Optimal control of viscous Burgers equation via an adaptive nonmonotone Barzilai–Borwein gradient method

An adaptive nonmonotone spectral gradient method for the solution of distributed optimal control problem (OCP) for the viscous Burgers equation is presented in a black-box framework. Regarding the implicit function theorem, the OCP is transformed into an unconstrained nonlinear optimization problem (UNOP). For solving UNOP, an adaptive nonmonotone Barzilai–Borwein gradient method is proposed in which to make a globalization strategy, first an adaptive nonmonotone strategy which properly controls the degree of nonmonotonicity is presented and then is incorporated into an inexact line search approach to construct a more relaxed line search procedure. Also an adjoint technique is used to effectively evaluate the gradient. The low memory requirement and the guaranteed convergence property make the proposed method quite useful for large-scale OCPs. The efficiency of the presented method is supported by numerical experiments.     

基于牛顿法的并行优化算法

针对非线性数值优化问题,提出一种在分布式环境下的基于牛顿法的并行算法。引入松弛变量,将不等式约束转换为等式约束,利用广义拉格朗日乘子将约束优化问题转换为无约束子优化问题。为了并行地求解这些子优化问题,将Newton迭代法中的Hessian矩阵进行适当的分裂,采用简单迭代法求解Newton法中的线性方程组。在理论上对该算法进行了收敛性分析。在HP rx2600集群上进行的数值实验结果表明并行效率达90%以上。     

On finite Newton method for support vector regression

In this paper, we propose a Newton iterative method of solution for solving an ε-insensitive support vector regression formulated as an unconstrained optimization problem. The proposed method has the advantage that the solution is obtained by solving a system of linear equations at a finite number of times rather than solving a quadratic optimization problem. For the case of linear or kernel support vector regression, the finite termination of the Newton method has been proved. Experiments were performed on IBM, Google, Citigroup and Sunspot time series. The proposed method converges in at most six iterations. The results are compared with that of the standard, least squares and smooth support vector regression methods and of the exact solutions clearly demonstrate the effectiveness of the proposed method.     

离散时间最优控制—–评论动态规划

阐述离散时间最优控制的特点.对比3种求解离散时间最优控制的解法,即:1)用非线性规划求解离散时间最优控制;2)用无约束优化求解离散时间最优控制;3)动态规划及其数值解.1)和2)都适用于多维静态优化,计算效率较高,是高级方法.在名义上,3)为动态优化.实际上,3)为一维分段无约束静态优化,计算效率较低,是初级方法.本文并用数字实例进一步阐明动态规划及其数值解在求解方面较差,故动态规划及其数值解已失去实用价值.在求解离散时间最优控制问题方面,无法与非线性规划求解相匹敌.     

State waypoint approach to continuous‐time nonlinear optimal control problems

In this paper, we propose an optimal control technique for a class of continuous‐time nonlinear systems. The key idea of the proposed approach is to parametrize continuous state trajectories by sequences of a finite number of intermediate target states; namely, waypoint sequences. It is shown that the optimal control problem for transferring the state from one waypoint to the next is given an explicit‐form suboptimal solution, by means of linear approximation. Thus the original continuous‐time nonlinear control problem reduces to a finite‐dimensional optimization problem of waypoint sequences. Any efficient numerical optimization method, such as the interior‐reflection Newton method, can be applied to solve this optimization problem. Finally, we solve the optimal control problem for a simple nonlinear system example to illustrate the effectiveness of this approach. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

An affine scaling interior trust-region method combining with nonmonotone line search filter technique for linear inequality constrained minimization

This paper proposes an affine scaling interior trust-region method in association with nonmonotone line search filter technique for solving nonlinear optimization problems subject to linear inequality constraints. Based on a Newton step which is derived from the complementarity conditions of linear inequality constrained optimization, a trust-region subproblem subject only to an ellipsoidal constraint is defined by minimizing a quadratic model with an appropriate quadratic function and scaling matrix. The nonmonotone schemes combining with trust-region strategy and line search filter technique can bring about speeding up the convergence progress in the case of high nonlinear. A new backtracking relevance condition is given which assures global convergence without using the switching condition used in the traditional line search filter technique. The fast local convergence rate of the proposed algorithm is achieved which is not depending on any external restoration procedure. The preliminary numerical experiments are reported to show effectiveness of the proposed algorithm.     

A new nonmonotone filter Barzilai–Borwein method for solving unconstrained optimization problems

In this paper, a finite filter is used in the structure of the Barzilai–Browein (BB) gradient method in order to propose a new modified BB algorithm for solving large-scale unconstrained optimization problems. Our algorithm is equipped with a relaxed nonmonotone line search technique which allows the algorithm to enjoy the nonmonotonicity properties from scratch. Under some suitable conditions, the global convergence property of the new proposed algorithm is established. Numerical results on some test problems in CUTEr library show the efficiency and effectiveness of the new algorithm in practice too.     

Newton-conjugate gradient （CG） augmented Lagrangian method for path constrained dynamic process optimization

In this paper, a Newton-conjugate gradient (CG) augmented Lagrangian method is proposed for solving the path constrained dynamic process optimization problems. The path constraints are simplified as a single final time constraint by using a novel constraint aggregation function. Then, a control vector parameterization (CVP) approach is applied to convert the constraints simplified dynamic optimization problem into a nonlinear programming (NLP) problem with inequality constraints. By constructing an augmented Lagrangian function, the inequality constraints are introduced into the augmented objective function, and a box constrained NLP problem is generated. Then, a linear search Newton-CG approach, also known as truncated Newton (TN) approach, is applied to solve the problem. By constructing the Hamiltonian functions of objective and constraint functions, two adjoint systems are generated to calculate the gradients which are needed in the process of NLP solution. Simulation examples demonstrate the effectiveness of the algorithm.     

混合快速细菌觅食算法求解非线性方程

对于非线性方程组的求解,传统方法有很多,如牛顿法、梯度下降法等,但这些算法存在要求方程组连续可微、初值的选取是否合适等缺点,根据以上缺点将求解的问题转化为优化的问题,提出了新的交叉优化算法,充分利用细菌觅食算法局部搜索能力和粒子群算法的全局搜索能力,充分发挥了这两个算法各自优点。数值实验表明,新的算法可以弥补粒子群算法局部搜索能力弱和细菌觅食算法的全局搜索能力的不足,是求解非线性方程的有效方法。     

A nonmonotone scaled conjugate gradient algorithm for large-scale unconstrained optimization

This paper proposes a nonmonotone scaled conjugate gradient algorithm for solving large-scale unconstrained optimization problems, which combines the idea of scaled memoryless Broyden–Fletcher–Goldfarb–Shanno preconditioned conjugate gradient method with the nonmonotone technique. An attractive property of the proposed method is that the search direction always provides sufficient descent step at each iteration. This property is independent of the line search used. Under appropriate assumptions, the method is proven to possess global convergence for nonconvex smooth functions, and R-linear convergence for strongly convex functions. Preliminary numerical results and related comparisons show the efficiency of the proposed method in practical computation.     

Efficient symmetric Hessian propagation for direct optimal control

Direct optimal control algorithms first discretize the continuous-time optimal control problem and then solve the resulting finite dimensional optimization problem. If Newton type optimization algorithms are used for solving the discretized problem, accurate first as well as second order sensitivity information needs to be computed. This article develops a novel approach for computing Hessian matrices which is tailored for optimal control. Algorithmic differentiation based schemes are proposed for both discrete- and continuous-time sensitivity propagation, including explicit as well as implicit systems of equations. The presented method exploits the symmetry of Hessian matrices, which typically results in a computational speedup of about factor 2 over standard differentiation techniques. These symmetric sensitivity equations additionally allow for a three-sweep propagation technique that can significantly reduce the memory requirements, by avoiding the need to store a trajectory of forward sensitivities. The performance of this symmetric sensitivity propagation is demonstrated for the benchmark case study of the economic optimal control of a nonlinear biochemical reactor, based on the open-source software implementation in the ACADO Toolkit.     

On the nonmonotonicity degree of nonmonotone line searches

The nonmonotone globalization technique is useful in difficult nonlinear problems, because of the fact that it may help escaping from steep sided valleys and may improve both the possibility of finding the global optimum and the rate of convergence. This paper discusses the nonmonotonicity degree of nonmonotone line searches for the unconstrained optimization. Specifically, we analyze some popular nonmonotone line search methods and explore, from a computational point of view, the relations between the efficiency of a nonmonotone line search and its nonmonotonicity degree. We attempt to answer this question how to control the degree of the nonmonotonicity of line search rules in order to reach a more efficient algorithm. Hence in an attempt to control the nonmonotonicity degree, two adaptive nonmonotone rules based on the morphology of the objective function are proposed. The global convergence and the convergence rate of the proposed methods are analysed under mild assumptions. Numerical experiments are made on a set of unconstrained optimization test problems of the CUTEr (Gould et al. in ACM Trans Math Softw 29:373–394, 2003) collection. The performance data are first analysed through the performance profile of Dolan and Moré (Math Program 91:201–213, 2002). In the second kind of analyse, the performance data are analysed in terms of increasing dimension of the test problems.     

Optimum design of plano-milling machine structure using finite element analysis

The problem of optimum design of plano-milling machine structure is formulated as a nonlinear mathematical programming problem with the objective of minimizing the structural weight. The plano-milling machine structure is idealized with triangular plate elements and three dimensional frame elements based on finite element displacement method. Constraints are placed on static deflections and principal stresses in the problem formulation. The optimization problem is solved by using an interior penalty function method in which the Davidon-Fletcher-Powell variable metric unconstrained minimization technique and cubic interpolation method of one dimensional search are employed. A numerical example is presented for demonstrating the effectiveness of the procedure outlined. The results of sensitivity analysis conducted with respect to design variables and fixed parameters about the optimum point are also reported.     

Some implementation issues associated with multidimensional interval Newton methods

This paper presents the development of an optimal interval Newton method for systems of nonlinear equations. The context of solving such systems of equations is that of optimization of nonlinear mathematical programs. The modifications of the interval Newton method presented in this paper provide computationally effective enhancements to the general interval Newton method. The paper demonstrates the need to compute an optimal step length in the interval Newton method in order to guarantee the generation of a sequence of improving solutions. This method is referred to as the optimal Newton method and is implemented in multiple dimensions. Secondly, the paper demonstrates the use of the optimal interval Newton method as a feasible direction method to deal with non-negativity constraints. Also, included in this implementation is the use of a matrix decomposition technique to reduce the computational effort required to compute the Hessian inverse in the interval Newton method. The methods are demonstrated on several problems. Included in these problems are mathematical programs with perturbations in the problem coefficients. The numerical results clearly demonstrate the effectiveness and efficiency of these approaches.     

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．     

Optimal worst case estimation for LPV–FIR models with bounded errors

This work presents an approximate solution method for the infinite-time nonlinear quadratic optimal control problem. The method is applicable to a large class of nonlinear systems and involves solving a Riccati equation and a series of algebraic equations. The conditions for uniqueness and stability of the resulting feedback policy are established. It is shown that the proposed approximation method is useful in determining the region in which the constrained and unconstrained optimal control problems are identical. A reactor control problem is used to illustrate the method.     

On the use of a modified Newton method for nonlinear finite element analysis

In this paper, the usefulness of modified Newton methods for solving certain minimization problems arising in nonlinear finite element analysis is investigated. The application considered is nonlinear elasticity, in particular geometrically nonlinear shells. On a test problem, it is demonstrated that a particular implementation of a modified Newton method using both descent directions and directions of negative curvature is able to identify a minimizer, whereas an unmodified Newton method and modified Newton methods using only descent directions fail to converge to the minimizer. The use of modified Newton methods is suggested as a useful complement to the present continuation methods used for nonlinear finite element analysis.     

一种非线性互补问题的信赖域算法收敛性

针对非线性互补问题求解困难,利用信赖域算法,并结合极大熵函数法给出该类问题的一种新的有效算法．该算法首先利用极大熵函数将非线性互补问题转化为一个无约束最优化问题,然后应用信赖域算法来优化该问题,并在一定条件下证明该算法具有全局收敛性。数值算例表明算法的有效性。     

A Fast 2D Shape Recovery Approach by Fusing Features and Appearance

In this paper, we present a fusion approach to solve the nonrigid shape recovery problem, which takes advantage of both the appearance information and the local features. We have two major contributions. First, we propose a novel progressive finite Newton optimization scheme for the feature-based nonrigid surface detection problem, which is reduced to only solving a set of linear equations. The key is to formulate the nonrigid surface detection as an unconstrained quadratic optimization problem that has a closed-form solution for a given set of observations. Second, we propose a deformable Lucas-Kanade algorithm that triangulates the template image into small patches and constrains the deformation through the second-order derivatives of the mesh vertices. We formulate it into a sparse regularized least squares problem, which is able to reduce the computational cost and the memory requirement. The inverse compositional algorithm is applied to efficiently solve the optimization problem. We have conducted extensive experiments for performance evaluation on various environments, whose promising results show that the proposed algorithm is both efficient and effective.     

Genetic evolving ant direction particle swarm optimization algorithm for optimal power flow with non-smooth cost functions and statistical analysis

This paper presents an evolving ant direction particle swarm optimization algorithm for solving the optimal power flow problem with non-smooth and non-convex generator cost characteristics. In this method, ant colony search is used to find a suitable velocity updating operator for particle swarm optimization and the ant colony parameters are evolved using genetic algorithm approach. To update the velocities for particle swarm optimization, five velocity updating operators are used in this method. The power flow problem is solved by the Newton–Raphson method. The feasibility of the proposed method was tested on IEEE 30-bus, IEEE 39-bus and IEEE-57 bus systems with three different objective functions. Several cases were investigated to test and validate the effectiveness of the proposed method in finding the optimal solution. Simulation results prove that the proposed method provides better results compared to classical particle swarm optimization and other methods recently reported in the literature. An innovative statistical analysis based on central tendency measures and dispersion measures was carried out on the bus voltage profiles and voltage stability indices.