and continuity of a smooth path for solving the mixed complementarity problem with a P ₀ function are discussed. Then we presented a predictor-corrector smoothing Newton algorithm to solve the MCP with a P ₀-function. The global convergence of the proposed algorithm is verified under mild conditions. And by using the smooth and semismooth technique, the local superlinear convergence of the method is proved under some suitable assumptions.     

Integrated Inventory Control and Facility Location Decisions in a Multi-Echelon Supply Chain Network with Hubs

This paper develops mathematical models to coordinate facility location and inventory control for a four-echelon supply chain network consisting of multiple suppliers, warehouses, hubs and retailers. The hubs help in reducing transportation costs by consolidating products from multiple warehouses and directing the larger shipments to the retailer. The integrated models studied in this paper simultaneously determines three types of decisions: (i) facility location—the number and location of warehouses and hubs, (ii) allocation—assignment of suppliers to located warehouses and retailers to located warehouses via the location hubs, and (iii) inventory control decisions at each located warehouse. The goal is to minimize the facility location, transportation and the inventory costs. A mixed integer nonlinear programming formulation is first presented. The nonlinear integer programming formulation is then transformed into a conic mixed integer program and a novel and compact conic mixed integer programming formulation. Computational runs are conducted using commercial solvers to compare the performance of the different formulations. The compact conic mixed integer programming formulation was found to significantly outperform the other formulations by achieving significant computational savings. The results demonstrate that large scale instances of certain multi-echelon supply chain network design problems can be solved using commercial solvers through intelligent reformulation of the model.     

Solving discretely constrained,mixed linear complementarity problems with applications in energy

This paper presents an approach to solving discretely constrained, mixed linear complementarity problems (DC-MLCPs). Such formulations include a variety of interesting and realistic models of which two are highlighted: a market-clearing auction typical in electric power markets but suitable in other more general contexts, and a network equilibrium suitable to energy markets as well as other grid-based industries. A mixed-integer, linear program is used to solve the DC-MLCP in which both complementarity as well as integrality are allowed to be relaxed. Theoretical and numerical results are provided to validate the approach.     

A General Equilibrium Framework for Integrated Assessment of Transport and Economic Impacts

The use of separate transport and economic models in urban planning provides a limited view of economic impacts, restricts the testing of network design options and lengthens the planning process. Furthermore, the standard methodology for economic appraisal assumes partial economic equilibrium and cannot determine the distribution of impacts from the transport sector to households. Computable general equilibrium (CGE) models can capture general equilibrium effects and measure welfare at the household level, but mostly lack integration with transport models and do not represent all trip generators. This paper develops an integrated traffic assignment and spatial CGE model in nonlinear complementarity form, casted as a framework for economic appraisal of urban transport projects. The CGE submodel generates commuting, shopping and leisure trips as inputs into the transport submodel, which then assigns trips to the network according to user equilibrium. The resulting travel times then feed back into household prices and freight margins. Households and firms fully account for travel times in decisions on where to shop, how much labour to supply and where to source production inputs. Calibration and applications of the model are demonstrated for 14 regions and 2 industries across Sydney using GAMS/PATH on the NEOS server. The welfare of various network improvements is measured using equivalent variations. The model can be calibrated to external strategic transport models, and be extended to simulate additional trip generators and land-use.     

供应链集成化模型与优化

本文建立了一个包括供应商、制造商、销售商 在内的集成化模型,这一模型含有线性状态方程和神经网络非线性模型,提出了库存成本、 供应成本、生产水平等在内的供应链目标函数和约束条件．对于供应链集成化模型采用模拟 退火方法进行优化,并以辽化公司为背景进行了初步条件的仿真工作．     

求解最优潮流KKT系统的一类新模型及算法设计

电力工业的市场化改革对最优潮流(optim al pow er flow,OPF)的计算精度和速度提出了更高的要求.本文针对OPF模型中存在大量的无功界约束的特性,把一般非线性不等式约束和界约束分开处理,通过引入一个对角矩阵和非线性互补函数,建立了与OPF问题的K arush-Kuhn-Tucker(KKT)系统等价的约束非光滑方程新模型.进一步,基于新建立的模型,提出了一类具有理论上收敛性保证的投影半光滑N ew ton型算法.相对于传统的解OPF的KKT系统和非线性互补函数方法,新方法一方面保持了非线性互补函数法无需识别有效集的优点,同时又减少了问题的维数,且投影计算保持了无功界约束的可行性.IEEE多个算例的数值试验显示本文所提出的模型和算法具有较好的计算效果.     

Multi‐product supply chain horizontal network integration: models,theory, and computational results

In this paper, we develop multi‐product supply chain network models with explicit capacities, before and after their horizontal integration. In addition, we propose a measure, which allows one to quantify and assess, from a supply chain network perspective, the synergy benefits associated with the integration of multi‐product firms through mergers/acquisitions. We utilize a system‐optimization perspective for the model development and provide the variational inequality formulations, which are then utilized to propose a computational procedure which fully exploits the underlying network structure. We illustrate the theoretical and computational framework with numerical examples. This paper is a contribution to the literatures of supply chain integration and mergers and acquisitions.     

基于L-M算法的反向传播网络的湿度传感器输出误差补偿研究

针对湿度传感器的输出非线性问题,提出了基于L-M算法建立BP神经网络进行补偿校正,实现电阻型湿度传感器的输入与输出非线性补偿,并与共轭梯度算法、拟牛顿算法所建立的神经网路模型进行对比,重点比较了模型迭代性能、标准偏差;最后发现当神经网络用L-M算法进行训练模拟时在迭代性能、标准偏差等方面具有更优异的表现,更适合湿度传感器的非线性特性的补偿校正。     

基于免疫粒子群算法的广义Nash均衡问题求解

针对广义Nash均衡求解问题, 提出了一种免疫粒子群算法。首先利用非线性互补问题, 将广义Nash均衡问题转换为非线性方程组问题, 然后把免疫算法中抗体的免疫记忆功能和抗体浓度抑制机制引入基本粒子群算法, 设计了一种免疫粒子群算法。最后通过数值实验表明, 该算法保持了粒子群种群多样性, 增强了粒子群算法的全局寻优能力, 加快了算法的收敛速度, 具有较好的性能。     

动态复杂供需网络局域演化模型的研究

为了深入理解供需网络的演化规律,研究了已有的复杂网络演化模型刻画供需网络生长过程的不足,提出了以星型网络表示初始网络,在局域世界中选择新增节点的连接节点,局域世界的选取,采用了依据节点之间的网络路径值作为选取局域世界的原则,同时定义了局域世界的规模动态增长,从而建立了复杂供需网络的动态演化模型,并给出了生成模型的算法。该模型在考虑网络动态增长的同时,也考虑到网络内部边的动态演化以及节点的退出,这与现实情况相吻合,因此更细致、真实地刻画了供需网络的动态演化特性。仿真结果表明,该模型所生成的网络模型具有无标度特性和小世界性。     

An interactive approach based on a discrete differential evolution algorithm for a class of integer bilevel programming problems

This paper presents an interactive approach based on a discrete differential evolution algorithm to solve a class of integer bilevel programming problems, in which integer decision variables are controlled by an upper-level decision maker and real-value or continuous decision variables are controlled by a lower-level decision maker. Using the Karush--Kuhn–Tucker optimality conditions in the lower-level programming, the original discrete bilevel formulation can be converted into a discrete single-level nonlinear programming problem with the complementarity constraints, and then the smoothing technique is applied to deal with the complementarity constraints. Finally, a discrete single-level nonlinear programming problem is obtained, and solved by an interactive approach. In each iteration, for each given upper-level discrete variable, a system of nonlinear equations including the lower-level variables and Lagrange multipliers is solved first, and then a discrete nonlinear programming problem only with inequality constraints is handled by using a discrete differential evolution algorithm. Simulation results show the effectiveness of the proposed approach.     

MPEC strategies for optimization of a class of hybrid dynamic systems

With the development and widespread use of large-scale nonlinear programming (NLP) tools for process optimization, there has been an associated application of NLP formulations with complementarity constraints in order to represent discrete decisions. In particular, these constraints arise frequently in equation-based formulations for real-time optimization. Also known as mathematical programs with equilibrium constraints (MPECs), these formulations can be used to model certain classes of discrete events and can be more efficient than a mixed integer formulation, particularly for large systems with many discrete decisions, such as dynamic systems with switches at any point in time. In this study, we consider and extend MPEC formulations for the optimization of a class of hybrid dynamic models, where the differential states remain continuous over time. These include differential inclusions of the Filippov type. Here, particular care is required in the formulation in order to preserve smoothness properties of the dynamic system. Results on three case studies, including process control examples, illustrate the effectiveness and accuracy of the proposed MPEC optimization methodology for a class of hybrid dynamic systems.     

Two aggregate-function-based algorithms for analysis of 3D frictional contact by linear complementarity problem formulation

Three dimensional frictional contact is formulated as linear complementarity problem (LCP) by using the parametric variational principle and quadratic programming method. Two aggregate-function-based algorithms, called respectively as self-adjusting interior point algorithm and aggregate function smoothing algorithm, are proposed for the solution of the LCP derived from the contact problems. A nonlinear finite element code is developed for numerical analysis of 3D multi-body contact problems. Four numerical examples are computed to demonstrate the applicability and computational efficiency of the methods proposed.