Globally convergent homotopy algorithm for solving the KKT systems to the principal-agent bilevel programming

In this paper, a constraint set swelling homotopy (CSSH) algorithm for solving the single-level non-convex programming problem with designing piecewise linear contractual function which is equivalent to the principal-agent model with integral operator is proposed, and the existence and global convergence is proven under some mild conditions. As a comparison, a piecewise constant contract is also designed for solving the single-level non-convex programming problem with the corresponding discrete distributions. And some numerical tests are done by the proposed homotopy algorithm as well as by using fmincon in Matlab, LOQO and MINOS. The numerical results show that the CSSH algorithm is robust, feasible and effective.     

A double homotopy method for decentralised control design

An iterative algorithm that is based on the idea of two homotopy paths is proposed for output feedback decentralised ?_∞ control design. The approach follows a bilinear matrix inequality (BMI) formulation of the decentralised control problem. Along one homotopy path the BMI problem, which is non-convex, is locally linearised and solved. Along the second homotopy path the controller is deformed at each step so that in the end it attains a decentralised structure. The proposed computational algorithm can also be applied to obtain reduced-order decentralised controllers. Numerical examples are used to demonstrate the efficacy of the proposed algorithm.     

面向欠约束几何系统的一种同伦求解方法

针对几何约束系统的数值求解过程中，经常发生的数值不稳定性问题，构造了一种面向欠约束系统的同伦方法，并将其与现有的求解与分解方法有机地结合起来，提出了一种牛顿－同伦混合方法，在牛顿迭代失败的位置自动调用欠约束同伦法，既提高了几何约束求解器的效率，同时又保证了求解的效率。     

不确定关联大系统分散鲁棒H∞输出反馈控制

针对一类状态矩阵和控制矩阵存在参数不确定性关联大系统,研究其分散鲁棒H∞输出反馈控制问题.基于有界实引理将其鲁棒分散H∞动态输出反馈控制器的解归结为一个非线性矩阵不等式(NLMI),先通过选取适当的同伦函数来表示该非线性矩阵不等式,再通过Schur补引理将其化为两个双线性矩阵不等式,最后通过迭代算法求解该控制器,使闭环大系统鲁棒渐进稳定,并且满足给定的H∞性能指标.     

非线性迭代学习控制问题的延拓修正牛顿法

对于非线性迭代学习控制问题,提出基于延拓法和修正Newton法的具有全局收敛性的迭代学习控制新方法.由于一般的Newton型迭代学习控制律都是局部收敛的,在实际应用中有很大局限性.为拓宽收敛范围,该方法将延拓法引入迭代学习控制问题,提出基于同伦延拓的新的Newton型迭代学习控制律,使得初始控制可以较为任意的选择.新的迭代学习控制算法将求解过程分成N个子问题,每个子问题由换列修正Newton法利用简单的递推公式解出.本文给出算法收敛的充分条件,证明了算法的全局收敛性.该算法对于非线性系统迭代学习控制具有全局收敛和计算简单的优点.     

The application of homotopy perturbation method for MHD flows of UCM fluids above porous stretching sheets

In this study, by means of homotopy perturbation method (HPM) an approximate analytical solution of the magnetohydrodynamic (MHD) boundary layer flow of an upper-convected Maxwell (UCM) fluid over a porous stretching sheet is obtained. The main feature of the HPM is that it deforms a difficult problem into a set of problems which are easier to solve. HPM produces analytical expressions for the solution of nonlinear differential equations. The obtained analytic solution is in the form of an infinite power series. In this work, the analytical solution obtained by using only two terms from HPM solution. The results reveal that the proposed method is very effective and simple and can be applied to other nonlinear problems. Also it is shown that this method coincides with homotopy analysis method (HAM) for the studied problem.     

一种改进的同伦算法与H∞鲁棒控制器设计

提出了一种具有阶次限制的鲁棒控制器设计方法, 该算法将控制系统的性能指标转化为灵敏度函数问题, 并利用Nevanlinna-Pick插值算法进行求解. 提出了一种改进的同伦算法, 将其用于求解由灵敏度函数产生的非线性方程. 基于改进同伦算法设计的鲁棒控制器 不仅避免了传统H∞控制中加权函数的选择问题, 而且克服了鲁棒控制器阶次较高的缺陷. 最后,文章以4阶系统为例, 设计了具有阶次限制的H∞鲁棒控制器, 通过与传统鲁棒控制器的比较可以看出, 基于本文方法设计的控制器不仅具有较低的阶次, 而且其控制性能也具有明显的优越性.     

A new approach to modified regularized long wave equation

In this paper, a homotopy analysis method employed to obtain approximate numerical solution of the modified regularized long wave (MRLW) equation with some specified initial conditions. The results show that the method converges rapidly and approximates the exact solution very accurately using only few iterates of the recursive scheme. The proposed technique solves the nonlinear problems without using Adomian polynomials and He’s polynomials, which can be considered as a clear advantage of this new algorithm over decomposition and the homotopy perturbation method. Numerical results are presented graphically, and tabular form reveals that the homotopy analysis method is an effective and convenient method to solve MRLW equation.     

Solving singular nonlinear boundary value problems by combining the homotopy perturbation method and reproducing kernel Hilbert space method

This paper investigates singular nonlinear boundary value problems (BVPs). The numerical solutions are developed by combining He's homotopy perturbation method (HPM) and reproducing kernel Hilbert space method (RKHSM). He's HPM is based on the use of traditional perturbation method and homotopy technique. The HPM can reduce a nonlinear problem to a sequence of linear problems and generate a rapid convergent series solution in most cases. RKHSM is also an analytical technique, which can solve powerfully singular linear BVPs. Therefore, we solve singular nonlinear BVPs using advantages of these two methods. Three numerical examples are presented to illustrate the strength of the method.     

A continuation approach to eigenvalue assignment

This paper describes a continuation approach to eigenvalue assignment. The method is a homotopy technique which embeds the control problem into a parameterized family of control problems. This parameterization describes a continuous deformation of a system with the desired spectrum into the original system. Based on this deformation, a differential equation is constructed whose solution trajectory has an end-point which is a constant output feedback matrix assigning the desired spectrum to the original system. The derivation of this differential equation and conditions which guarantee the existence of a solution are given. Also, two examples of its numerical implementation are detailed.     

Approximate analytical solutions to thermo-poroelastic equations by means of the iterated homotopy analysis method

A new approach of the homotopy analysis method (HAM), named iterated homotopy analysis method (I-HAM) is used to find approximate analytical solutions to thermo-poroelastic equations. The homotopy analysis method contains the auxiliary parameter ?, which provides us with a simple way to adjust and control the convergence region of the series solution. This method is reliable and manageable. The I-HAM solutions are compared with numerical solutions. To the best of our knowledge, this kind of analytic solutions has been never reported. Also, in each example, the comparison between I-HAM and traditional HAM is done, which shows the efficiency of I-HAM.     

Calculating luminosity distance versus redshift in FLRW cosmology via homotopy perturbation method

We propose an efficient analytical method for estimating the luminosity distance in a homogenous Friedmann-Lemaître-Robertson-Walker (FLRW) model of the Universe. This method is based on the homotopy perturbation method (HPM) which has a high accuracy in many nonlinear problems and can be easily implemented. For an analytical calculation of the luminosity distance, we suggest to proceed not from computation of the integral which determines it but from the solution of a certain differential equation with the corresponding initial conditions. Solving this equation by means of HPM, we obtain approximate analytical expressions for the luminosity distance as a function of the redshift for two different types of homotopy. A possible extension of this method to other cosmological models is also discussed.     

Solution Of Bounded Nonlinear Systems Of Equations Using Homotopies With Inexact Restoration

Nonlinear systems of equations often represent mathematical models of chemical production processes and other engineering problems. Homotopic techniques (in particular, the bounded homotopies introduced by Paloschi) are used for enhancing convergence to solutions, especially when a good initial estimate is not available. In this paper, the homotopy curve is considered as the feasible set of a mathematical programming problem, where the objective is to find the optimal value of the homotopic parameter. Inexact restoration techniques can then be used to generate approximations in a neighborhood of the homotopy, the size of which is theoretically justified. Numerical examples are given.     

A new spectral-homotopy analysis method for the MHD Jeffery-Hamel problem

In this paper a novel hybrid spectral-homotopy analysis technique developed by Motsa et al. (2009) and the homotopy analysis method (HAM) are compared through the solution of the nonlinear equation for the MHD Jeffery-Hamel problem. An analytical solution is obtained using the homotopy analysis method (HAM) and compared with the numerical results and those obtained using the new hybrid method. The results show that the spectral-homotopy analysis technique converges at least twice as fast as the standard homotopy analysis method.     

Approximate solutions to a parameterized sixth order boundary value problem

In this paper, the homotopy analysis method (HAM) is applied to solve a parameterized sixth order boundary value problem which, for large parameter values, cannot be solved by other analytical methods for finding approximate series solutions. Convergent series solutions are obtained, no matter how large the value of the parameter is.     

Path homotopy invariants and their application to optimal trajectory planning

We consider the problem of optimal path planning in different homotopy classes in a given environment. Though important in robotics applications, path-planning with reasoning about homotopy classes of trajectories has typically focused on subsets of the Euclidean plane in the robotics literature. The problem of finding optimal trajectories in different homotopy classes in more general configuration spaces (or even characterizing the homotopy classes of such trajectories) can be difficult. In this paper we propose automated solutions to this problem in several general classes of configuration spaces by constructing presentations of fundamental groups and giving algorithms for solving the word problem in such groups. We present explicit results that apply to knot and link complements in 3-space, discuss how to extend to cylindrically-deleted coordination spaces of arbitrary dimension, and also present results in the coordination space of robots navigating on an Euclidean plane.     

A Log-Linear Homotopy Approach to Initialize the Parameterized Expectations Algorithm

In this paper I present a proposal to obtain appropriate initial conditions while solving general equilibrium rational expectations models with the Parameterized Expectations Algorithm. The proposal is based on a log-linear approximation for the model under study, so that it can be a particular variant of the homotopy approach. The main advantages of the proposal are: (i) it guarantees the ergodicity of the initial time series used as an input to the Parameterized Expectations Algorithm; (ii) it performs well in regard to the speed of convergence when compared to some homotopy alternatives; (iii) it is easy to implement. The claimed advantages are successfully illustrated in the framework of the Cooley and Hansen (1989) model with indivisible labor and money demand motivated via a cash-in-advance constraint, as compared to a procedure based on the standard implementation of homotopy principles.     

Robustness of declarative modeling languages: Improvements via probability-one homotopy

Robustness issues with steady-state initialization remain a barrier in the practical use of declarative modeling languages for multi-domain modeling of large, complex, and heterogeneous technical systems. The objective of this paper is to illustrate how probability-one homotopy, an established method from topology, can solve this issue. This is achieved by establishing a framework for application-specific probability-one homotopy in declarative modeling languages. The analysis is based on domain-specific probability-one homotopy maps, which were reformulated in a declarative fashion. Additionally, a novel probability-one homotopy map and associated coercivity proof is introduced for a class of thermo-fluid dynamics problems. It was found that the approach enables robust initialization for declarative modeling languages on several test cases and leads to a concise declarative problem formulation.     

An efficient approach for solving the Riccati equation with fractional orders

The present study introduces a novel and simple analytical method for the solution of fractional order Riccati differential equation. In this approach, the solution considered as a Taylor series expansion converges rapidly to the nonlinear problem. New homotopy perturbation method (NHPM) depends only on two components of the homotopy series. The method is illustrated by applications and the results obtained are compared with those of the exact solution. Moreover, comparing the methodology with some known techniques shows that the present approach is relatively easy and efficient.